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• 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
  14 KB (2,104 words) - 23:56, 28 August 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
  9 KB (1,689 words) - 02:54, 4 September 2024
• 1969 IMO Problems/Problem 5
  ...elect one of the points (C) which minimizes the angle BAC. You cannot have two of them because then they would be collinear. Also no points lie inside the ...3 points, you can create a convex quadrilateral containing two of them and two of A,B,C.
  10 KB (1,755 words) - 20:10, 4 August 2024
• 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) - 16:54, 28 July 2024
• 2014 AMC 10A Problems/Problem 21
  ..., \dots</math>. Therefore, we see that, the only values common between the two sequences are <math>-1, -5, -\frac13,-\frac53</math>, and adding them up, w ==Video Solution by Pi Academy==
  2 KB (328 words) - 20:43, 7 October 2024
• 2014 AMC 10A Problems/Problem 23
  ...e top of the paper, draw lines to each endpoint of the dotted line. These two lines plus the dotted line form a triangle which is the double-layered port == Video Solution by Pi Academy ==
  4 KB (605 words) - 20:44, 7 October 2024
• 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
• 2014 AMC 12A Problems/Problem 12
  Two circles intersect at points <math>A</math> and <math>B</math>. The minor a <cmath>\dfrac{\pi x^2}{\pi y^2} = \dfrac{x^2}{y^2} </cmath>
  7 KB (1,191 words) - 23:37, 23 June 2022
• 2014 AMC 12B Problems
  ...consists of triangles whose sides have integer lengths less than 5, and no two elements of <math>S</math> are congruent or similar. What is the largest nu ...itive solutions of <cmath>2\cos(2x) \left(\cos(2x) - \cos\left( \frac{2014\pi^2}{x} \right)\right) = \cos(4x) - 1</cmath>
  13 KB (2,066 words) - 14:08, 1 November 2022
• Mock AIME 3 2006-2007 Problems/Problem 11
  (one value of <math>a</math> which produces this maximum is <math>a=\frac{\pi}{4}</math>) ...^2+y^2</math>. Adding and subtracting the first two equations, dividing by two, and setting <math>\mu = 2\lambda</math> for convenience yields:
  4 KB (699 words) - 01:53, 30 April 2022
• 2014 AIME II Problems
  ...ee factors, the probability that a randomly selected man has exactly these two risk factors (but not the third) is 0.14. The probability that a randomly s ...es of length <math>a</math> can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the
  8 KB (1,410 words) - 00:04, 29 December 2021
• 2014 AIME I Problems/Problem 10
  Let <math>F</math> be the new tangency point of the two disks. The smaller disk rolled along minor arc <math>\overarc{AF}</math> on ...the smaller disk, so <math>6\alpha = 2\pi</math>, or <math>\alpha = \frac{\pi}{3}</math>.
  5 KB (854 words) - 20:02, 4 September 2021

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