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• The Golden Ratio or phi
  #redirect [[Phi]]
  17 bytes (2 words) - 13:58, 1 January 2024

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• Phi
  ...ase, either <math>\phi</math> or <math>\varphi</math>; capitalized, <math>\Phi</math>) is the 21st letter in the Greek alphabet. It is used frequently in <math>\phi</math> appears in a variety of different mathematical contexts: it is the l
  2 KB (302 words) - 14:04, 1 January 2024
• 1988 AIME Problems/Problem 13
  ...<math>x^2-x-1</math> are <math>\phi</math> (the Golden Ratio) and <math>1-\phi</math>. These two must also be roots of <math>ax^{17}+bx^{16}+1</math>. Thu a\phi^{17}+b\phi^{16}+1=0, \\
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1997 AIME Problems/Problem 9
  ...tely guess-able that <math>a = \phi = \frac{1+\sqrt{5}}2</math> (the [[phi|golden ratio]]) is the answer. The following is the way to derive that: ...have also used other properties of <math>\phi</math> like <math>\phi^3 = 2\phi + 1</math>.
  4 KB (586 words) - 21:53, 30 December 2023
• Ratio
  *[[Phi | The golden ratio]]
  666 bytes (100 words) - 20:25, 29 December 2023
• 2009 PUMaC Problems
  ...<math>k</math> such that <math>\phi ( \sigma ( 2^k)) = 2^k</math>. (<math>\phi(n)</math> denotes the number of positive integers that are smaller than <ma ...h>\phi_d</math>, where <math>\frac1{\phi_d} = \phi_d - d</math> (so <math>\phi = \phi_1</math>). Given that <math>\phi_{2009} = \frac{a + \sqrt{b}}{c}</m
  25 KB (4,154 words) - 16:27, 2 September 2011
• 2021 Fall AMC 12A Problems/Problem 13
  The value of <math>k</math> is known as the <b>Golden Ratio</b>: <math>\phi=\frac{1+\sqrt{5}}{2}\approx 1.61803398875.</math> ...hi has many properties and is related to the [[Fibonacci sequence]]. See [[Phi]].
  16 KB (2,526 words) - 00:53, 6 May 2023
• 2023 AMC 12B Problems/Problem 25
  \cos{36^\circ} &= \dfrac{a_s}{r_s} = \dfrac{\phi}{2} = \dfrac{\sqrt{5}+1}{4}\\ &=\left(\dfrac{a_s}{\dfrac{\phi}{2} r_b}\right)^2 \left(1+\sqrt{5}\right)\\
  19 KB (2,967 words) - 16:56, 24 February 2024