Math Olympiad Question | Can you Solve this Problem ? | Golden Ratio

[@raghvendrasingh1289]

x= (1+√5)/2
x^3 = (1+3√5 + 3× 5 +5√5)/8
= 2+√5
x^6 = 9+4√5
x^12 = 81+80+72√5 = 161+72√5

[@giuseppemalaguti435]

x^2-x-1=0..x^2=x+1...x^6=x^3+3x^2+3x+1=x(x^2+3)+3x^2+1=...8x+5...x^12=(8x+5)^2=64(x+1)+25+80x=89+144x=89+72(1+√5)=161+72√5

[@RealQinnMalloryu4]

{1x+1x ➖ }+{5x+5x ➖ }={2x^2+10x^2}/2=12x^4/2=6x^4 2^3x^2^2 1^3x1^2 3x^2 (x ➖ 3x+2). x^2^6 x^2^2^3x^1^2^1 x^2^1(x ➖ 2x+1).