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Fun with Math Olympiad
India
เข้าร่วมเมื่อ 5 มี.ค. 2012
Channel for cracking Math problems for Olympiad.
Olympiad Engineering Entrance IOQM Part 10
𝑇𝑤𝑜 𝑐𝑖𝑟𝑐𝑙𝑒𝑠, 𝑆1 𝑎𝑛𝑑 𝑆2, 𝑜𝑓 𝑟𝑎𝑑𝑖𝑖 6 𝑢𝑛𝑖𝑡𝑠 𝑎𝑛𝑑 3 𝑢𝑛𝑖𝑡𝑠 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦, 𝑎𝑟𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟, 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙𝑙𝑦. 𝐿𝑒𝑡 𝐴𝐶 𝑎𝑛𝑑 𝐵𝐷 𝑏𝑒 𝑡ℎ𝑒𝑖𝑟 𝑑𝑖𝑟𝑒𝑐𝑡 𝑐𝑜𝑚𝑚𝑜𝑛 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑠 𝑤𝑖𝑡ℎ 𝐴 𝑎𝑛𝑑 𝐵 𝑜𝑛 𝑆1, 𝑎𝑛𝑑 𝐶 𝑎𝑛𝑑 𝐷 𝑜𝑛 𝑆2. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑞𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝐵𝐷𝐶 𝑡𝑜 𝑡ℎ𝑒 𝑛𝑒𝑎𝑟𝑒𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟. [𝐼𝑂𝑄𝑀]
𝐿𝑒𝑡 𝐴𝐵𝐶 𝑏𝑒 𝑎𝑛 𝑒𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑤𝑖𝑡ℎ 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 10. 𝐴 𝑠𝑞𝑢𝑎𝑟𝑒 𝑃𝑄𝑅𝑆 𝑖𝑠 𝑖𝑛𝑠𝑐𝑟𝑖𝑏𝑒𝑑 𝑖𝑛 𝑖𝑡, 𝑤𝑖𝑡ℎ 𝑃 𝑜𝑛 𝐴𝐵. 𝑄, 𝑅 𝑜𝑛 𝐵𝐶 𝑎𝑛𝑑 𝑆 𝑜𝑛 𝐴𝐶. 𝐼𝑓 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑃𝑄𝑅𝑆 𝑖𝑠 𝑚 + 𝑛√𝑘 𝑤ℎ𝑒𝑟𝑒 𝑚, 𝑛 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎𝑛𝑑 𝑘 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑒𝑛 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 √((𝑚+𝑛)/𝑘^2 ) [𝑰𝑶𝑸𝑴]
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝐵𝑖𝑠𝑒𝑐𝑡𝑜𝑟𝑠 𝐴𝐼 𝑎𝑛𝑑 𝐶𝐼 𝑚𝑒𝑒𝑡 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑢𝑚𝑐𝑖𝑟𝑐𝑙𝑒 𝑜𝑓 ∆𝐴𝐵𝐶 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡𝑠 𝐴1, 𝐶1 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦. 𝑇ℎ𝑒 𝑐𝑖𝑟𝑐𝑢𝑚𝑐𝑖𝑟𝑐𝑙𝑒 𝑜𝑓 ∆𝐴𝐼𝐶1 𝑚𝑒𝑒𝑡𝑠 𝐴𝐵 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝐶0; 𝑝𝑜𝑖𝑛𝑡 𝐴0 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦. 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝐴0, 𝐴1, 𝐶0, 𝐶1 𝑎𝑟𝑒 𝑐𝑜𝑙𝑖𝑛𝑒𝑎𝑟.
𝐴 𝑠𝑒𝑚𝑖𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑝𝑎𝑝𝑒𝑟 𝑖𝑠 𝑓𝑜𝑙𝑑𝑒𝑑 𝑎𝑙𝑜𝑛𝑔 𝑎 𝑐ℎ𝑜𝑟𝑑 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑓𝑜𝑙𝑑𝑒𝑑 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑎𝑟𝑐 𝑖𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑚𝑖𝑐𝑖𝑟𝑐𝑙𝑒. 𝑇ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑚𝑖𝑐𝑖𝑟𝑐𝑙𝑒 𝑖𝑠 4 𝑢𝑛𝑖𝑡𝑠 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑡𝑎𝑛𝑔𝑒𝑛𝑐𝑦 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑡ℎ𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜 7:1. 𝐼𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑟𝑒𝑎𝑠𝑒 (𝑡ℎ𝑒 𝑑𝑜𝑡𝑡𝑒𝑑 𝑙𝑖𝑛𝑒 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑔𝑢𝑟𝑒)𝑖𝑠 𝑙 𝑡ℎ𝑒𝑛 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑙^2. [𝐼𝑂𝑄𝑀 𝐾𝑉]
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑑 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 ∆ 𝐴𝐵𝐶 𝑤ℎ𝑜𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝐴𝐶 𝑖𝑠 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ 1. 𝑇ℎ𝑒 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 ∠𝐴𝐶𝐵 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑠 𝐴𝐵 𝑎𝑡 𝐷. 𝐼𝑓 𝐵𝐶 𝑖𝑠 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ 𝑥, 𝑡ℎ𝑒𝑛 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐶𝐷?
𝐹𝑜𝑟 𝑠𝑜𝑚𝑒+𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑛, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 110𝑛^3 ℎ𝑎𝑠 110 𝑑𝑖𝑣𝑖𝑠𝑜𝑟𝑠 𝑖𝑛𝑐𝑙𝑢𝑑𝑖𝑛𝑔 1 𝑎𝑛𝑑 𝑖𝑡𝑠𝑒𝑙𝑓. 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦+𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑑𝑖𝑣𝑖𝑠𝑜𝑟𝑠 𝑑𝑜𝑒𝑠 81𝑛 ℎ𝑎𝑣𝑒?
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑓𝑜𝑟𝑚𝑒𝑑 𝑏𝑦 𝑟𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑖𝑔𝑖𝑡𝑠 𝑜𝑓 234578 𝑎𝑟𝑒 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 55?
𝑂𝑢𝑡 𝑜𝑓 (2𝑛+1) 𝑡𝑖𝑐𝑘𝑒𝑡𝑠 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒𝑙𝑦 𝑛𝑢𝑚𝑏𝑒𝑟𝑒𝑑, 𝑡ℎ𝑟𝑒𝑒 𝑎𝑟𝑒 𝑑𝑟𝑎𝑤𝑛 𝑎𝑡 𝑟𝑎𝑛𝑑𝑜𝑚.
𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑐𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑛 𝑡ℎ𝑒𝑚 𝑎𝑟𝑒 𝑖𝑛 𝐴𝑃.
[𝐼𝑆𝐼𝐴𝑇] 𝐼𝑓 𝑎0 =1/2 𝑎𝑛𝑑 𝑎𝑛 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑒𝑙𝑦 𝑎𝑠 𝑎𝑛 =√((1+𝑎𝑛−1)/2), 𝑛 greater than 1. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑓𝑜𝑟 𝑛 greater than 0, 𝑎𝑛 =cos(𝑥𝑛)𝑓𝑜𝑟 0 less than 𝑥𝑛 less than 90
𝑆 = (𝜃 sin(𝜋𝜃/((1+𝜃) )),1/𝜃 cos(𝜋𝜃/(1+𝜃))) : 𝜃∈𝑅, 𝜃 greater than 0 𝑎𝑛𝑑 𝑇=(𝑥,𝑦):𝑥,𝑦∈𝑅,𝑥𝑦=1/2 How many elements does S∩𝑇 ℎ𝑎𝑣𝑒?
𝐿𝑒𝑡 𝐴𝐵𝐶 𝑏𝑒 𝑎𝑛 𝑒𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑤𝑖𝑡ℎ 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 10. 𝐴 𝑠𝑞𝑢𝑎𝑟𝑒 𝑃𝑄𝑅𝑆 𝑖𝑠 𝑖𝑛𝑠𝑐𝑟𝑖𝑏𝑒𝑑 𝑖𝑛 𝑖𝑡, 𝑤𝑖𝑡ℎ 𝑃 𝑜𝑛 𝐴𝐵. 𝑄, 𝑅 𝑜𝑛 𝐵𝐶 𝑎𝑛𝑑 𝑆 𝑜𝑛 𝐴𝐶. 𝐼𝑓 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑃𝑄𝑅𝑆 𝑖𝑠 𝑚 + 𝑛√𝑘 𝑤ℎ𝑒𝑟𝑒 𝑚, 𝑛 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎𝑛𝑑 𝑘 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑒𝑛 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 √((𝑚+𝑛)/𝑘^2 ) [𝑰𝑶𝑸𝑴]
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝐵𝑖𝑠𝑒𝑐𝑡𝑜𝑟𝑠 𝐴𝐼 𝑎𝑛𝑑 𝐶𝐼 𝑚𝑒𝑒𝑡 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑢𝑚𝑐𝑖𝑟𝑐𝑙𝑒 𝑜𝑓 ∆𝐴𝐵𝐶 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡𝑠 𝐴1, 𝐶1 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦. 𝑇ℎ𝑒 𝑐𝑖𝑟𝑐𝑢𝑚𝑐𝑖𝑟𝑐𝑙𝑒 𝑜𝑓 ∆𝐴𝐼𝐶1 𝑚𝑒𝑒𝑡𝑠 𝐴𝐵 𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝐶0; 𝑝𝑜𝑖𝑛𝑡 𝐴0 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦. 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝐴0, 𝐴1, 𝐶0, 𝐶1 𝑎𝑟𝑒 𝑐𝑜𝑙𝑖𝑛𝑒𝑎𝑟.
𝐴 𝑠𝑒𝑚𝑖𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑝𝑎𝑝𝑒𝑟 𝑖𝑠 𝑓𝑜𝑙𝑑𝑒𝑑 𝑎𝑙𝑜𝑛𝑔 𝑎 𝑐ℎ𝑜𝑟𝑑 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑓𝑜𝑙𝑑𝑒𝑑 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑎𝑟𝑐 𝑖𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑚𝑖𝑐𝑖𝑟𝑐𝑙𝑒. 𝑇ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑚𝑖𝑐𝑖𝑟𝑐𝑙𝑒 𝑖𝑠 4 𝑢𝑛𝑖𝑡𝑠 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑡𝑎𝑛𝑔𝑒𝑛𝑐𝑦 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑡ℎ𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑟𝑎𝑡𝑖𝑜 7:1. 𝐼𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑟𝑒𝑎𝑠𝑒 (𝑡ℎ𝑒 𝑑𝑜𝑡𝑡𝑒𝑑 𝑙𝑖𝑛𝑒 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑔𝑢𝑟𝑒)𝑖𝑠 𝑙 𝑡ℎ𝑒𝑛 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑙^2. [𝐼𝑂𝑄𝑀 𝐾𝑉]
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑑 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 ∆ 𝐴𝐵𝐶 𝑤ℎ𝑜𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝐴𝐶 𝑖𝑠 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ 1. 𝑇ℎ𝑒 𝑏𝑖𝑠𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 ∠𝐴𝐶𝐵 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑠 𝐴𝐵 𝑎𝑡 𝐷. 𝐼𝑓 𝐵𝐶 𝑖𝑠 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ 𝑥, 𝑡ℎ𝑒𝑛 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐶𝐷?
𝐹𝑜𝑟 𝑠𝑜𝑚𝑒+𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑛, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 110𝑛^3 ℎ𝑎𝑠 110 𝑑𝑖𝑣𝑖𝑠𝑜𝑟𝑠 𝑖𝑛𝑐𝑙𝑢𝑑𝑖𝑛𝑔 1 𝑎𝑛𝑑 𝑖𝑡𝑠𝑒𝑙𝑓. 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦+𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑑𝑖𝑣𝑖𝑠𝑜𝑟𝑠 𝑑𝑜𝑒𝑠 81𝑛 ℎ𝑎𝑣𝑒?
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑓𝑜𝑟𝑚𝑒𝑑 𝑏𝑦 𝑟𝑒𝑎𝑟𝑟𝑎𝑛𝑔𝑖𝑛𝑔 𝑡ℎ𝑒 𝑑𝑖𝑔𝑖𝑡𝑠 𝑜𝑓 234578 𝑎𝑟𝑒 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 55?
𝑂𝑢𝑡 𝑜𝑓 (2𝑛+1) 𝑡𝑖𝑐𝑘𝑒𝑡𝑠 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒𝑙𝑦 𝑛𝑢𝑚𝑏𝑒𝑟𝑒𝑑, 𝑡ℎ𝑟𝑒𝑒 𝑎𝑟𝑒 𝑑𝑟𝑎𝑤𝑛 𝑎𝑡 𝑟𝑎𝑛𝑑𝑜𝑚.
𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑐𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑛 𝑡ℎ𝑒𝑚 𝑎𝑟𝑒 𝑖𝑛 𝐴𝑃.
[𝐼𝑆𝐼𝐴𝑇] 𝐼𝑓 𝑎0 =1/2 𝑎𝑛𝑑 𝑎𝑛 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑒𝑙𝑦 𝑎𝑠 𝑎𝑛 =√((1+𝑎𝑛−1)/2), 𝑛 greater than 1. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑓𝑜𝑟 𝑛 greater than 0, 𝑎𝑛 =cos(𝑥𝑛)𝑓𝑜𝑟 0 less than 𝑥𝑛 less than 90
𝑆 = (𝜃 sin(𝜋𝜃/((1+𝜃) )),1/𝜃 cos(𝜋𝜃/(1+𝜃))) : 𝜃∈𝑅, 𝜃 greater than 0 𝑎𝑛𝑑 𝑇=(𝑥,𝑦):𝑥,𝑦∈𝑅,𝑥𝑦=1/2 How many elements does S∩𝑇 ℎ𝑎𝑣𝑒?
มุมมอง: 40
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Thanks I’m your number one fan
Thank you very much.
you made the best video😍😍
@@sovannarotnun7561 Thank you very much. These words helps a lot to make more such videos.
Speak in hindi
Thank you and sorry.. I can not speak that good Hindi to explain these complex problems. I may go wrong.
@@psk99999 oh!! But explanation in English is not up to the mark needs to be improved and keep it up
@@SagnikBiswas_69 Sure.. I will try to improve.. Thanks
Good explanation sir
Thank you for your kind words. We just started and we are trying to improve. We want more kids to take part in Mathematics Olympiad. We hope this will reach to kids who are really interested in excelling.
Thank you sir, you make tough questions look like a piece of cake.
You are most welcome. We are trying to help students and create interest in Olympiad. It's a great test.
You are most welcome. We are trying to help students and create interest in Olympiad. It's a great test.
Please note correction for the 7th Question. 4C2 = 3! = 6. But I mistakenly wrote as 3!/2. Correct answer below. Question: 𝑈𝑛𝑐𝑜𝑛𝑣𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 𝑑𝑖𝑐𝑒 𝑎𝑟𝑒 𝑡𝑜 𝑏𝑒 𝑑𝑒𝑠𝑖𝑔𝑛𝑒𝑑 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑠𝑖𝑥 𝑓𝑎𝑐𝑒𝑠 𝑎𝑟𝑒 𝑚𝑎𝑟𝑘𝑒𝑑 𝑤𝑖𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑓𝑟𝑜𝑚 1 𝑡𝑜 6 𝑤𝑖𝑡ℎ 1 𝑎𝑛𝑑 2 𝑎𝑝𝑝𝑒𝑎𝑟𝑖𝑛𝑔 𝑜𝑛 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑓𝑎𝑐𝑒𝑠. 𝐹𝑢𝑟𝑡ℎ𝑒𝑟, 𝑒𝑎𝑐ℎ 𝑓𝑎𝑐𝑒 𝑖𝑠 𝑐𝑜𝑙𝑜𝑟𝑒𝑑 𝑒𝑖𝑡ℎ𝑒𝑟 𝑟𝑒𝑑 𝑜𝑟 𝑦𝑒𝑙𝑙𝑜𝑤 𝑤𝑖𝑡ℎ 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑓𝑎𝑐𝑒𝑠 𝑎𝑙𝑤𝑎𝑦𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑐𝑜𝑙𝑜𝑟. 𝑇𝑤𝑜 𝑑𝑖𝑐𝑒 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒𝑑 𝑡𝑜 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑑𝑒𝑠𝑖𝑔𝑛 𝑖𝑓 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑚 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑜𝑡𝑎𝑡𝑒𝑑 𝑡𝑜 𝑜𝑏𝑡𝑎𝑖𝑛 𝑎 𝑑𝑖𝑐𝑒 𝑡ℎ𝑎𝑡 ℎ𝑎𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑛𝑑 𝑐𝑜𝑙𝑜𝑟𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑓𝑎𝑐𝑒𝑠 𝑎𝑠 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑜𝑛𝑒. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑖𝑠𝑡𝑖𝑛𝑐𝑡 𝑑𝑖𝑐𝑒 𝑡ℎ𝑎𝑡 𝑐𝑎𝑛 𝑏𝑒 𝑑𝑒𝑠𝑖𝑔𝑛𝑒𝑑. Correct Answer: 1, 2 𝑎𝑟𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑐𝑒. 3, 4, 5, 6 𝑐𝑎𝑛 𝑏𝑒 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑑 𝑖𝑛 𝑎 𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟 𝑓𝑎𝑠ℎ𝑖𝑜𝑛 𝑖𝑛 4𝐶2 𝑤𝑎𝑦𝑠 𝑜𝑟 3!=6 𝐴𝑛𝑦 𝑜𝑓 𝑡ℎ𝑒 2 𝑐𝑜𝑙𝑜𝑟𝑠 𝑐𝑎𝑛 𝑏𝑒 𝑝𝑎𝑖𝑛𝑡𝑒𝑑 𝑜𝑛 𝑜𝑓 𝑑𝑖𝑐𝑒=2 𝑥 2 𝑥 2 (3 𝑜𝑝𝑝𝑜 𝑠𝑖𝑑𝑒𝑠 𝑎𝑛𝑑 2 𝑐𝑜𝑙𝑜𝑟𝑠 𝑒𝑎𝑐ℎ) 𝑫𝒊𝒔𝒕𝒊𝒏𝒄𝒕 𝒅𝒊𝒄𝒆 𝒘𝒉𝒊𝒄𝒉 𝒄𝒂𝒏 𝒃𝒆 𝒅𝒆𝒔𝒊𝒈𝒏𝒆𝒅 =𝟔𝒙𝟐𝒙𝟐𝒙𝟐=𝟒𝟖
To complete the problem, you need to find sum of squares of digits of N. Who will give me final answer?
Please refer to the full video for the detailed explanation. th-cam.com/video/RU6HysQsEwA/w-d-xo.html
It would be helpful if you mention the class for which the video is intended
Hi, IMO test is common for class 8 to class 10. The cut-off marks are decided by IMO depending on the paper's difficulty level. Ideally, these are for 8th to 10th grades.
Please refer to the full video for a detailed explanation. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for the detailed explanation. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for a detailed explanation. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for a detailed explanation. th-cam.com/video/M7iK6LpnERs/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/OIBefssjOwY/w-d-xo.htmlsi=GOauEXYMVm5e6wIB
Please refer to the full video for the detailed explanation. th-cam.com/video/OIBefssjOwY/w-d-xo.htmlsi=GOauEXYMVm5e6wIB
Math-62 Please refer to the full video for detailed explaination. th-cam.com/video/j4HfVFlNqPg/w-d-xo.html
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Please refer to the full video for detailed explaination. th-cam.com/video/nPLqt2am_qE/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/3F5zdCyvZvY/w-d-xo.html
Please refer to the full video for detailed explaination. th-cam.com/video/Q1QKC5nT9Vw/w-d-xo.htmlsi=_W8hRthCK7qcZGDo
Suprise to see first question in this video. I have solved and explained this problem exactly in my previous video sometime back. Check the following video. The problem is exactly same, even the numericals. th-cam.com/video/t69M_J-fuRc/w-d-xo.htmlsi=9BC3os6OzxjApLIr The question is: 𝐴 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑚 ℎ𝑎𝑠 𝑡ℎ𝑒 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦 𝑡ℎ𝑎𝑡 𝑚^2 𝑖𝑠 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑙𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 4𝑛^2−5𝑛+16 𝑤ℎ𝑒𝑟𝑒 𝑛 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 (𝑜𝑓 𝑎𝑛𝑦 𝑠𝑖𝑔𝑛). 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 |𝑚−𝑛|.
For the Last Question (Binary sequence) The correct recurring equation is : F(n+1) = F(n) + F(n-2) + F(n-3). Please use this for calculation. The calculation is done based on this equation. Not the one mentioned in the video. All the values are correct. Just wrote the recurring equation wrong.
For problem "For a positive integer n let n denote the perfect square integer closest to n. For example, (74) = 81, (18) = 16 . If N is the smallest positive integer such that (91). (120) . (143). (180). (N) = 91.120.143 .180 .N Find the sum of the square of the digits of N." The final answer is 4^2 + 6^2 + 2^2 = 56
Thanks for completing
Wow
Thanks
Iam 45th subsciber
Thanks
Not that it really affects the solution, but under the given conditions point D lies outside of the circle.
Thanks Andy. Yes, I did not check that. But as you said, solution is same.
Awesome, simplified like anything, Awesome..!
Thank you!
Wow Awesome...!
Many many thanks
There are a few typos in the 3rd problem. Digit 7 came wrongly in a few places and -2 is missing in the final solution set. Listen to the explanation, that is correct.
Nice presentation
Thanks a lot
Congruence relation in number theory should be explained lucidly
Sure... I will make a video shortly explaining that.. Give me sometime
Please check the video on Congruence basics. th-cam.com/video/b3v-W-r6NnY/w-d-xo.html Also check all problems on play list
Wow. That's a nice , simple and good solution
Thank you. If you have any interesting problems to solve, please put in the comments. I will make video.
I want to make reminder theorem problem-solving videos. Please post in the comment section if you want to solve any reminder theorem problems till the 10th standard.
Planning to make reminder theorem problem-solving videos. Please post in the comment section, if you want to solve any reminder theorem problems till the 10th standard.