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• 2006 Canadian MO Problems/Problem 2
  ...s on <math>A'M</math>, as desired. Reversing the argument proves the other direction for a locus, and we are done.
  2 KB (416 words) - 20:00, 21 September 2014
• 2007 USAMO Problems/Problem 2
  <cmath>a=b+c+d+2 \sqrt{bc+cd+db}</cmath> Now clearly, we have <math>b+c+d \le \frac 35</math>, and <math>bc+cd+db\le \frac 3{25}</math>.
  5 KB (754 words) - 03:41, 7 August 2014
• 2008 AMC 10B Problems/Problem 24
  ...g parallel lines <math>AB</math> and <math>DE</math> and transversal <math>DB</math>, we see that angle <math>BDE</math> is also <math>65^\circ</math>, a ...subtracted from the angle since it heads that amount off from the original direction.
  12 KB (1,944 words) - 18:31, 28 August 2024
• Spiral similarity
  ...\angle CBD = \angle ACE</math> and <math>k = \frac {|AF|}{|AB|} = \frac {|DB|}{|BC|} = \frac {|EC|}{|CA|}.</math> A vector has two parameters, modulo and direction. It is not tied to a center of the spiral similarity. Therefore
  28 KB (4,863 words) - 00:29, 16 December 2023
• 1952 AHSME Problems
  ...that angle <math> C = 90^{\circ} </math>,<math> \overline{AD} = \overline{DB} </math>,<math> DE \perp AB </math>, <math> \overline{AB} = 20 </math>, and ...me, would be together in <math>r</math> hours if they traveled in the same direction, but would pass each other in <math>t</math> hours if they traveled in oppo
  23 KB (3,556 words) - 15:35, 30 December 2023
• Symmetry
  ...>AI \implies BC'</math> is symmetrical to segment <math>DB' \implies BC' = DB'.</math> Symilarly, <cmath>BA' = EB' \implies EB' = DB'. \blacksquare.</cmath>
  19 KB (3,291 words) - 13:44, 6 October 2024
• 2020 AIME I Problems
  ...nd <math>B</math> on side <math>\overline{AB}</math> such that <math>AE=ED=DB=BC.</math> The degree measure of <math>\angle ABC</math> is <math>\tfrac{m} ...ates <math>60^\circ</math> counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close t
  7 KB (1,257 words) - 17:51, 5 February 2022
• 2024 AIME II Problems/Problem 12
  b(1 - \frac{x}{a}) = (b + db)(1 - \frac{x}{a + da}) ...>a^2 + b^2 = 1^2</math> by the Pythagorean theorem. Solving this for <math>db</math>, we get
  17 KB (2,843 words) - 08:26, 11 September 2024