China | Can you solve this ? | A Nice Math Olympiad Radical Simplification

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[@gustavosalcedo3092]

Greetings professor. Thank you for sharing your knowledge. You are truly a very intelligent man!

[@SaucyCinemaFilm]

Whew! Tour de force!! Buried within here is that (1 + √5)/2 is the number PHI, aka, “The Golden Ratio” which does reduce to x^2 - x = 1. And furthermore phi^n - phi^(n-1) = phi^(n-2)

[@thomashorine9198]

You can use polynomial division to write x^12 as q(x)(x^2-x-1) +r(x), where r(x) is at most linear. Then the value sought is r evaluated at phi.

[@cyruschang1904]

((1 + ✓5)/2)^2 = (3 + ✓5)/2 ((1 + ✓5)/2)^4 = ((3 + ✓5)/2)^2 = (7 + 3✓5)/2 ((1 + ✓5)/2)^6 = ((3 + ✓5)/2)((7 + 3✓5)/2) = (9 + 4✓5) ((1 + ✓5)/2)^12 = (9 + 4✓5)^2 = 161 + 72✓5

[@keizerantares]

The final answer is 144×PHI + 89 Reason is that you en with (PHI^4)^3 which gives you (3PHI + 2)^3 This yields: 27PHI^3 + 54PHI^2 + 36PHI + 8 Which gives in turn: 27(2PHI + 1) + 90PHI + 62 This gives: 54PHI + 27 + 90PHI + 62 And we get 144PHI + 89

[@MuhammadSharjeelAnsari-s8w]

Professor use complex method.

[@migueldia9771]

cool

[@NouraNor-h1x]

تبارك الله عليك

[@mathforallbyn.r6653]

❤❤❤ #naweenraaj

[@kambizkhosravi6138]

set it equal to x and take the Log.

[@NouraNor-h1x]

متابعة من المغرب

[@MathSync]

i ❤ Mathematics