One of the best problems I've seen on this channel and anywhere, really. Didnt expect that you'd be able to solve for the actual polynomial, and the solution is super elegant.
@generalframework😂😂 where are you from mate this is a highschool question I dont think in your country’s universty exam questions are not hard like that
@generalframeworkwell no, most countries’ highschool math curriculums aren’t as advanced and in my opinion it shouldn’t be anyways. even if you think that it should, you have to implement it in a very balanced and correct way, which turkey fails to do. our education system has been declining for a while now, and students are not getting the education these questions are demanding for.
As a Turk, I can say asking harder questions and try to teach many subjects in a year doesn't make us learn Math better. Because you forget them all together. That is why we forget the simplest things.
In Turkey University exam is an elemination exam. So main target is eleminating students not to detect the level of students. So elemination exams are like hurdle race.
@@darkprofile Yes but the problem is that you have to study those kind of questions for years at school as well. They used to plan curriculums according to those hard questions. I don't know how it is now though.
@@enesa6489 Such questions can’t be taught at school. Every year there must be few new question types which never asked before. It is for eleminating hard wıorking students who studied all question types from intelligent students who can find a way to solve a new question type.
this specific type of question is beneficiary if you study mechanical engineering or similar...sometimes you have to come up with a mathematical model based on practical findings.
zaten mesele öğrendiklerini hatırlamak değil. Beceriyi ölçmek. O sınava çalışan herkes aynı sonucu almıyor. Ayrıca 5kya girmiş bi mühendislik öğrencisi olarak yks dönemimde analitik düşünce becerim çok yükseldi. Analitik düşünce becerisi kolay kolay kaybolmayacak bi nitelik. Gerçi artık böyle zor da sormuyor ösym. Son sınavlar çok daha basit hazırlanmış
I was one of the rare students who answered all 40 Math questions correctly in the exam. The hardness of the exam doesn't come from the questions themselves, it comes from the time pressure and stress you're experiencing. You're whole future is being determined by a 3 hour exam. I remember it felt like as if there was a gun on my head and people were yelling at me to solve the questions. Thank you so much for remarking the hardness of the exam.
I was also one of the students who correctly answered all the math questions on uni exam during my time but i dont aggree with your views, one can always attend a university exam in the followings years if he/she thinks his/her current success is not satisfactory. i have seen people attending uni exams during their retirement just for the fun of it. so it is not like "Your whole future is being determined by a 3 hour exam" besides economic or social success in life is not related much with one's academic success
@@servetc1970that doesnt work for everybody and seocnd exam is a hell year for sure since i just experienced it last year. I have high stress tolarance and still constant stress made me sick not just the exam but the durstion till the exam for like 3 months. Terrible experience fors ure
@@lavolpe_irl8997 that is another way of "elimination". eventually this is an exam for selection and you need reasons for elimination. not being able to correctly answer questions is a reason, but not being able to adequately cope with stress and pressure is also another way of elimination.
In my opinion, this question is actually very good and mind-blowing, even though it is very hard and not suitable for the university entrance exam. In the exam, I was able to solve the question thanks to background on mathematical olympiad.
In the last years questions started to become like iq tests rather than questioning your knowledge. 10 years ago it wasnt this bad but since youth unemployment rose and need for high education in good unis rose too.This caused a surging competition that only grew bigger as the economic crisis got worse. As everyone knows once the competition climbs it is really difficult to get less difficult questions in exams. Korean and Japanese exams are a nice example to that but Turkey's education system is not good as them, in fact it has been steadily declining for at least 7-8 years.
ıan fact, it is not like a logical IQ test, it is based entirely on memorizing the question type and just studying the same thing over and over again, and this is actually the sad part.Because after years u not learn anything and not memorize mostly
@@Kaholens Believe me as a doctor who won medical school and got 89 in Ales two years ago memorizing is not a good way to solve any Turkish and math question nowadays. In the past you needed to solve a lot of different type of questions over and over again to be able to think when you see a slightly different type of question come to you but nowadays, no matter what type of questions you solve they manage to find a much more different question type that you are not ready and if your IQ and knowledge isnt enough you are fucked. When I go into uni it wasn't this intense and I saw an intensity spike when I get into Ales two years ago. They were just Iq questions rather than "solve a lot of questions types and you are good to go" questions I was accustomed to.
I like your method. To me the obvious thing to do was to use the fact that the derivative of P(x) had to be 1 at x=1 and x=3 to complete the system of equations. But simultaneously solving 5 equations by hand is laborious, and your approach avoided this.
I used this technique, and reducing the augmented matrix by rows wasn't all that bad, just don't bother trying to get a diagonal matrix. Instead just reduce it to a triangular matrix and then solve up the triangle as you get each coefficient value.
A similar trick can be used to find the unique line tangent to a quartic at two points (if the quartic has 3 local extreme values). If you start with y=x^4+ax^3+bx^2+cx+d, you can find u and v such that (x^4+ax^3+bx^2+cx+d)-(ux+v) is a perfect square. Alternatively, you can first shift x to get rid of the x^3 term, then subtract the equation of the tangent. That gives a perfect square with no x^3 term, so it has to be (x^2+k)^2 for some constant k. Then it's easy to compute k from comparing coefficients.
When student learns this trick and solves the question without actually understanding how and why, we get serious issues in Turkey with IT,Med anything engineering related. This system is wrong. They even memorize AI prompts now!
Old Turkish student here. This is not the method of solving this type of questions. Because a student is expected solve the hard questions under 2-3 minutes, you cannot apply all the logic to solve the questions step by step. The magic lies in your preparation. High calibre students often solve thousands of (not kidding) questions just for some topics in the exam such as calculus part. So a Turkish student should recognise the pattern of the question and skip few steps. That’s how you ace turkish uni entrance exam :)
For the question in the video, you should immediately see that 1 and 3 are double roots without thinking over the graph or behaviour of the equation. For real, the exam is mostly about pattern recognition and precision rather than pure knowledge.
Whenever I see an international channel mention this exam I feel so weird because I was one of the many students who participated this exam and solved this questions.I was lucky enough to solve most of the questions. (regarding to the fact mathematic was my favorite lesson I can even say I was lucky that they did math extra difficult) I can appreciate hard Harvard-Oxford or Olimpiad questions but oddly enough seeing my exam's questions in this channels gives me a strange out of reality feeling... Idk why... Nonetheless thank you for bringing attention to this exam and solution... I enjoyed watching it :)
@@elidrissii well, I never said this was the hardest question... Also there are many factors which makes an exam difficult you cannot claim it is easy because of one question. This question is from second part of the exam. First part and second are not require the same thing from us. First part known as TYT requires basic knowledge and high understanding skills while second part (AYT) requires branch spesific knowledge, you have to have deep understanding of the topic(in high school level of course) to solve many of that questions... At that time it was not a new system that we didn't know but what made this one special it was out of the border. ÖSYM(the authority that prepares this exam) used to ask some similar question types that we are familiar with and trained for but that time they turned the tables around and did completely different exam without prior notice. There were olimpiad questions in both science and maths. Also as I said before there are many factors that make an exam difficult. This was University entrance exam that we took in pandemic... We had to study with online lessons, courses were closed, our relatives even ourselves infected with virus and we had to deal with all of that along with the stress an important exam gives us. In our previous year eventhough the exam was the easiest they had done this far they added extra time for the first part. (Normally it is 135 min for 120 questions but they added extra 30 min for them) But logically(!) for us they made it harder without adding any seconds.... So, for me it was hard and I think I have every right to think that way :)
@@elidrissii I also want to thank you for your kindness... When I read what I wrote for the second time, I felt like it may came a little bit harsh... I didn't want to sound that way, I just wanted to explain a bit 😅... I English is not my first language and I am not so very good at it... So if I offended you in any way I am so sorry... And again thank you for your kindness :) ..
@@elidrissiithis question alone might not be very hard but imagine having to solve questions like this one for 39 more math questions, and don't forget you can't lose much time on one question bc you still have the science part left
I like the approach to the first problem, and it's more efficient than mine. I stuck with p(x), realising that it had to be tangent to y=x at x=1 and x=3. So you have 5 equations for the 5 unknowns in a x^4 + b x^3 + c x^2 + d x + e, knowing p(x) at 1, 2 and 3 and that p'(x) = 1 at x = 1 and 3. I wrote them in matrix form and inverted the matrix to get p(x) = 2 x^4 - 16 x^3 + 44 x^2 - 47 x + 18, which has value 22 at x=4.
Gençler 3 milyon öğrencinin girdiği üniversite sınavı bir seviye tespit sınavı değil bir eleme sınavıdır. Bu sınavda az sayıda aşırı zor ve aşırı kolay soru sorulur. Sınavın genel zorluğunu belirleyen ise orta ve zor kategorideki soruların seviyesidir. 3 milyon öğrenciyi sıralamak gerektiğinde soruların zorluğunu ayarlamak sorun haline gelir. Kolay bir sınav yapıldığında herkes bir birine yakın netler yapacağı için çok iyiyi iyiden, iyiyi de ortadan ayıramaz hale gelirsiniz. Yani adil sıralama olmaz. Aşırı zor sınavda da kötü ve orta düzeydekileri ayırmak zorlaşır. Sınava hazırlanan disiplinli öğrenciler önceki yıllarda çıkan tüm soru tiplerini ezberlerler. Çalışkan disiplinli çocuklar ile zeki çocukları ayırmak için daha önce hiç sorulmamış böyle zor ve yeni soru tipleri üretilir. Zeka yeni koşullara uyum sağlama becerisidir. Yeni soru tipini zorluk derecesine göre daha zeki olanlar çözebilir ve zeka üzerinden eleme yapılır. Yani eğitim bilimlerinde ölçme değerlendirme zor bir iştir. 3 milyon öğrenciye yüksek eğitim vermek için sıralama yapmak gerektiğinde size zor ve saçma gelen bu sınav en adil yoldur. Düşünün bu ülkede memur olmak için KPSS ye giren adaylardan yüksek puanlılar mülakatta elenip yerlerine düşük puanlılar alınabiliyor. Üniversite sınavının zorluk düzeyinin yüksek olması herkese aynı zorluk uygulandığı ve daha çok doğrusu olanı yukarı taşıdığı için adildir. Hepinize başarılar dilerim...
Peki hocam işlenen onca konuya onca üniteye rağmen bu kadar az sorunun olması sizce de adaletsiz değil mi? Biz biliyoruz ki bir verinin miktarı ne kadar fazla olursa sonucu da doğruya o kadar yaklaşır. Sınavda çalıştığımız kısmın neredeyse yüzde 60ı bizim karşımıza çıkıyor. Çıkan sorularda da birden fazla konuyu bir soruya sıkıştırıyorlar, diyelim ki sen o sorudaki 5 konunun 4ünü çok iyi biliyosun bir tane öncülü yapamadın diye o soruyu iki şıkka indirip sallayan adamdan daha geriye düşüyosun. Dediğiniz gibi sınavlarda zor sorular kesinlikle olmalı çünkü giren sayısı çok fazla ve bu insanların bir şekilde elenmesi lazım fakat soru sayısının azlığı adaletsizlik yaratıyor bana göre
@@emirhanylmaz946 Güzel kardeşim bu sorular belirli araştırmalara göre optimize ediliyor. Orta öğretimdeki her konu bu sınavda çıkamaz. Çünkü amaç seviye tespiti değil eleme yapmak. Eleme sınavları seviye tespit sınavından farklıdır. Seviye tespit sınavlarında dediğin gibi sorular konu ağırlığına uygun oranda sorulur. Yani sımavın içerk tutarlılığı yüksektir. Ancak eleme sınavları 400 metre koşusundan ziyade 400 metre engelli koşusu gibidir. Senin ne kadar hızlı koştuğuna değil engelleri ne kadar hızlı geçtiğine bakılır. Bahsettiğin farklı konuları tek bir soruda sormak da eleme sınavlarının engellerinden biridir. Bir geometri sorusu aslında köklü sayılarla işlem yapma sorusudur. Bir kenarı 3 verse herkesin yapabileceği soruyu o kenarı 3 kök 2 verince sınava girenlerin yarısı elenir. Yani amaç eleme yapmak olunca çok basit bir soru 15 cümlelik paragrafın içine gömülür. Aslında herkesin doğru yapacağı bu soru yavaş okuyanlara zaman kaybettirmek çok test çözmeyenleri yorum odaklanmalarını düşürmek için konuluyor. Yani değerli kardeşim eleme sınavlarının doğası böyledir. 3 milyon kişi sınava girerken aslında hedeflenen doğru düzgün üniversiteler ilk 100-150 bin kişinin girebileceği yerlerdir. Yani aslında sınavda ama %1-5 arası en yüksek puanlı üniversitelere sıralama yapmak. Kalan bölümler zaten o %5’e yerleşemeyenlerin tercihleri. Maalesef kalabalık bir ülkeyiz. Ve parasız üniversite eğitimi için maalesef bir eleme sınavına ihtiyaç var. Ve 3 milyon kişinin girdiği sınavda uzun sorular, birden fazla konu alanını kontrol eden sorular ve daha önce hiç sorulmamış tip sorular olması gerekiyor. Senin istediğin gibi soru sayısını arttırarak eleme yapmak pratik değil. Sınav süresi zaten yeterince uzun. Daha fazla soru ve daha uzun sınavda tuvalete bile çıkamadan çocuklardan performans alınamaz.
İnsanların şikayeti bu sorunun zorluğundan veya sınavın yönteminden değil. İnsanların şikayeti milli eğitimin bu tür soruları çözecek kaynak, öğretim veya ortam sağlamıyor olması
@@azovianace İşte mesajımda tam da bunu anlatmak istedim değerli kardeşim. Eleme yapacağın zaman bir kaç tane eğitimini vermediğin, öğrenciye nasıl çözebileceği öğretilmemiş soru sorarsın. Bu soru çok zeki çocuk ile zeki çocuğu ayıran sorudur, sınav birincilerini belirleyen sorudur. Siz hayatta karşınıza çıkacak tüm soruları nasıl çözeceğinizin size bilgi olarak verilmesini bekliyorsunuz. Hayat böyle bir şey değil. Hayatta daha önce kimsenin çözemediği sorunları çözenler ya da bir sorunu başkalarından daha iyi hızlı verimli çözenler başarılı olur. Yoksa herkes kendine öğretilen bir konuyu ve soruyu çözebilir.
Thanks for this nice problem. To my surprise, I found it quite straight forward. Let f(x)=P(x)-x, a fourth degree polynomial that is always non-negative. Then f(1)=P(1)-1=0 f(2)=P(2)-2=2 f(3)=P(3)-3=0 So f has zeros at 1 & 3. These must be roots of even order as f is always non-negative (if a polynomial f has a root α of odd order n, then f(x)=(x-α)ⁿg(x) where (x-α)ⁿ changes sign at x=α but g doesn't, so f changes sign at x=α). Hence each root is of order 2. So f(x)=a(x-1)²(x-3)² for some real a≥0. From f(2)=2 we get a(2-1)²(2-3)²=2 So a=2 f(x)=2(x-1)²(x-3)² P(x)=2(x-1)²(x-3)²+x P(4)=2(4-1)²(4-3)²+4=2×3²×1²+4=18+4=22
@@berdigylychrejepbayev7503 Indeed, my solution (which I wrote before watching the video) is very similar to the solution in the video, but I thought it worth adding anyway because of the justification of the double roots, which I think is clearer than in the video.
@@MichaelRothwell1 Exactly the way I did it; also before watching, half hoping to see Presh use some other technique. I also didn't call on the derivative, but just noted that both 1 and 3 had to be double roots (any higher even order, and P would have degree > 4). It's a very pretty problem. Fred
always feels weird when non-turkish people acknowledge the difficulty of turkish uni entrance exams. it both feels sad because we're tormented like this, and kinda boosts one's ego because "yes, i did manage to solve this!" good video as always, love from turkey!
You are not tormented, you are challenged. There has to be difficult questions otherwise among the top students many would solve a similar amount and it would be difficult to get a good grading among them.
@@chief4180 yeah i don't think we need to be dealing with this high level stuff in high school. thankfully i'm planning to either go abroad or study humanities here, and my math is sufficient for the TYT math questions. still, this system isn't humane or pedagogically sound.
There is a way to non use calculus to complete the exercise and do it using pure algebraic properties... Just note that Q(x)= P(x)-x has roots at x=1and x=3, this means that Q(x)= (x-1)(x-3)H(x), with H a 2 degree polynomial, then given that P(x)>=x, we see that Q(x)>=0 meaning that H(x) Is a quadratic polynomial that just happens to match the signs of (x-1)(x-3) to be always positive, given that (x-1)(x-3) is negative in ]1,3[ and positive outside that interval, H is forced to match this while being a polynomial of degree 2, negative in ]1,3[ and positive outside that interval, meaning that is actually h*(x-1)(x-3) with h>0, now evaluate Q(x) in x=2 to get the value of h and you get the result. Is less clean but can be technically solved without using calculus tools.
Regardless of the whole min/max analysis, it is obvious that Q'(1)=Q'(3)=0 because Q(x) cannot cross the zero. So setting Q(x)=(x-1)(x-3)(ax^2+b.x+c) and solving the system Q'(1)=Q'(3)=0 and Q(2)=2 gives a, b and c right away.
in turkey you have 40 math (30math+10geometry) and 40(physics, chemistry, biology) questions and have 120 minutes to solve these, nearly all math questions are same difficulty as this or most propably higher difficulties, and you almost have 1:30 minutes to read understand and solve each question, and every high school senior has to take this exam in order to join an university, and every university has a limit of points
@@saurongrows its 160 and its depending on your choice of grades, 120 if you chose literature you solve 120(40 literature, 40 social stuides, 40math) and if you chose science you solve 80 (40 math, 40 science (14 physics, 13 chemistry, 13 biology)
@@das-panzer-maus if you dont know, then dont talk about it. every student that participate in AYT solve 80 questions, literature students dont solve math.
This question is to eliminate best ones in that test. You have to be in top 2000 to be in best universities, and there are 2 million candidates. So if you are good , you have to solve easier questions so fast and you can have time to solve hard problems like this.Otherwise you have 3-4 mins for every questions. But you can't solve this question in 3 mins if you are not a genius and and ur not lucky and you are not calm. Hard to stay calm, because its very stressful
I rather liked this problem! I feel like your justification of the multiplicity of the two roots could have been slimmed down to simply that polynomial roots either cross or touch and turn. Since crossing would make the polynomial go negative, they are touch and turn roots. Since the degree is 4, that makes them both multiplicity 2. No derivative or talking about turning points needed!
Exactly! This is a pre-calculus problem that really requires careful thinking about the nature of roots, and of the relationship of polynomial difference.
ayt bu oldukça zaman var ki soruyu çözmek aslında o kadar uzun sürmüyor. videoyu izlediysen de görmüşsündür daha kısa bir şekilde nasıl çözülür onu da anlattı
Well here is how I would approach it. The condition P(x) >= x motivates the definition of Q(x) = P(x) - x Q(x) is positive everywhere thus every roots are at least double (because there are extrema. 1 and 3 are roots of Q. Thus, Q must be of the form: Q(x)=C*(x-1)^2(x-3)^2 with C a real constant Given P(2) = 4 Q(2)=2, we deduce C=2 Thus, we can compute Q(4) = 2*3^2*1^2= 18 Thus, P(4) = Q(4) + 4 = 22
Hello there, the only reason behind the average of 7.6 out of 40 isn't the hardness of the test but also the worsening education system of Turkey. At my year (2011) I got 39 out of 40 and the average was 13,17 out of 40.
Math is a political weapon in many countries now. After all, rich kids are in private school anyway so the politicians are happy to lower the bar for public school.
@@adam.gizlin The guy did not even mention Erdoğan in the context. He rather said it is "the worsening education system" which MAY be Erdoğan's fault but he did not blame anyone.
Pre-watch: Actually pretty easy, via the following insights. Given: P(x) ≥ x Now define Q(x) = P(x) - x Then Q(x) = P(x) - x ≥ x - x = 0 Q(x) ≥ 0 And from the given values of P, P(1,2,3) = (1, 4, 3) we know that Q(1,2,3) = (1, 4, 3) - (1,2,3) = (0, 2, 0) But these facts mean that 1 and 3 are both double roots of Q [otherwise Q would have to be negative somewhere in the neighborhood of both]. Thus, Q(x) = a(x-1)²(x-3)² where a > 0. Thus, 2 = Q(2) = a, meaning that Q(x) = 2(x-1)²(x-3)² P(x) = 2(x-1)²(x-3)² + x P(4) = 2·3²·1² + 4 = 18 + 4 = 22 Now let's check out Presh's solution . . . Fred
Yes, exactly! Since P(x) can not pass below the line y=x, this means that P(x) is tangent at the points x=1 and x=3. In addition, the slope at those two points must equal 1. This gives you two additional equations so that you can solve this problem easily with 5 equations and 5 unknowns.😀 I got P(x) = 2x^4 - 16x^3 + 44x^2 - 47x + 18
Same. The given solution is better though; it took me half an hour to solve the 5-equation system, and if I hadn't used a spreadsheet to do it I almost certainly would have dropped a digit somewhere and gotten the wrong answer.
I just bashed it with calc. P(x) = ax^4 + bx^3 + cx^2 + dx + e. You have 5 unknowns and are given 3 equations. But notice P(3) = 3 and P(1) = 1 and P(x) >= x. This means P(x) must touch the line y=x when x=3 or x=1 but not cross it, otherwise you will have P(x) < x. This means y=x is the tangent line at x=3 and x=1. So you have P'(3) = 1. P'(1) = 1. This gives you two extra equations, so you can solve. Took way too long tho, have to bash 5 linear equations. Don't recommend
lmao, as a math grad I looked at this and came to the same conclusion but also realized in an exam it would certainly not be possible to do in time, despite the saviness of it. I STILL BRUTE FORCED IT ANYWAYS
I am a Turkish person going through this kind of education system, people might think that We turkish children are smart and have very high academic success, but no, we don't. In fact, this system causes many children with un-explored talents to fail, and literally drop out of school, then go to an average job like an adult. in age of maybe 14. Our system relies mostly on obedient students, not the ones that can actually question and think. Many students started to kill themselves also because of sociological structure and mindset we have. The government has also started to slowly assimilate children politically and religiously, I dont wish to talk about these topics though, I might be enjoying my last happy moments
I recently noticed some suicidal tendencies on some of my friends, Im concerned honestly. Yeni eğitim bakanı içimizden geçti zaten daha neler bekliyo bizi bakalım💀🤯
In an exam which must sort 3 millions of students, there can be few extremely high level questions which eleminates very good students from genius students...
Sayısal 200e girmiş biri olarak yazıyorum 2021 soruları abartıydı, soruların ilk bin ile geri kalanı ayıracak zorlukta olması gerekiyor; bu ölçüyü tutturdukları sınav görmedim galiba.
@@darkprofile Çünkü ilk bindeki öğrencilerin düzeyleri genel olarak aynı, sıralamalarda yukarı çıktıkça şans faktörü daha önemli oluyor. İlk iki binin dışına çıkmaya başladığımızda öğrenciler arası fark daha barizleşiyor. İlk bindeki potansiyeli yüksek öğrencilerin diğerlerinden ayrışması için zor sınavlar lazım. Sınav kolaylaştıkça yeteneği yüksek öğrencilerin elenme ihtimali de yükseliyor. Benim bu sözümdeki amaç bu öğrencilerin istediği okula girmek için kendilerini harap edip sonra sınavın kolaylığı yüzünden israf olmalarını engellemek.
turkish med student here, it was one of the most comprehensible ones in the pool of hard turkish exam questions imo (entered the exam in 2021, the same year this was asked btw)
After plotting p(x)=f(x)-x, which leaves you with a function that must be greater or equal to zero everywhere, which implies slope of zero at 1 and 3, you can look at it and realize the result must be symmetric around x=2 and write the function p(x)=2-a*(x-2)^2+b*(x-2)^4, then solving for a and b is easy. The odd terms in (x-2) are zero because the square and forth power terms are symmetric in (x-2). The 2 is at x-2=0, the slope and value at x=1 determine the rest.
For Q to be positive, each of the two roots has to have multiplicity 2. This can be checked with the derivative. This determines Q up to a constant, which is determined by the value at x=2.
This question is actually easy if you solve something like this in your 4 year highschool education. The real hard problems are not the one solved with methods, methods can easily be memorized. The real hard problems are generally lie in the problem section, so few students be able to do problem section
This question was easily one of the hardest if not the hardest question of the 2021 AYT exam. There weren't " a lot " of questions harder than this. I'm saying this as a person who took the exam in 2021.
Others have given very nice solutions. If someone wants to learn more about such problems, it is called an __Hermite interpolation__ problem: Given m equations on a polynomial of the form P(x_i) = y_i or P^{k_i}(x_i) = y_i (denoting k derivations), then there exists a unique polynomial of degree at most m-1 that satisfies all equations.
1:30 The first thought was about making a graph of known data, it immediately help to realize that needed P(x) tangent to f(x) = x in {1; 3}. After that I chose the harder way: made a system of 5 equations and 5 variables (coefficients of polinomial) and solved it via Gauss' Method) (and double check it in Wolfram and Desmos) After graphing P(x) - x in Desmos I realize the better and faster approach, but it was too late... Know, let's continue watching))
that is possiable but remember you have a time limit at the exam, it is 3 hours for 80 questions but if the exam is filled with these kind of questions you dont acctualy have time to do that
They probably assume that you've solved so many questions that you'd already know the steps of the solution by memory. To be honest they expect you to memorize things.
Elegant. The way I solved it was that we have 5 points: P(1)=1, P(2)=4, P(3)=3, and by the P(x)>=x we have that x=1, x=3 are points of tangency with gradient one: P’(1)=1, P’(3)=1. Then use Gaussian elimination to solve the 5x5 system. A bit rough, but it works
this was a break-through question in 2021 ayt. before that, math tests were relatively easy. after 2020 being too easy, they decide to make it a little bit harder in 2021. and it made the students to prepare entirely different for the exam. since it is an elegant question and we don't get to solve questions like that at school because teachers are not qualified enough, students started to bail out of school and study with private tutors or all by themselves. and turkish officials got really mad because students eventually realized the educational system is bad, so they made continuity in school mandatory recently. so long story short, students are stuck in school with teachers who can not even solve this question, yet students are expected to solve this question.
Before watching, here's how I solved it: Let Q(x) = P(x)-x, then Q(1)=Q(3)=0, Q(2)=2, deg(Q) = 4 And it satisfies Q(x) ≥ 0 for all x. Therefore, Q(x) = (x-1)(x-3)R(x) for R being a second degree polynomial. Since Q(x) ≥ 0, the points x= 1,3 are local minimums and therefore Q'(1)=Q'(3)=0 Note that Q'(x) = (x-3)R(x) + (x-1) (R(x) + (x-3)R'(x)) Plug in x= 1, -2R(1) = 0 -> R(1) = 0 Similarly, plug in x = 3, 2R(3)=0 -> R(3) =0 Therefore, R(x) = A(x-1)(x-3) And Q(x)= A(x-1)^2(x-3)^2 Plug x=2, Q(2) = A•1•1 = A But Q(2)=2, therefore A = 2 Finally, P(4) = 2(4-1)^2•(4-3)^2+4 = 22 Edit: looks like my solution is very similar, down to the name and definition of Q. But he used a visual argument, while mine was more algebraic.
If you transform by subtracting x and shifting left and down by 1, you get a new poly Q(x)=Cx^2(x-2)^2 as the poly that touches y=0 twice without ever being less. Solve for C, evaluate at 3 and transform back to get P(4). I skipped ahead and the ending suggests something similar is how it's done here too. The two roots of even multiplicity and the deg 4 don't leave room for additional freedom.
The horrible thing is that students only have 3 hours for THIS QUESTION + 159 QUESTIONS IN TOTAL ! Even tho you dont have to answer all 160 questions (There are the lessons of "Turkish, Math, Science and History" 40 questions each. You solve the lesson(s) that are related to your goal profession.), its still so BRUTAL !
let Q(x) = P(x) - x there for Q(1) = 0 Q(2) = 2 Q(3) = 0 and Q(x) >= 0 so we have 2 roots and those roots are also extremums (so on 1 and 3 Q’(x) = 0) so they are both double solution so Q(x) = k[(x-1)(x-3)]^2 replace with 2 we got Q(2) = k[1*(-1)]^2 so 2 = k and now replace with 4 Q(4) = 2[3*1]^2 = 2*9 = 18 so P(4) = Q(4) + 4 = 18 + 4 = 22
Exclamation, the reason avg score is so low is not just because the difficulty but because of the “fields” system, not all of the 2 million students are responsible of the math part but some try it anyways thus lowering the average. 30 and more out of 40 is common for students who are responsible for math part and it is not surprising to see someone making a perfect score.
This is a selective question in this exam. There are about 5 questions at this level, and the remaining 35 are more straightforward; however, it is bizarre to expect 18-19-year-old students' to solve this in a few minutes with that exam pressure.
In 2006, I took that profeciency test for University. I had only 4 correct answer. And years I was blaming my self because of that a disaster and now thanking to the God at least I accepted and graduated from University..
Since P(x) is a 4th degree polynomial with real coefficients and satisfies the condition P(x) >= x for all real values of x, we can conclude that P(4) is greater than or equal to 4. However, without more specific information about the polynomial, we can't determine an exact value for P(4)
I’m med student from turkey and I took this AYT exam. This exam was really hard because I have 2 minutes for every questions. And this questions answered by 400 students out of 2 millions and this statistic shows this question is really difficult. If you like this question you should check 2021 AYT’s derivative and permutation questions. And meanwhile I became 4664th in 2022 AYT exam out of 2.5 millions student
Atıp tutturan bile 1000 kişi vardır, o 400 sayısını nerden buldun bilmiyorum. Zor soruydu ben de cozememistim diye hatırlıyorum sjdksn ama 400 kişi olma ihtimali yok
i can still remember how i was working my brain off for this bloody question in that exam.When i was done with reading that question what came out from my mouth was just “i’m fucked up” i had no clue at all
are the Turkish students supposed to solve these problems right after high school?? i am fron india and here they don't have dedicated cubic of quartic equation lessons. so it's a bit bizzare seeing such a problem
Without having watched beyond the problem statement; you are given that P(x)>=x for all real x, and that P(1)=1 and P(3)=3. Consider the polynomial Q(x)=P(x)-x. Then Q(x)>=0 for all real x and Q(1)=0, and Q(3)=0, which means that Q(x) has a double root at x=1 and at x=3. This means Q(x)=(x-1)^2(x-3)^2R(x) for some polynomial R(x). Because Q(x) is quartic you know that R(x) is in fact a constant. Then P(2)=4 tells you that Q(2)=2 and so R(2)=2, so Q(x)=2(x-1)^2(x-3)^2 and hence Q(4)=18 and P(4)=22. This is about 2 minutes of work, and all you need to know is that if a polynomial is tangent to the x-axis at a point t, then it has a factor (x-t)^2.
Yes we do. we dont really have dedicated classes for each subject. The same math teacher will explain everything from log's to integrals in a years time steadly going over them in (roughly) a pre determaned speed set by the goverment body of MEB (National education something İdk)
The point of this exam is to differentiate the students. This was one of the hardest questions in the test, so it was expected that only the best students would be able to solve it.
I am a Turkish student, I remember the first time that I solved this question without any help, I was super excited that I was about to cry lol! But it really improved my problem solving skills.
One possibility is by the use of matrices. Let A be the matrix [[1,1,1,1,1], [1,2,4,8,16], [1,3,9,27,81]] and B be the column matrix [[1], [4], [3]], and let C be the column matrix of the coefficients of the 4th degree polynomial [[a],[b],[c],[d],[e]], then one could solve for C as, C = B.INV(A) , where INV(A) is the inverse of matrix A.
If b^2 - 4c < 0 then the polynomial q(x) = a(x-1)(x-3)(x^2+ bx + c) has opposite sign of a for x between 1 and 3 . So if a>0, then this polynomial expression is _negative_ for 1
Q(x)=P(x)-x Q(1)=0 Q(2)=2 Q(3)=0 Q(x)>=0 So Q'(1)=0 and Q'(3)=0 so Q can't dive under 0. The curve looks like a W. Also by symmetry Q'(2)=0 and Q(4)=Q(0) So we only need to find "e" in Q(x)=ax⁴+bx³+cx²+dx+e And P(4)=e+4
For anyone who got this, how did you know to create Q(x)? Did it stand to you from the 1 and 3 values? Have you seen similar questions? Did you know intuitively you could solve if you had a function touching the x axis? Did you try different things first?
This is mostly because ministers think making the questions harder instead of teaching stem subjects rather than increasing the islamist propaganda will make students learn better. We really need to ditch religious bs and focus on a better education system. Watching you solve this question as an engineer made me realize the severity of this issue
@@user-re2vi9rp6r believe me its true, turkeys gov trying their hardest to make society more ignorant and more obedient to them by using religious propaganda and the whole country is just going backwards to dark ages.
Imamhatips (religious schools that literally brainswashes people) should be closed. Religion being a part of education is just dissappointing. Religion is not an obligation in education, a person can do their religious duties outside of the school. Discussing belief systems in schools should not be allowed as they dont allow politics either. Time to make all of my friends atheists.
@poumybeloved when I was in high school (about 10 years ago), I made all religion teachers life miserable for the lecture hour they had with my class. Don't show mercy kardeş.
This has to be *HARD*. 9:52 video to solve this assuming you know what you are doing and it's only ONE question on the exam. How much time do you even get for the entire exam of 40 questions?
In this case of AYT 2021, we had to solve 40 questions of biology, chemistry and physics in addition to the 40 math questions, so 80 total. We had 3 hours without any break in between (this made it hard to focus on the exam after some time). So 2.5 minutes for each question.
I must admit, that as a test problem for a university entrance test, it is in the tough end, because it relies on a trick. Nevertheless I found the problem both easy and funny. Funny because it requires a trick, and easy because the trick was really easy to guess. The information that P(x) >= x, P(1)=1 and P(3)=3, drew immediately the attention to the polynomium Q(x) = P(x) - x. Knowing that Q(x) >= 0, it is clear that ALL roots of Q MUST be double roots, and the rest is easy. If you are not familiar with the double root rule, here it follows an argument in a general context: Theorem: Suppose P(x) is a polynomial of degree n, and suppose r is both a root and an extremum of P. Then r is at least a double root. Proof: P(r) = 0, because r is a root of P, such that we may write P(x) = (x-r)*Q(x), where Q is a polynomial of degree n-1. Take the derivative of P, such that P'(x) = Q(x) + (x-r) * Q'(x), and thus P'(r) = Q(r) + (r-r) * Q'(r) = Q(r). As an extremum we have 0 = P'(r) = Q(r), and therefore r is ALSO a root in Q. This way r is at least a double root.
I figured out that when you subtract x from P(X), it must be a perfect square for the minimum to be 0. Then, a quadratic R(x) with R^2(x) = P(x)-x would make it so that: R(1) = R(3) = 0, R(2) = sqrt(2). Then you can solve it.
now imagine the stress of all your future depending on it, it being surrounded by many others that are even worse than this, and also you just have 1.30 minutes
The limits are (plus or minus) infinity because P(x) (and hence Q(x)) is a (fourth degree) _polynomial_ function. This results in that the limit of the quotient P(x)/(x^4) is finite but nonzero as x goes to +/- infinity. x=1 and x=3 are roots of Q(x), because Q(1) = 0 and Q(3) = 0. So Q(x) = (x-1)(x-2)*S(x) for some second-degree polynomial function S(x). Furthermore, we must have Q'(x) = 0 at x=1 and x=3 ; otherwise the graph of Q(x) is _passing through_ the x-axis at those points (rather than merely tangentially touching) which would mean Q(x) becomes negative in a neighbourhood of those points, which goes against the stipulated premise that Q(x) ≥ 0 . And since therefore Q'(1)=0 and Q'(3)=0 , it can be shown that we must also have S(1)=0 and S(3)=0 . (Just determine the derivative of Q(x) = (x-1)(x-3)*S(x) , and set it equal to 0 for x=1 and x=3 .) This means S(x) = K(x-1)(x-3), for some constant K, and hence Q(x) = K(x-1)(x-3)(x-1)(x-3) .
Because the 2 roots lie on the horizontal axis so they are double root. The limits are infinity because 4th degree polynomial's domain is R so we have to take the +- inf limit
@@NovaH00 "Because the 2 roots lie on the horizontal axis so they are a double root." -- That sentence makes no sense. Roots _always_ lie on the horizontal axis.
yerenchu already gave a correct mathematical explanation. Let me give a more 'geometrical' explanation. Q(x) ≥ 0 for all x. I hope you agree with that. And if you also agree with the general graph of Q(x) as depicted, then you know that x=1 and x=3 are the only zeroes of Q(x). Now suppose that we shift the graph of Q(x) a tiny little bit below. That means that we decrease the constant term of Q(x) by a tiny amount. Then all of a sudden Q(x) has 4 real roots ('obvious' from the graph of Q(x)). If the shift is really tiny, I hope you agree they are clustered around x=1 and x=3. But still 4 distinct roots. 'Undoing' the shift, these 4 distinct roots will (eventually) again cluster around x=1 and x=3. Hence, the double roots. Hope this makes sense. (to make this rigurous, I would have to prove that the roots of a fourth order polynomial depend - in this case - 'continuously' on the constant factor of Q(x), but I hope you will forgive me in this case and trust me on the 'graphical view'). The limits for x → +/--∞ for Q(x) depend on the sign of the coefficient of x^4 of this polynomial. Since Q(x) ≥ 0, this means that the sign of the coefficient of the 4th power of Q(x) must be positive (can't be zero, since then Q(x) would then not be a fourth order polynomial). Hope this helps....
Im sorry but im not shure there is only 1 possible answer in terms of specific coefficients: the polinomial interpolation of the points (1,1), (2,4), (3,3), (0,10) and (4,1) respect all conditions so 1 its a possible solution
@@s.p.a.3583 Pretty sure that still doesn't work. While the extra condition is satisfied at the specified points, it needs to be satisfied EVERYWHERE. The set of points you give cannot be described by a polynomial of degree 4 that is everywhere greater than x. What polynomial did you find? Plot it and the line y=x. I guarantee they will have a non-tangential intersection.
I tried with any value for P(4)) >= 4, there are infinite solutions. The issue I see is that Persh as many others assume integer coefficients while doing the Q(x) function, while the problem accepts real coefficients, hence there are infinite solution by just setting P(4) to anything >= 4. Here it is for P(4)=5 : x^4-53*x^3/6+26*x*x-169*x/6+11
@@ikocheratcr Did you PLOT that function and LOOK at it? That function has a local minimum at P(3.444)=2.231, which violates the condition P(x)≥x, just like I said. You can't just check the values of the function at integer inputs. You need to satisfy P'(1)=P'(3)=1, which ensures that P(x) touches, but does not cross, the line y=x.
This problem can be solved with basic knowledge about quadratic function and fundamental laws about polynomial factorization derivative knowledge is not needed. Q(x) = (x - 1)(x - 3)f(x) where f(x) is a quadratic function since Q(x) >= 0 for all x, f(x) is negative in (1, 3) and positive in (-inf, 1)U(3, +inf) therefore f(x) must be 0 at 1 and 3 therefore Q(x) = (x-1)(x-3)* a(x-1)(x-3)
Its not that difficult, it’s just a polynomial function so it’s continous and always admits derivative, we have 5 coefficients to find and five equations: 3 equations from p(1)=1 p(3)=3 and p(2)=4 and 2 equations from p’(1)=p’(3)=1 because p(x)>=x (the function has to be above the function y=x so in (1,1) and (3,3) must be tangent to y=x).
I know you westerners can't really relate to this but to us 3rd world dwellers this feels like a really harsh reality check. this is what we have to go through for a decent education in a less than decent country. every competent student in turkey has to sacrifice at least a year or two to their studies and still it's not guaranteed you will accomplish ANYTHING. we are just chained to the system our parents and our grandparents saw fit to us, to suffer throughout our teen years to have go to waste. it sucks man
As the saying goes:
"You don't enter the exam, the exam enters you"
Can relate as a Turkish student
One of the best problems I've seen on this channel and anywhere, really. Didnt expect that you'd be able to solve for the actual polynomial, and the solution is super elegant.
We learn about polynomials in 10th grade which 2nd grade of highschool
@generalframework😂😂 where are you from mate this is a highschool question I dont think in your country’s universty exam questions are not hard like that
@lowlytarnished8667 This question is from a country where 80% are creationists, Ah I see they question culled them out!
@generalframework where do you study? i'm curious about the country not the school itself
@generalframeworkwell no, most countries’ highschool math curriculums aren’t as advanced and in my opinion it shouldn’t be anyways. even if you think that it should, you have to implement it in a very balanced and correct way, which turkey fails to do. our education system has been declining for a while now, and students are not getting the education these questions are demanding for.
Çok şükür kendi ülkemin sorusunun kaliteli bir kanal tarafından çözüldüğünü görebildim😊
Çok haklısın
Tibees var.
Bende
@@thehuntermansdaha ben de yazmayı bilmiyorsunuz abicim nasıl öğrencisiniz siz ?
Bu ne la biz okulda bunları görmedik ki.
As a Turk, I can say asking harder questions and try to teach many subjects in a year doesn't make us learn Math better. Because you forget them all together. That is why we forget the simplest things.
In Turkey University exam is an elemination exam. So main target is eleminating students not to detect the level of students. So elemination exams are like hurdle race.
@@darkprofile Yes but the problem is that you have to study those kind of questions for years at school as well. They used to plan curriculums according to those hard questions. I don't know how it is now though.
@@enesa6489 Such questions can’t be taught at school. Every year there must be few new question types which never asked before. It is for eleminating hard wıorking students who studied all question types from intelligent students who can find a way to solve a new question type.
this specific type of question is beneficiary if you study mechanical engineering or similar...sometimes you have to come up with a mathematical model based on practical findings.
zaten mesele öğrendiklerini hatırlamak değil. Beceriyi ölçmek. O sınava çalışan herkes aynı sonucu almıyor. Ayrıca 5kya girmiş bi mühendislik öğrencisi olarak yks dönemimde analitik düşünce becerim çok yükseldi. Analitik düşünce becerisi kolay kolay kaybolmayacak bi nitelik. Gerçi artık böyle zor da sormuyor ösym. Son sınavlar çok daha basit hazırlanmış
I was one of the rare students who answered all 40 Math questions correctly in the exam. The hardness of the exam doesn't come from the questions themselves, it comes from the time pressure and stress you're experiencing. You're whole future is being determined by a 3 hour exam. I remember it felt like as if there was a gun on my head and people were yelling at me to solve the questions. Thank you so much for remarking the hardness of the exam.
Usta hangi üni geldi
@@beperseverance520 Hakkari Peşmerge Yüksekokulu
I was also one of the students who correctly answered all the math questions on uni exam during my time but i dont aggree with your views, one can always attend a university exam in the followings years if he/she thinks his/her current success is not satisfactory. i have seen people attending uni exams during their retirement just for the fun of it. so it is not like "Your whole future is being determined by a 3 hour exam" besides economic or social success in life is not related much with one's academic success
@@servetc1970that doesnt work for everybody and seocnd exam is a hell year for sure since i just experienced it last year. I have high stress tolarance and still constant stress made me sick not just the exam but the durstion till the exam for like 3 months. Terrible experience fors ure
@@lavolpe_irl8997 that is another way of "elimination". eventually this is an exam for selection and you need reasons for elimination. not being able to correctly answer questions is a reason, but not being able to adequately cope with stress and pressure is also another way of elimination.
I followed you for years and I never expected a question from my homeland, let alone an exam question I had to solve in 2021 💜 thank you
Geçmiş olsun sıralaman nasıldı ve netlerin nasıldı (sormamı sakınca etmezsen)
Shout-out to all Turkish people keeping it real 🇹🇷
very appreciated bro, i'm gonna take this exam at the end of the june and hopefully can start to study at the university i want
@@krpzsvrmSelam! Çalışmalarında başarılar dilerim ben de YKS için hazırlanıyorum😍 (Türk buldum yuppi🇹🇷😍😭)
@@OrangeAutumn13 teşekkür ederim sana başarılar diliyorum umarım emeklerimiz karşılıksız kalmaz
this questions was super easy
@@shivx3295well who are you to judge when an actual person who studied mathematics and economy at the university says it
In my opinion, this question is actually very good and mind-blowing, even though it is very hard and not suitable for the university entrance exam. In the exam, I was able to solve the question thanks to background on mathematical olympiad.
Yav bırak. Şıklar olduğu için çözebidin
@@ToxicTurtleIsMad kdbdbdbsjsbbsn kardeşim şıklar var diye herkes full mu çekiyor
@@ToxicTurtleIsMadabi araştırmanı öneririm (genel)
In the last years questions started to become like iq tests rather than questioning your knowledge. 10 years ago it wasnt this bad but since youth unemployment rose and need for high education in good unis rose too.This caused a surging competition that only grew bigger as the economic crisis got worse. As everyone knows once the competition climbs it is really difficult to get less difficult questions in exams. Korean and Japanese exams are a nice example to that but Turkey's education system is not good as them, in fact it has been steadily declining for at least 7-8 years.
ıan fact, it is not like a logical IQ test, it is based entirely on memorizing the question type and just studying the same thing over and over again, and this is actually the sad part.Because after years u not learn anything and not memorize mostly
@@Kaholens Believe me as a doctor who won medical school and got 89 in Ales two years ago memorizing is not a good way to solve any Turkish and math question nowadays. In the past you needed to solve a lot of different type of questions over and over again to be able to think when you see a slightly different type of question come to you but nowadays, no matter what type of questions you solve they manage to find a much more different question type that you are not ready and if your IQ and knowledge isnt enough you are fucked.
When I go into uni it wasn't this intense and I saw an intensity spike when I get into Ales two years ago. They were just Iq questions rather than "solve a lot of questions types and you are good to go" questions I was accustomed to.
skill issue
@@purplewine7362wow you did so much by typing that
@@EliminationSquare They solved our education system just with one comment. Who knew it would be this easy?!
We need more Turkish questions.
I like your method. To me the obvious thing to do was to use the fact that the derivative of P(x) had to be 1 at x=1 and x=3 to complete the system of equations. But simultaneously solving 5 equations by hand is laborious, and your approach avoided this.
I used this technique, and reducing the augmented matrix by rows wasn't all that bad, just don't bother trying to get a diagonal matrix. Instead just reduce it to a triangular matrix and then solve up the triangle as you get each coefficient value.
A similar trick can be used to find the unique line tangent to a quartic at two points (if the quartic has 3 local extreme values). If you start with y=x^4+ax^3+bx^2+cx+d, you can find u and v such that (x^4+ax^3+bx^2+cx+d)-(ux+v) is a perfect square. Alternatively, you can first shift x to get rid of the x^3 term, then subtract the equation of the tangent. That gives a perfect square with no x^3 term, so it has to be (x^2+k)^2 for some constant k. Then it's easy to compute k from comparing coefficients.
When student learns this trick and solves the question without actually understanding how and why, we get serious issues in Turkey with IT,Med anything engineering related. This system is wrong. They even memorize AI prompts now!
My father taught me this trick
Bu soruyu çözmek için verilen süre maksimum 2.5 dakika
Old Turkish student here. This is not the method of solving this type of questions. Because a student is expected solve the hard questions under 2-3 minutes, you cannot apply all the logic to solve the questions step by step. The magic lies in your preparation. High calibre students often solve thousands of (not kidding) questions just for some topics in the exam such as calculus part. So a Turkish student should recognise the pattern of the question and skip few steps. That’s how you ace turkish uni entrance exam :)
For the question in the video, you should immediately see that 1 and 3 are double roots without thinking over the graph or behaviour of the equation. For real, the exam is mostly about pattern recognition and precision rather than pure knowledge.
Whenever I see an international channel mention this exam I feel so weird because I was one of the many students who participated this exam and solved this questions.I was lucky enough to solve most of the questions. (regarding to the fact mathematic was my favorite lesson I can even say I was lucky that they did math extra difficult)
I can appreciate hard Harvard-Oxford or Olimpiad questions but oddly enough seeing my exam's questions in this channels gives me a strange out of reality feeling... Idk why...
Nonetheless thank you for bringing attention to this exam and solution... I enjoyed watching it :)
If this is the hardest question in the exam, I'm surprised the average is 20%, because it's not that hard.
@@elidrissii well, I never said this was the hardest question...
Also there are many factors which makes an exam difficult you cannot claim it is easy because of one question.
This question is from second part of the exam.
First part and second are not require the same thing from us. First part known as TYT requires basic knowledge and high understanding skills while second part (AYT) requires branch spesific knowledge, you have to have deep understanding of the topic(in high school level of course) to solve many of that questions... At that time it was not a new system that we didn't know but what made this one special it was out of the border. ÖSYM(the authority that prepares this exam) used to ask some similar question types that we are familiar with and trained for but that time they turned the tables around and did completely different exam without prior notice. There were olimpiad questions in both science and maths.
Also as I said before there are many factors that make an exam difficult. This was University entrance exam that we took in pandemic... We had to study with online lessons, courses were closed, our relatives even ourselves infected with virus and we had to deal with all of that along with the stress an important exam gives us. In our previous year eventhough the exam was the easiest they had done this far they added extra time for the first part. (Normally it is 135 min for 120 questions but they added extra 30 min for them) But logically(!) for us they made it harder without adding any seconds....
So, for me it was hard and I think I have every right to think that way :)
@@delta4325 Thanks for elaborating.
@@elidrissii I also want to thank you for your kindness...
When I read what I wrote for the second time, I felt like it may came a little bit harsh... I didn't want to sound that way, I just wanted to explain a bit 😅... I English is not my first language and I am not so very good at it... So if I offended you in any way I am so sorry... And again thank you for your kindness :) ..
@@elidrissiithis question alone might not be very hard but imagine having to solve questions like this one for 39 more math questions, and don't forget you can't lose much time on one question bc you still have the science part left
I like the approach to the first problem, and it's more efficient than mine. I stuck with p(x), realising that it had to be tangent to y=x at x=1 and x=3. So you have 5 equations for the 5 unknowns in a x^4 + b x^3 + c x^2 + d x + e, knowing p(x) at 1, 2 and 3 and that p'(x) = 1 at x = 1 and 3. I wrote them in matrix form and inverted the matrix to get p(x) = 2 x^4 - 16 x^3 + 44 x^2 - 47 x + 18, which has value 22 at x=4.
Gençler 3 milyon öğrencinin girdiği üniversite sınavı bir seviye tespit sınavı değil bir eleme sınavıdır. Bu sınavda az sayıda aşırı zor ve aşırı kolay soru sorulur. Sınavın genel zorluğunu belirleyen ise orta ve zor kategorideki soruların seviyesidir.
3 milyon öğrenciyi sıralamak gerektiğinde soruların zorluğunu ayarlamak sorun haline gelir. Kolay bir sınav yapıldığında herkes bir birine yakın netler yapacağı için çok iyiyi iyiden, iyiyi de ortadan ayıramaz hale gelirsiniz. Yani adil sıralama olmaz.
Aşırı zor sınavda da kötü ve orta düzeydekileri ayırmak zorlaşır.
Sınava hazırlanan disiplinli öğrenciler önceki yıllarda çıkan tüm soru tiplerini ezberlerler. Çalışkan disiplinli çocuklar ile zeki çocukları ayırmak için daha önce hiç sorulmamış böyle zor ve yeni soru tipleri üretilir. Zeka yeni koşullara uyum sağlama becerisidir. Yeni soru tipini zorluk derecesine göre daha zeki olanlar çözebilir ve zeka üzerinden eleme yapılır.
Yani eğitim bilimlerinde ölçme değerlendirme zor bir iştir. 3 milyon öğrenciye yüksek eğitim vermek için sıralama yapmak gerektiğinde size zor ve saçma gelen bu sınav en adil yoldur. Düşünün bu ülkede memur olmak için KPSS ye giren adaylardan yüksek puanlılar mülakatta elenip yerlerine düşük puanlılar alınabiliyor. Üniversite sınavının zorluk düzeyinin yüksek olması herkese aynı zorluk uygulandığı ve daha çok doğrusu olanı yukarı taşıdığı için adildir.
Hepinize başarılar dilerim...
Peki hocam işlenen onca konuya onca üniteye rağmen bu kadar az sorunun olması sizce de adaletsiz değil mi? Biz biliyoruz ki bir verinin miktarı ne kadar fazla olursa sonucu da doğruya o kadar yaklaşır. Sınavda çalıştığımız kısmın neredeyse yüzde 60ı bizim karşımıza çıkıyor. Çıkan sorularda da birden fazla konuyu bir soruya sıkıştırıyorlar, diyelim ki sen o sorudaki 5 konunun 4ünü çok iyi biliyosun bir tane öncülü yapamadın diye o soruyu iki şıkka indirip sallayan adamdan daha geriye düşüyosun. Dediğiniz gibi sınavlarda zor sorular kesinlikle olmalı çünkü giren sayısı çok fazla ve bu insanların bir şekilde elenmesi lazım fakat soru sayısının azlığı adaletsizlik yaratıyor bana göre
@@emirhanylmaz946 Güzel kardeşim bu sorular belirli araştırmalara göre optimize ediliyor.
Orta öğretimdeki her konu bu sınavda çıkamaz. Çünkü amaç seviye tespiti değil eleme yapmak. Eleme sınavları seviye tespit sınavından farklıdır. Seviye tespit sınavlarında dediğin gibi sorular konu ağırlığına uygun oranda sorulur. Yani sımavın içerk tutarlılığı yüksektir.
Ancak eleme sınavları 400 metre koşusundan ziyade 400 metre engelli koşusu gibidir. Senin ne kadar hızlı koştuğuna değil engelleri ne kadar hızlı geçtiğine bakılır.
Bahsettiğin farklı konuları tek bir soruda sormak da eleme sınavlarının engellerinden biridir. Bir geometri sorusu aslında köklü sayılarla işlem yapma sorusudur. Bir kenarı 3 verse herkesin yapabileceği soruyu o kenarı 3 kök 2 verince sınava girenlerin yarısı elenir.
Yani amaç eleme yapmak olunca çok basit bir soru 15 cümlelik paragrafın içine gömülür. Aslında herkesin doğru yapacağı bu soru yavaş okuyanlara zaman kaybettirmek çok test çözmeyenleri yorum odaklanmalarını düşürmek için konuluyor.
Yani değerli kardeşim eleme sınavlarının doğası böyledir. 3 milyon kişi sınava girerken aslında hedeflenen doğru düzgün üniversiteler ilk 100-150 bin kişinin girebileceği yerlerdir. Yani aslında sınavda ama %1-5 arası en yüksek puanlı üniversitelere sıralama yapmak. Kalan bölümler zaten o %5’e yerleşemeyenlerin tercihleri.
Maalesef kalabalık bir ülkeyiz. Ve parasız üniversite eğitimi için maalesef bir eleme sınavına ihtiyaç var. Ve 3 milyon kişinin girdiği sınavda uzun sorular, birden fazla konu alanını kontrol eden sorular ve daha önce hiç sorulmamış tip sorular olması gerekiyor. Senin istediğin gibi soru sayısını arttırarak eleme yapmak pratik değil. Sınav süresi zaten yeterince uzun. Daha fazla soru ve daha uzun sınavda tuvalete bile çıkamadan çocuklardan performans alınamaz.
İnsanların şikayeti bu sorunun zorluğundan veya sınavın yönteminden değil. İnsanların şikayeti milli eğitimin bu tür soruları çözecek kaynak, öğretim veya ortam sağlamıyor olması
@@azovianace İşte mesajımda tam da bunu anlatmak istedim değerli kardeşim. Eleme yapacağın zaman bir kaç tane eğitimini vermediğin, öğrenciye nasıl çözebileceği öğretilmemiş soru sorarsın. Bu soru çok zeki çocuk ile zeki çocuğu ayıran sorudur, sınav birincilerini belirleyen sorudur.
Siz hayatta karşınıza çıkacak tüm soruları nasıl çözeceğinizin size bilgi olarak verilmesini bekliyorsunuz.
Hayat böyle bir şey değil. Hayatta daha önce kimsenin çözemediği sorunları çözenler ya da bir sorunu başkalarından daha iyi hızlı verimli çözenler başarılı olur.
Yoksa herkes kendine öğretilen bir konuyu ve soruyu çözebilir.
Harika bir yorum olay tamamen bu
Here’s the ‘Before 3 million subscribers button’
Yh
Here
Hard to believe that many, looking at the views videos get
Here
Same
Thanks for this nice problem.
To my surprise, I found it quite straight forward.
Let f(x)=P(x)-x, a fourth degree polynomial that is always non-negative.
Then
f(1)=P(1)-1=0
f(2)=P(2)-2=2
f(3)=P(3)-3=0
So f has zeros at 1 & 3. These must be roots of even order as f is always non-negative (if a polynomial f has a root α of odd order n, then f(x)=(x-α)ⁿg(x) where (x-α)ⁿ changes sign at x=α but g doesn't, so f changes sign at x=α).
Hence each root is of order 2.
So f(x)=a(x-1)²(x-3)² for some real a≥0.
From f(2)=2 we get
a(2-1)²(2-3)²=2
So a=2
f(x)=2(x-1)²(x-3)²
P(x)=2(x-1)²(x-3)²+x
P(4)=2(4-1)²(4-3)²+4=2×3²×1²+4=18+4=22
i think you wrote this comment before he solved, right?
@@berdigylychrejepbayev7503 Indeed, my solution (which I wrote before watching the video) is very similar to the solution in the video, but I thought it worth adding anyway because of the justification of the double roots, which I think is clearer than in the video.
nicee, this is how we solved this problem too (also approved by our teacher) (i''m a turkish student studying for the entrance exam)
@@MichaelRothwell1 Exactly the way I did it; also before watching, half hoping to see Presh use some other technique.
I also didn't call on the derivative, but just noted that both 1 and 3 had to be double roots (any higher even order, and P would have degree > 4).
It's a very pretty problem.
Fred
Dont get me wrong im not trying to be rude but how long did it take for you to solve it genuinely asking
always feels weird when non-turkish people acknowledge the difficulty of turkish uni entrance exams. it both feels sad because we're tormented like this, and kinda boosts one's ego because "yes, i did manage to solve this!"
good video as always, love from turkey!
You are not tormented, you are challenged. There has to be difficult questions otherwise among the top students many would solve a similar amount and it would be difficult to get a good grading among them.
@@chief4180 yeah i don't think we need to be dealing with this high level stuff in high school. thankfully i'm planning to either go abroad or study humanities here, and my math is sufficient for the TYT math questions. still, this system isn't humane or pedagogically sound.
At the end, it's a "ranking" exam so questions being hard wouldn't change anything
@@eda6654 If you don't think you are smart enough to solve it, then don't solve. Not everyone needs to go to college.
@@chief4180 this isn't about being smart enough. you need to dedicate your life to it and it's so strange for you to assume my intelligence
There is a way to non use calculus to complete the exercise and do it using pure algebraic properties...
Just note that Q(x)= P(x)-x has roots at x=1and x=3, this means that Q(x)= (x-1)(x-3)H(x), with H a 2 degree polynomial, then given that P(x)>=x, we see that Q(x)>=0 meaning that H(x) Is a quadratic polynomial that just happens to match the signs of (x-1)(x-3) to be always positive, given that (x-1)(x-3) is negative in ]1,3[ and positive outside that interval, H is forced to match this while being a polynomial of degree 2, negative in ]1,3[ and positive outside that interval, meaning that is actually h*(x-1)(x-3) with h>0, now evaluate Q(x) in x=2 to get the value of h and you get the result.
Is less clean but can be technically solved without using calculus tools.
This was the desired solution because there is not much time during the exam. Practical and fast solutions are needed.
well done!
Regardless of the whole min/max analysis, it is obvious that Q'(1)=Q'(3)=0 because Q(x) cannot cross the zero. So setting Q(x)=(x-1)(x-3)(ax^2+b.x+c) and solving the system Q'(1)=Q'(3)=0 and Q(2)=2 gives a, b and c right away.
in turkey you have 40 math (30math+10geometry) and 40(physics, chemistry, biology) questions and have 120 minutes to solve these, nearly all math questions are same difficulty as this or most propably higher difficulties, and you almost have 1:30 minutes to read understand and solve each question, and every high school senior has to take this exam in order to join an university, and every university has a limit of points
in ayt we have 180 minutes and 80 questions, in tyt we have 120 questions ond 165 minutes
Not 120 , its 180 for ayt
@@saurongrows its 160 and its depending on your choice of grades, 120 if you chose literature you solve 120(40 literature, 40 social stuides, 40math) and if you chose science you solve 80 (40 math, 40 science (14 physics, 13 chemistry, 13 biology)
@@das-panzer-maus if you dont know, then dont talk about it. every student that participate in AYT solve 80 questions, literature students dont solve math.
This question is to eliminate best ones in that test. You have to be in top 2000 to be in best universities, and there are 2 million candidates. So if you are good , you have to solve easier questions so fast and you can have time to solve hard problems like this.Otherwise you have 3-4 mins for every questions. But you can't solve this question in 3 mins if you are not a genius and and ur not lucky and you are not calm. Hard to stay calm, because its very stressful
I rather liked this problem! I feel like your justification of the multiplicity of the two roots could have been slimmed down to simply that polynomial roots either cross or touch and turn. Since crossing would make the polynomial go negative, they are touch and turn roots. Since the degree is 4, that makes them both multiplicity 2. No derivative or talking about turning points needed!
Yes, exactly how I tackled it.
nothing wrong with getting fancy lol, he definitely knew that, he just wanted to mention it
Exactly! This is a pre-calculus problem that really requires careful thinking about the nature of roots, and of the relationship of polynomial difference.
"touch and turn" is a turning point, but the 3rd turning point isn't important.
You are expected to solve this within a minute.😅
180/80 yapamıyosan niye videoyu izliyonuz 2 dk dan fazla düşüyor 1 soruyo
no, you are not
sallama aq
bizim cocuklara yazik ya
ayt bu oldukça zaman var ki soruyu çözmek aslında o kadar uzun sürmüyor. videoyu izlediysen de görmüşsündür daha kısa bir şekilde nasıl çözülür onu da anlattı
I made 40/40 at maths this year and video title make me proud but since i chose medicine faculty i wont need maths for my all life ;(
Well here is how I would approach it.
The condition P(x) >= x motivates the definition of Q(x) = P(x) - x
Q(x) is positive everywhere thus every roots are at least double (because there are extrema.
1 and 3 are roots of Q. Thus, Q must be of the form:
Q(x)=C*(x-1)^2(x-3)^2 with C a real constant
Given P(2) = 4 Q(2)=2, we deduce C=2
Thus, we can compute Q(4) = 2*3^2*1^2= 18
Thus, P(4) = Q(4) + 4 = 22
thats what i did
@@punjabigaming146 yeah I commented before seeing the rest of the video… so yeah, sometimes it is just the same or almost the same
Hello there, the only reason behind the average of 7.6 out of 40 isn't the hardness of the test but also the worsening education system of Turkey. At my year (2011) I got 39 out of 40 and the average was 13,17 out of 40.
şimdi ne yapıyorsun peki
Should we blame Mr. Erdogan for that? Math is political now :--]
Ngl those years were kinda easy 😏😏😏
Math is a political weapon in many countries now. After all, rich kids are in private school anyway so the politicians are happy to lower the bar for public school.
@@adam.gizlin The guy did not even mention Erdoğan in the context. He rather said it is "the worsening education system" which MAY be Erdoğan's fault but he did not blame anyone.
Pre-watch:
Actually pretty easy, via the following insights. Given:
P(x) ≥ x
Now define Q(x) = P(x) - x
Then Q(x) = P(x) - x ≥ x - x = 0
Q(x) ≥ 0
And from the given values of P,
P(1,2,3) = (1, 4, 3)
we know that
Q(1,2,3) = (1, 4, 3) - (1,2,3) = (0, 2, 0)
But these facts mean that 1 and 3 are both double roots of Q [otherwise Q would have to be negative somewhere in the neighborhood of both]. Thus,
Q(x) = a(x-1)²(x-3)²
where a > 0. Thus,
2 = Q(2) = a, meaning that
Q(x) = 2(x-1)²(x-3)²
P(x) = 2(x-1)²(x-3)² + x
P(4) = 2·3²·1² + 4 = 18 + 4 = 22
Now let's check out Presh's solution . . .
Fred
I graphed P(x), then noticed that for P(x) >= x, P'(1) = 1 = P'(3). This gives 5 equations with 5 unknowns.
Me too! You have to verify that the 5 equations are linearly independent.. but they are 😁
Yes, exactly! Since P(x) can not pass below the line y=x, this means that P(x) is tangent at the points x=1 and x=3. In addition, the slope at those two points must equal 1. This gives you two additional equations so that you can solve this problem easily with 5 equations and 5 unknowns.😀 I got P(x) = 2x^4 - 16x^3 + 44x^2 - 47x + 18
So did I - but inverting a 5x5 matrix takes more time than the solution presented here. Using double-root theory is more efficient.
Same. The given solution is better though; it took me half an hour to solve the 5-equation system, and if I hadn't used a spreadsheet to do it I almost certainly would have dropped a digit somewhere and gotten the wrong answer.
@@fralfa95 I mean, if you can solve the system they are linearly independent...
It feels so surreal this is the first time I’m able to fully understand the question and know how to answer it on my own
I just bashed it with calc. P(x) = ax^4 + bx^3 + cx^2 + dx + e. You have 5 unknowns and are given 3 equations. But notice P(3) = 3 and P(1) = 1 and P(x) >= x. This means P(x) must touch the line y=x when x=3 or x=1 but not cross it, otherwise you will have P(x) < x. This means y=x is the tangent line at x=3 and x=1. So you have P'(3) = 1. P'(1) = 1. This gives you two extra equations, so you can solve. Took way too long tho, have to bash 5 linear equations. Don't recommend
lmao, as a math grad I looked at this and came to the same conclusion but also realized in an exam it would certainly not be possible to do in time, despite the saviness of it. I STILL BRUTE FORCED IT ANYWAYS
From Turkey. Funny, how we have 2 or 3 minutes to solve this. XD
@@SA-xe4bfyalan değil ki 180/80 dk soru başına düşüyor
@@SA-xe4bf haklısın da sonuç olarak bu soru için 5 6 dk bile az bence
@@SA-xe4bf tamam kanka en zeki sensin
I am a Turkish person going through this kind of education system, people might think that We turkish children are smart and have very high academic success, but no, we don't. In fact, this system causes many children with un-explored talents to fail, and literally drop out of school, then go to an average job like an adult. in age of maybe 14. Our system relies mostly on obedient students, not the ones that can actually question and think. Many students started to kill themselves also because of sociological structure and mindset we have.
The government has also started to slowly assimilate children politically and religiously, I dont wish to talk about these topics though, I might be enjoying my last happy moments
I recently noticed some suicidal tendencies on some of my friends, Im concerned honestly. Yeni eğitim bakanı içimizden geçti zaten daha neler bekliyo bizi bakalım💀🤯
Welcome to Turkish secondary education :) This is asked at the end of the high school in the university entrance exam to be answered in a few minutes.
In an exam which must sort 3 millions of students, there can be few extremely high level questions which eleminates very good students from genius students...
Amaç o kanka zaten herkese orta seviye soru sorarlarsa herkesin puanı aynı olur ekstrem sorular sorarak 3 milyon kişi içinden insan elersin
Sayısal 200e girmiş biri olarak yazıyorum 2021 soruları abartıydı, soruların ilk bin ile geri kalanı ayıracak zorlukta olması gerekiyor; bu ölçüyü tutturdukları sınav görmedim galiba.
@@h3z33g9 Soruların ilk bin kişi ile kalanları ayıracak zorlukta olması gerektiği fikrinin dayanağı nedir?
@@darkprofile Çünkü ilk bindeki öğrencilerin düzeyleri genel olarak aynı, sıralamalarda yukarı çıktıkça şans faktörü daha önemli oluyor. İlk iki binin dışına çıkmaya başladığımızda öğrenciler arası fark daha barizleşiyor. İlk bindeki potansiyeli yüksek öğrencilerin diğerlerinden ayrışması için zor sınavlar lazım. Sınav kolaylaştıkça yeteneği yüksek öğrencilerin elenme ihtimali de yükseliyor. Benim bu sözümdeki amaç bu öğrencilerin istediği okula girmek için kendilerini harap edip sonra sınavın kolaylığı yüzünden israf olmalarını engellemek.
Sizin şaşırdığınız şeyler bizim dramımız
turkish med student here, it was one of the most comprehensible ones in the pool of hard turkish exam questions imo (entered the exam in 2021, the same year this was asked btw)
hangi üni
yaptınmı
It’s easier to consider P(x)-x. P(1)-1 and P(3)-3 must both be double roots to avoid P(x)
Solved it this exact way, but I think thats whats happening in the video as well
Easier than what? Your approach is exactly what happens in the video explanation.
@@TheStickManPainter I commented before watching the video. Presh did take the same route more verbosely.
@@yurenchusolving for P(x)-x is easier than solving for P(x) directly.
After plotting p(x)=f(x)-x, which leaves you with a function that must be greater or equal to zero everywhere, which implies slope of zero at 1 and 3, you can look at it and realize the result must be symmetric around x=2 and write the function p(x)=2-a*(x-2)^2+b*(x-2)^4, then solving for a and b is easy. The odd terms in (x-2) are zero because the square and forth power terms are symmetric in (x-2). The 2 is at x-2=0, the slope and value at x=1 determine the rest.
For Q to be positive, each of the two roots has to have multiplicity 2. This can be checked with the derivative. This determines Q up to a constant, which is determined by the value at x=2.
I got 35 right in tyt and 31 right answers in ayt for maths in 2019. It was pure hell just studying
i entered the same 2021 one and had 19 rght and 6 wrong still enough for a good üni tho lol
This question is actually easy if you solve something like this in your 4 year highschool education. The real hard problems are not the one solved with methods, methods can easily be memorized. The real hard problems are generally lie in the problem section, so few students be able to do problem section
You are absolutely on the right track of thinking.
There are a lot of questions that are as hard or even harder in our exams... This is just one question
This question was easily one of the hardest if not the hardest question of the 2021 AYT exam. There weren't " a lot " of questions harder than this. I'm saying this as a person who took the exam in 2021.
@@ruz4668ikiniz de Türksunuz la
Ciddi misin?
Others have given very nice solutions.
If someone wants to learn more about such problems, it is called an __Hermite interpolation__ problem:
Given m equations on a polynomial of the form P(x_i) = y_i or P^{k_i}(x_i) = y_i (denoting k derivations), then there exists a unique polynomial of degree at most m-1 that satisfies all equations.
Turkey is where people work really hard and get really nothing in return. (Maybe except sorrow.)
1:30
The first thought was about making a graph of known data, it immediately help to realize that needed P(x) tangent to f(x) = x in {1; 3}. After that I chose the harder way: made a system of 5 equations and 5 variables (coefficients of polinomial) and solved it via Gauss' Method) (and double check it in Wolfram and Desmos)
After graphing P(x) - x in Desmos I realize the better and faster approach, but it was too late...
Know, let's continue watching))
My approach as well.
But Presh's solution is more elegant since he did not have to deal with a system of 5 linear equations.
that is possiable but remember you have a time limit at the exam, it is 3 hours for 80 questions but if the exam is filled with these kind of questions you dont acctualy have time to do that
@@Emir_DuzlerWasn't it 160 minutes? Are they giving more time in the new exams?
Welp, it means they will ask harder. We will be damned...
keep in mind that you’re given approximately 2-3 minutes to solve this in the actual exam
Üniversite sınavında bu kadar zamanımız yok
They probably assume that you've solved so many questions that you'd already know the steps of the solution by memory. To be honest they expect you to memorize things.
Elegant. The way I solved it was that we have 5 points:
P(1)=1, P(2)=4, P(3)=3, and by the P(x)>=x we have that x=1, x=3 are points of tangency with gradient one: P’(1)=1, P’(3)=1.
Then use Gaussian elimination to solve the 5x5 system.
A bit rough, but it works
this was a break-through question in 2021 ayt. before that, math tests were relatively easy. after 2020 being too easy, they decide to make it a little bit harder in 2021. and it made the students to prepare entirely different for the exam. since it is an elegant question and we don't get to solve questions like that at school because teachers are not qualified enough, students started to bail out of school and study with private tutors or all by themselves. and turkish officials got really mad because students eventually realized the educational system is bad, so they made continuity in school mandatory recently. so long story short, students are stuck in school with teachers who can not even solve this question, yet students are expected to solve this question.
Before watching, here's how I solved it:
Let Q(x) = P(x)-x, then Q(1)=Q(3)=0, Q(2)=2, deg(Q) = 4
And it satisfies Q(x) ≥ 0 for all x.
Therefore, Q(x) = (x-1)(x-3)R(x) for R being a second degree polynomial.
Since Q(x) ≥ 0, the points x= 1,3 are local minimums and therefore Q'(1)=Q'(3)=0
Note that Q'(x) = (x-3)R(x) + (x-1) (R(x) + (x-3)R'(x))
Plug in x= 1, -2R(1) = 0 -> R(1) = 0
Similarly, plug in x = 3, 2R(3)=0 -> R(3) =0
Therefore, R(x) = A(x-1)(x-3)
And Q(x)= A(x-1)^2(x-3)^2
Plug x=2, Q(2) = A•1•1 = A
But Q(2)=2, therefore A = 2
Finally, P(4) = 2(4-1)^2•(4-3)^2+4 = 22
Edit: looks like my solution is very similar, down to the name and definition of Q. But he used a visual argument, while mine was more algebraic.
If you transform by subtracting x and shifting left and down by 1, you get a new poly Q(x)=Cx^2(x-2)^2 as the poly that touches y=0 twice without ever being less. Solve for C, evaluate at 3 and transform back to get P(4).
I skipped ahead and the ending suggests something similar is how it's done here too. The two roots of even multiplicity and the deg 4 don't leave room for additional freedom.
The horrible thing is that students only have 3 hours for THIS QUESTION + 159 QUESTIONS IN TOTAL ! Even tho you dont have to answer all 160 questions (There are the lessons of "Turkish, Math, Science and History" 40 questions each. You solve the lesson(s) that are related to your goal profession.), its still so BRUTAL !
80 soru 180 dk ideal kanka fen uzun sürmüyor zaten
let Q(x) = P(x) - x
there for
Q(1) = 0
Q(2) = 2
Q(3) = 0
and Q(x) >= 0
so we have 2 roots and those roots are also extremums (so on 1 and 3 Q’(x) = 0) so they are both double solution so Q(x) = k[(x-1)(x-3)]^2
replace with 2 we got Q(2) = k[1*(-1)]^2
so 2 = k
and now replace with 4
Q(4) = 2[3*1]^2 = 2*9 = 18
so P(4) = Q(4) + 4 = 18 + 4 = 22
1:07 before I watch any further im gonna guess that this is simultaneous equations with a general quadratic equation with the x values subbed in
Exclamation, the reason avg score is so low is not just because the difficulty but because of the “fields” system, not all of the 2 million students are responsible of the math part but some try it anyways thus lowering the average. 30 and more out of 40 is common for students who are responsible for math part and it is not surprising to see someone making a perfect score.
beautiful question with fantastic solution
Im taking that exam ( TYT - AYT ) in 2 years and i hope i will get a really good mark .
is it then so that if a quartic function has more than 3 turning points, the function will be of a complex function?
This is a selective question in this exam. There are about 5 questions at this level, and the remaining 35 are more straightforward; however, it is bizarre to expect 18-19-year-old students' to solve this in a few minutes with that exam pressure.
In 2006, I took that profeciency test for University. I had only 4 correct answer. And years I was blaming my self because of that a disaster and now thanking to the God at least I accepted and graduated from University..
Love from Turkey, thanks for the video.❤️🩹
Yks nin gelmiş geçmiş en zor sorusu olabilir ama çözümü gerçekten çok güzel
hangi senenin sorusu bu ?
@@BlueSky-ho6dy2021 diyor videoda
@@mhm6421 aynen 2021 ayt
En zor mu? Keşke 😃
@@_cran cidden en zoru bu olabilir ki olimpiyat sorusu bu direkt TYT ya da AYT'de daha zorunu görmedim ben
Since P(x) is a 4th degree polynomial with real coefficients and satisfies the condition P(x) >= x for all real values of x, we can conclude that P(4) is greater than or equal to 4. However, without more specific information about the polynomial, we can't determine an exact value for P(4)
I’m med student from turkey and I took this AYT exam. This exam was really hard because I have 2 minutes for every questions. And this questions answered by 400 students out of 2 millions and this statistic shows this question is really difficult. If you like this question you should check 2021 AYT’s derivative and permutation questions. And meanwhile I became 4664th in 2022 AYT exam out of 2.5 millions student
Bunlar 2022 nin sorusu muydu hatırlayamadım
@@Geraltofrivia1012 2021
bu sorunun sadece 400 kişi tarafından cevaplanmış olma şansı yok nerden duyduysan doğru değil.
Atıp tutturan bile 1000 kişi vardır, o 400 sayısını nerden buldun bilmiyorum. Zor soruydu ben de cozememistim diye hatırlıyorum sjdksn ama 400 kişi olma ihtimali yok
@@r_voltaire7211 sen de yap sen de 4 hane yaz kanka 💀
i can still remember how i was working my brain off for this bloody question in that exam.When i was done with reading that question what came out from my mouth was just “i’m fucked up” i had no clue at all
1 4 3
3 -1
4
So , 1 * (n-1 choose 0) + 3* (n-1 choose 1) +4* (n-1 choose 2)
Put n= 4 to get ans as 1+9+12 =22
are the Turkish students supposed to solve these problems right after high school??
i am fron india and here they don't have dedicated cubic of quartic equation lessons. so it's a bit bizzare seeing such a problem
yeap if they want to place in a nice faculty
Without having watched beyond the problem statement; you are given that P(x)>=x for all real x, and that P(1)=1 and P(3)=3. Consider the polynomial Q(x)=P(x)-x. Then Q(x)>=0 for all real x and Q(1)=0, and Q(3)=0, which means that Q(x) has a double root at x=1 and at x=3. This means Q(x)=(x-1)^2(x-3)^2R(x) for some polynomial R(x). Because Q(x) is quartic you know that R(x) is in fact a constant. Then P(2)=4 tells you that Q(2)=2 and so R(2)=2, so Q(x)=2(x-1)^2(x-3)^2 and hence Q(4)=18 and P(4)=22.
This is about 2 minutes of work, and all you need to know is that if a polynomial is tangent to the x-axis at a point t, then it has a factor (x-t)^2.
Yes we do. we dont really have dedicated classes for each subject. The same math teacher will explain everything from log's to integrals in a years time steadly going over them in (roughly) a pre determaned speed set by the goverment body of MEB (National education something İdk)
The point of this exam is to differentiate the students.
This was one of the hardest questions in the test, so it was expected that only the best students would be able to solve it.
I graduate as a science student from high school, but thanks to these kind of questions now I study ELT.
Same...
Actually, most difficult part is not the problem itself. It is trying to solve this problem with a small time limit.
I am a Turkish student, I remember the first time that I solved this question without any help, I was super excited that I was about to cry lol! But it really improved my problem solving skills.
One possibility is by the use of matrices. Let A be the matrix
[[1,1,1,1,1], [1,2,4,8,16], [1,3,9,27,81]] and B be the column matrix [[1], [4], [3]], and let C be the column matrix of the coefficients of the 4th degree polynomial [[a],[b],[c],[d],[e]], then one could solve for C as,
C = B.INV(A) , where INV(A) is the inverse of matrix A.
Why is 1 and 3 double roots? The polynomial can be written as a(x-1)(x-3)(x^2+bx+c) where b^2-4c
Because x=1 and x=3 are roots of Q and Q' simultaneously
And because it bounces off zero
If b^2 - 4c < 0 then the polynomial q(x) = a(x-1)(x-3)(x^2+ bx + c) has opposite sign of a for x between 1 and 3 . So if a>0, then this polynomial expression is _negative_ for 1
Elegant solution, as always!
Q(x)=P(x)-x
Q(1)=0
Q(2)=2
Q(3)=0
Q(x)>=0
So Q'(1)=0 and Q'(3)=0 so Q can't dive under 0.
The curve looks like a W.
Also by symmetry Q'(2)=0
and Q(4)=Q(0)
So we only need to find "e" in
Q(x)=ax⁴+bx³+cx²+dx+e
And P(4)=e+4
For anyone who got this, how did you know to create Q(x)? Did it stand to you from the 1 and 3 values? Have you seen similar questions? Did you know intuitively you could solve if you had a function touching the x axis? Did you try different things first?
Greetings from Turkey!🤍Love your videos, keep going!
Hi there. Love your math videos. I learn something new everyday.
It's not too difficult once you see the trick. And that's the problem.
Hello why Q(x) has double roots and why not complex roots ? Because Q(x)>=0. If Q(x) has no double roots, Q(x) can’t be >=0
Thanks for making a video about AYT and TYT ❤❤❤
This is mostly because ministers think making the questions harder instead of teaching stem subjects rather than increasing the islamist propaganda will make students learn better.
We really need to ditch religious bs and focus on a better education system. Watching you solve this question as an engineer made me realize the severity of this issue
I like how the topic changed from math to religion real fast
@@user-re2vi9rp6r believe me its true, turkeys gov trying their hardest to make society more ignorant and more obedient to them by using religious propaganda and the whole country is just going backwards to dark ages.
@user-re2vi9rp6r yeah, they aren't separated, unfortunately. That's the whole issue
Imamhatips (religious schools that literally brainswashes people) should be closed. Religion being a part of education is just dissappointing. Religion is not an obligation in education, a person can do their religious duties outside of the school. Discussing belief systems in schools should not be allowed as they dont allow politics either. Time to make all of my friends atheists.
@poumybeloved when I was in high school (about 10 years ago), I made all religion teachers life miserable for the lecture hour they had with my class.
Don't show mercy kardeş.
This has to be *HARD*. 9:52 video to solve this assuming you know what you are doing and it's only ONE question on the exam. How much time do you even get for the entire exam of 40 questions?
In this case of AYT 2021, we had to solve 40 questions of biology, chemistry and physics in addition to the 40 math questions, so 80 total. We had 3 hours without any break in between (this made it hard to focus on the exam after some time). So 2.5 minutes for each question.
I honestly didn’t think this problem was too hard, and I often have a hard time with the problems you post.
This was one of the easier questions in that exam, you can find the full exam if you search for 2021 AYT questions, but you'll need to translate.
Interesting analysis of the problem!
I must admit, that as a test problem for a university entrance test, it is in the tough end, because it relies on a trick.
Nevertheless I found the problem both easy and funny. Funny because it requires a trick, and easy because the trick was really easy to guess.
The information that P(x) >= x, P(1)=1 and P(3)=3, drew immediately the attention to the polynomium Q(x) = P(x) - x.
Knowing that Q(x) >= 0, it is clear that ALL roots of Q MUST be double roots, and the rest is easy.
If you are not familiar with the double root rule, here it follows an argument in a general context:
Theorem: Suppose P(x) is a polynomial of degree n, and suppose r is both a root and an extremum of P. Then r is at least a double root.
Proof: P(r) = 0, because r is a root of P, such that we may write P(x) = (x-r)*Q(x), where Q is a polynomial of degree n-1.
Take the derivative of P, such that P'(x) = Q(x) + (x-r) * Q'(x), and thus P'(r) = Q(r) + (r-r) * Q'(r) = Q(r). As an extremum we have 0 = P'(r) = Q(r), and therefore r is ALSO a root in Q. This way r is at least a double root.
Bouncing off of the x axis = double root? Never heard of
The tangent of Q at its zeros is also zero. Otherwiese Q would not be >=0. Therefore they are roots of order 2 at least.
I think this one is surprisingly easy if you see the trick. Just trying it in my head again, I think the answer is…
22?
I figured out that when you subtract x from P(X), it must be a perfect square for the minimum to be 0. Then, a quadratic R(x) with R^2(x) = P(x)-x would make it so that:
R(1) = R(3) = 0,
R(2) = sqrt(2).
Then you can solve it.
even his faster solution is not fast enough for that exam. See what kind of hell Turkish students go through
Living in here is enough to call it hell yk..
This problem was also in a seminar in Slovakia and I managed to solve it :)
now imagine the stress of all your future depending on it, it being surrounded by many others that are even worse than this, and also you just have 1.30 minutes
@@eda6654 yes ıf you want to become a doctor. First you will become a intel proccessor to solve this questions in 1.5 minutes.
Why cant one root be cubed and the other just grade one? Why the limits are infinity and not to zero?
The limits are (plus or minus) infinity because P(x) (and hence Q(x)) is a (fourth degree) _polynomial_ function. This results in that the limit of the quotient P(x)/(x^4) is finite but nonzero as x goes to +/- infinity.
x=1 and x=3 are roots of Q(x), because Q(1) = 0 and Q(3) = 0. So Q(x) = (x-1)(x-2)*S(x) for some second-degree polynomial function S(x).
Furthermore, we must have Q'(x) = 0 at x=1 and x=3 ; otherwise the graph of Q(x) is _passing through_ the x-axis at those points (rather than merely tangentially touching) which would mean Q(x) becomes negative in a neighbourhood of those points, which goes against the stipulated premise that Q(x) ≥ 0 . And since therefore Q'(1)=0 and Q'(3)=0 , it can be shown that we must also have S(1)=0 and S(3)=0 . (Just determine the derivative of Q(x) = (x-1)(x-3)*S(x) , and set it equal to 0 for x=1 and x=3 .) This means S(x) = K(x-1)(x-3), for some constant K, and hence Q(x) = K(x-1)(x-3)(x-1)(x-3) .
Because the 2 roots lie on the horizontal axis so they are double root. The limits are infinity because 4th degree polynomial's domain is R so we have to take the +- inf limit
@@NovaH00 "Because the 2 roots lie on the horizontal axis so they are a double root." -- That sentence makes no sense. Roots _always_ lie on the horizontal axis.
@@yurenchubut are double roots if they do not cross the axis.
yerenchu already gave a correct mathematical explanation.
Let me give a more 'geometrical' explanation.
Q(x) ≥ 0 for all x. I hope you agree with that. And if you also agree with the general graph of Q(x) as depicted, then you know that x=1 and x=3 are the only zeroes of Q(x).
Now suppose that we shift the graph of Q(x) a tiny little bit below. That means that we decrease the constant term of Q(x) by a tiny amount.
Then all of a sudden Q(x) has 4 real roots ('obvious' from the graph of Q(x)). If the shift is really tiny, I hope you agree they are clustered around x=1 and x=3. But still 4 distinct roots.
'Undoing' the shift, these 4 distinct roots will (eventually) again cluster around x=1 and x=3. Hence, the double roots.
Hope this makes sense.
(to make this rigurous, I would have to prove that the roots of a fourth order polynomial depend - in this case - 'continuously' on the constant factor of Q(x), but I hope you will forgive me in this case and trust me on the 'graphical view').
The limits for x → +/--∞ for Q(x) depend on the sign of the coefficient of x^4 of this polynomial. Since Q(x) ≥ 0, this means that the sign of the coefficient of the 4th power of Q(x) must be positive (can't be zero, since then Q(x) would then not be a fourth order polynomial).
Hope this helps....
Very good analysis!
Im sorry but im not shure there is only 1 possible answer in terms of specific coefficients: the polinomial interpolation of the points (1,1), (2,4), (3,3), (0,10) and (4,1) respect all conditions so 1 its a possible solution
You forgot the extra condition: P(x) >= x. Your last point (4,1) doesn’t satisfy that condition.
@@eroraf8637 u true, I wrote the wrong numbers: (1,1),(2,4),(3,3),(4,5),(0,1); this respect all conditions, included P(x)>=x
@@s.p.a.3583 Pretty sure that still doesn't work. While the extra condition is satisfied at the specified points, it needs to be satisfied EVERYWHERE. The set of points you give cannot be described by a polynomial of degree 4 that is everywhere greater than x. What polynomial did you find? Plot it and the line y=x. I guarantee they will have a non-tangential intersection.
I tried with any value for P(4)) >= 4, there are infinite solutions. The issue I see is that Persh as many others assume integer coefficients while doing the Q(x) function, while the problem accepts real coefficients, hence there are infinite solution by just setting P(4) to anything >= 4. Here it is for P(4)=5 : x^4-53*x^3/6+26*x*x-169*x/6+11
@@ikocheratcr Did you PLOT that function and LOOK at it? That function has a local minimum at P(3.444)=2.231, which violates the condition P(x)≥x, just like I said. You can't just check the values of the function at integer inputs. You need to satisfy P'(1)=P'(3)=1, which ensures that P(x) touches, but does not cross, the line y=x.
the fact that this is only a beginning...
This problem can be solved with basic knowledge about quadratic function and fundamental laws about polynomial factorization
derivative knowledge is not needed.
Q(x) = (x - 1)(x - 3)f(x) where f(x) is a quadratic function
since Q(x) >= 0 for all x, f(x) is negative in (1, 3) and positive in (-inf, 1)U(3, +inf)
therefore f(x) must be 0 at 1 and 3
therefore Q(x) = (x-1)(x-3)* a(x-1)(x-3)
Exactly my solution!
Hi, from Turkey. Our students have to solve these questions in order to enter university.
Its not that difficult, it’s just a polynomial function so it’s continous and always admits derivative, we have 5 coefficients to find and five equations: 3 equations from p(1)=1 p(3)=3 and p(2)=4 and 2 equations from p’(1)=p’(3)=1 because p(x)>=x (the function has to be above the function y=x so in (1,1) and (3,3) must be tangent to y=x).
=> P(x)-x has a double root at x=1, 3. Good one!
I know you westerners can't really relate to this but to us 3rd world dwellers this feels like a really harsh reality check. this is what we have to go through for a decent education in a less than decent country. every competent student in turkey has to sacrifice at least a year or two to their studies and still it's not guaranteed you will accomplish ANYTHING. we are just chained to the system our parents and our grandparents saw fit to us, to suffer throughout our teen years to have go to waste. it sucks man