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• 2011 USAJMO Problems/Problem 3
  ...th> is on the circumcircle of <math>\Delta XYZ</math>, and similarly <math>I</math> is on the circumcircle of <math>\Delta XYZ</math>. Therefore <math>\ ...th>HM=HB/2=HZ/2</math>. Therefore <math>\Delta HMZ</math> is a <math>30-60-90</math> right triangle and so <math>\angle ZHB=60^{\circ}</math>. So <math>\
  15 KB (2,593 words) - 13:37, 29 January 2021
• 1985 AHSME Problems
  if the sum of the remaining terms is to equal <math>1</math>? int i,j;
  17 KB (2,488 words) - 03:26, 20 March 2024
• 2012 AMC 12B Problems
  '''Note:''' 1 foot is equal to 12 inches. ...the first <math>10</math> positive integers such that for each <math>2 \le i \le 10</math> either <math>a_i+1</math> or <math>a_i-1</math> or both appea
  20 KB (2,681 words) - 09:47, 29 June 2023
• 2012 AIME I Problems
  {{AIME Problems|year=2012|n=I}} [[2012 AIME I Problems/Problem 1|Solution]]
  10 KB (1,617 words) - 14:49, 2 June 2023
• 1991 AHSME Problems
  ...<math>t=0</math> to time <math>t=1</math> a population increased by <math>i\%</math>, and from time <math>t=1</math> to time <math>t=2</math> the popul <math>\textbf{(A) } (i+j)\% \qquad
  16 KB (2,451 words) - 04:27, 6 September 2021
• 2014 AIME I Problems
  {{AIME Problems|year=2014|n=I}} path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,
  9 KB (1,472 words) - 13:59, 30 November 2021
• 1973 AHSME Problems
  Two congruent 30-60-90 are placed so that they overlap partly and their hypotenuses coincide. If t Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean),
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1987 AHSME Problems
  have a solution <math>(x, y)</math> inside Quadrant I if and only if \textbf{(E)}\ 90\pi </math>
  16 KB (2,291 words) - 13:45, 19 February 2020
• 1983 AHSME Problems
  ...angle at <math>C</math>. If <math>\sin A = \frac{2}{3}</math>, then <math>\tan B</math> is If <math>\tan{\alpha}</math> and <math>\tan{\beta}</math> are the roots of <math>x^2 - px + q = 0</math>, and <math>\co
  15 KB (2,309 words) - 23:43, 2 December 2021
• 1978 AHSME Problems
  ...then <math>(a+ b+ c-d) + (a + b- c+ d) +(a-b+ c+d)+ (-a+ b+c+d)</math> is equal to ...etween <math>\mathit{A}</math> and <math>\mathit{B}</math> is less than or equal to the distance between <math>\mathit{A}</math>
  15 KB (2,432 words) - 01:06, 22 February 2024
• 2007 IMO Problems/Problem 4
  pair I=incenter(A,B,C); ...90+\dfrac{C}{2}</math>, and similarly <math>\angle{RPK} = 90+\angle{PCK} = 90+\dfrac{C}{2}</math>. Therefore, <math>\angle{RQL} = \angle{RPK}</math>. Usi
  8 KB (1,480 words) - 14:52, 5 August 2022
• 2015 AIME II Problems/Problem 15
  ...the areas of <math>\triangle DBA</math> and <math>\triangle ACE</math> are equal. This common area is <math>\frac{m}{n}</math>, where <math>m</math> and <ma ...midpoint of <math>\overline{BC},</math> implying that <math>\angle{BAC} = 90.</math> Now draw <math>\overline{PA}, \overline{PB}, \overline{PM},</math>
  31 KB (5,086 words) - 19:15, 20 December 2023
• 2016 AIME I Problems/Problem 9
  ...os(m)\cos(n)=\frac{1}{2}(\cos(m+n)+\cos(m-n))</math>. <math>\cos(m+n)=\sin(90-m-n)=\sin(BAC)=\frac{1}{5}</math>. The maximum possible value of <math>\cos ...at <math>\sin \alpha = \frac{1}{5}</math>. Since <math>\sin \alpha = \cos (90- \alpha)</math>,
  9 KB (1,526 words) - 02:31, 29 December 2021
• 2016 AIME II Problems/Problem 14
  ...and <math>KQ=y</math>, assuming WLOG <math>x>y</math>, we must have <math>\tan(120^{\circ})=-\sqrt{3}=\dfrac{\dfrac{x+y}{100 \sqrt{3}}}{\dfrac{30000-xy}{3 ...qrt{33}}{2}+\frac{15\pm\sqrt{33}}{7\pm\sqrt{33}} &= 2d\end{align*}</cmath> I'll use <math>a=\frac{9+\sqrt{33}}{2}</math> because both values should give
  15 KB (2,560 words) - 01:44, 1 July 2023
• 2020 AMC 12A Problems
  for (int i = 0; i < 3; ++i) { pair A = (j,i);
  14 KB (2,073 words) - 15:15, 21 October 2021
• 2018 AIME I Problems/Problem 13
  ...rac A2 \sin\frac B2\sin\frac C2,</cmath> with equality when <math>\alpha = 90^\circ</math>, that is, when <math>X</math> is the foot of the perpendicular ..., and that occurs when <math>\alpha=45^{\circ}</math>, or <math>\angle AXB=90^{\circ}</math>. Ergo, <math>X</math> is the foot of the altitude from <math
  13 KB (2,200 words) - 21:36, 6 January 2024
• 2006 iTest Problems/Problem U6
  The smallest possible value of <math>\frac{y}{x}</math> is equal to <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are rel ...r(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs);
  4 KB (722 words) - 20:53, 27 March 2019
• 2019 AIME II Problems
  Call a positive integer <math>n</math> <math>k</math>-<i>pretty</i> if <math>n</math> has exactly <math>k</math> positive divisors and <math>n ...th> such that for nonnegative integers <math>n,</math> the value of <math>\tan(2^n\theta)</math> is positive when <math>n</math> is a multiple of <math>3<
  7 KB (1,254 words) - 14:45, 21 August 2023
• 2019 AIME II Problems/Problem 1
  label("$9$",(A+B)/2,dir(-90)); label("$6$",(A+E)/2,dir(-90));
  11 KB (1,794 words) - 15:32, 14 January 2024
• 2020 AIME I Problems/Problem 13
  label("$I$", (-3.7912668488110084,8.257242529248508), NE * labelscalefactor); pair A, B, C, D, E, F, I, P, MA, MB, MC;
  35 KB (5,215 words) - 23:08, 29 October 2023

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