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• 2013 AIME II Problems/Problem 4
  ...here we require a clockwise rotation, so we multiply by <math>e^{-\frac{i\pi}{3}}</math> to obtain C. Upon averaging the coordinates of A, B, and C, we We construct point <math>C</math> by drawing two circles with radius <math>r = AB = \sqrt{13}</math>. One circle will be ce
  8 KB (1,268 words) - 14:10, 31 January 2024
• 2013 AIME II Problems/Problem 8
  A=expi(pi); B=expi(pi-a); C=expi(a); D=expi(0); E=expi(-a); F=expi(pi+a); ...342x^2-2420x=0</cmath>, it is easy to see that <math>x=-10, x=0</math> are two solutions for the equation, so we can factorize that into <cmath>x(x+10)(x^
  8 KB (1,357 words) - 09:23, 11 March 2024
• 2003 AMC 12B Problems/Problem 10
  ...de by attaching two equilateral triangles to the regular pentagon ABCDE in two of the five positions shown. How many non-congruent figures can be construc real r = 5/dir(54).x, h = 5 tan(54*pi/180);
  1 KB (237 words) - 23:06, 3 February 2020
• 2014 AIME II Problems/Problem 7
  Let <math>f(x)=(x^2+3x+2)^{\cos(\pi x)}</math>. Find the sum of all positive integers <math>n</math> for which First, let's split it into two cases to get rid of the absolute value sign
  3 KB (436 words) - 02:10, 12 December 2023
• 2014 AIME II Problems/Problem 10
  ...ted to put <math>z</math> in any other location.) Then there are precisely two possible distinct locations for <math>w</math>; one is obtained by going 12 Let the two possible locations for <math>w</math> be <math>W_1</math> and <math>W_2</ma
  6 KB (1,045 words) - 13:08, 21 January 2024
• Geometry Solutions
  We see that both the radii are the two shorter sides of the triangle, making this a isosceles 45-45-90 triangle. We also see that the height that we drew is half the hypotenuse(note the two smaller 45-45-90 isosceles triangles).
  23 KB (3,182 words) - 12:30, 5 April 2014
• Symmetry
  <cmath>P = AI \cap \Omega \implies PB = PC = PI \implies</cmath> It is known that the composition of two axial symmetries with non-parallel axes is a rotation centered at
  19 KB (3,292 words) - 13:04, 13 May 2024
• ASIA TEAM Problems
  Kelvin the frog creates an infinite sequence of rational numbers. He chooses two starting terms <math>a_1=\tfrac{n_1}{d_1}</math> and <math>a_2=\tfrac{n_2}{ Kelvin the frog wants to find all positive integers <math>n</math> not over <math>9000</math> such that the order of <math>n \pmod{1000}</math> is <mat
  10 KB (1,710 words) - 23:23, 10 January 2020
• 2016 AMC 10A Problems
  ...thmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is <math>1</math> foot wide on all four sides. What is the l ...lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie h
  13 KB (1,979 words) - 11:34, 27 June 2024
• 1967 IMO Problems/Problem 4
  Let <math>A_0B_0C_0</math> and <math>A_1B_1C_1</math> be any two acute-angled triangles. Consider all triangles <math>ABC</math> that are si ...nt <math>P</math> inside <math>A_0B_0C_0</math> s.t. <math>\angle X_0PY_0=\pi-\angle X_1Z_1Y_1</math>, where <math>X,Y,Z</math> are a permutation of <mat
  2 KB (368 words) - 13:15, 29 January 2021
• 1971 IMO Problems/Problem 5
  ...2,\ldots, w_m</math>. we claim that for some choice of <math>0\leq\theta<2\pi</math>, <math>v_i+e^{i\theta}w_j</math> will do the job(a suitable rotation ...y set and adding vectors to it, one by one. We just need to make sure that two thing are respected:
  4 KB (749 words) - 14:09, 29 January 2021
• 1972 IMO Problems/Problem 6
  ...is the <math>3</math>-dimensional version of this), and then complete the two regular tetrahedra <math>A_1A_2A_3P_1,A_1A_2A_3P_2</math> having <math>A_1A ..._1,P_2</math>, these will be continuous functions of the angle that <math>\pi</math> makes with <math>\pi_i</math>, and for one of the points <math>P_1,P
  7 KB (1,370 words) - 15:42, 29 January 2021
• 1962 AHSME Problems
  <math>\textbf{(A)}\ \pi + 2 \qquad \textbf{(B)}\ \frac{2 \pi + 1}{2} \qquad
  17 KB (2,459 words) - 22:40, 10 April 2023
• 2013 AMC 8 Problems
  The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, <math>30 = 6\times5</mat A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
  15 KB (2,162 words) - 20:05, 8 May 2023
• 1994 AHSME Problems
  A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectang each term is the sum of the two terms to its left. Find <math>a</math>.
  14 KB (2,124 words) - 13:39, 19 February 2020
• 1981 AHSME Problems
  ...D</math>. If <math>EB</math> has length one and <math>EC</math> has length two, then the area of the square is ...the smaller and the difference between the numbers is 8, the the larger of two numbers is
  17 KB (2,633 words) - 15:44, 16 September 2023
• 1968 AHSME Problems
  ...increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diameter. Then <math>P</math> equals: ...pi\quad\text{(C) } \frac{\pi^2}{2}\quad\text{(D) } \pi^2\quad\text{(E) } 2\pi</math>
  16 KB (2,571 words) - 14:13, 20 February 2020
• 2014 AMC 10A Problems
  ...morning for <math>\textdollar 2.50</math> each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges on The two legs of a right triangle, which are altitudes, have lengths <math>2\sqrt3</
  15 KB (2,190 words) - 15:21, 22 December 2020
• 1994 AHSME Problems/Problem 17
  ...qquad\textbf{(C)}\ 4\pi-4 \qquad\textbf{(D)}\ 2\pi+4 \qquad\textbf{(E)}\ 4\pi-2 </math> ...length <math>\sqrt{2}</math>. We deduce that the two triangles formed by two radii of circle <math>O</math> and the segment of the rectangle are 45-45-9
  2 KB (287 words) - 03:07, 28 May 2021
• 1960 AHSME Problems
  ...r{10,000}</math> the difference between a discount of <math>40</math>% and two successive discounts of <math>36</math>% and <math>4</math>%, Each of two angles of a triangle is <math>60^{\circ}</math> and the included side is <m
  21 KB (3,242 words) - 21:27, 30 December 2020

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