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• 2020 AMC 10B Problems/Problem 18
  ...taining just one black card. We then insert <math>n</math> red cards one-by-one into the deck at random positions. It is easy to see using induction, th ...t how many choices there are for the four steps. No matter what the <math>k-th</math> step involves <math>k+1</math> numbers to choose from.
  11 KB (1,928 words) - 17:00, 8 June 2024
• 2020 AMC 10B Problems/Problem 15
  <ol start="0" style="margin-left: 1.5em;"> <ol start="0" style="margin-left: 1.5em;">
  10 KB (1,471 words) - 13:57, 30 October 2023
• 2020 AMC 10B Problems/Problem 24
  <cmath>(n-400)(n-2500) = 0</cmath> ...t <math>k^{2}+30k+225\leq70k+50\implies k^{2}-40k+175\leq 0\implies (k-5)(k-35)\leq0\implies 5\leq k\leq 35</math>.
  10 KB (1,550 words) - 10:26, 9 March 2024
• 2020 AMC 10B Problems/Problem 21
  real y=2sqrt(16-8sqrt(2))-4+2sqrt(2); real z=2sqrt(8-4sqrt(2));
  11 KB (1,850 words) - 23:33, 27 March 2024
• 2020 AMC 10B Problems/Problem 11
  {{AMC10 box|year=2020|ab=B|num-b=10|num-a=12}}
  4 KB (528 words) - 14:53, 6 June 2023
• 2020 AMC 10B Problems/Problem 14
  ...olored white, each with an area of <math>\frac{\sqrt{3}}{4}</math>, and one-sixth of a circle with radius <math>1</math> colored white, with an area of ...f the shaded region, and let <math>y</math> be the area of one of the 'leaf-shaped' regions.
  17 KB (2,392 words) - 12:36, 24 December 2023
• 1982 AHSME Problems/Problem 24
  ...O=circumcenter(G,H,J); dot(A^^B^^C^^D^^E^^F^^G^^H^^J); draw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7)); label("$A$", A, N); label("$B$", B, di ...ds a(a + x) = 6(6 + 7) = 78. Also, we know that because a, b, and x are non-intersecting segments of the side of the triangle, a + b + x = 16.
  3 KB (449 words) - 16:58, 11 October 2020
• 1972 AHSME Problems/Problem 23
  label("$2-x$",(1,45/32),E); ...ine{EF}</math> is <math>1+1=2</math>, <math>\overline{FK}</math> is <math>2-x</math>. Then, <math>\triangle FGK</math> and <math>\triangle AEK</math> ar
  2 KB (322 words) - 01:14, 26 May 2024
• 2006 iTest Problems/Problem 10
  {{iTest box|year=2006|num-b=9|num-a=11|ver=[[2006 iTest Problems/Problem U1|U1]] '''•''' [[2006 iTest Proble
  4 KB (612 words) - 23:33, 3 November 2023
• 2008 Indonesia MO Problems/Problem 2
  By the [[Cauchy-Schwarz Inequality]], <math>(1+1)(1+x) \ge (1 + \sqrt{x})^2</math> and <math ...+x} = \frac{1}{1+y}</math>, so <math>x = y</math>. Additionally, by the AM-GM Inequality, <math>\frac12 \cdot (\frac{x+1}{2} + \frac{y+1}{2}) \ge \sqrt
  3 KB (473 words) - 00:09, 4 December 2021
• G285 2021 Fall Problem Set
  ...5</math> problems, <math>10</math> multiple-choice, and <math>5</math> free-response.
  1 KB (247 words) - 21:43, 18 July 2021
• 2004 Pan African MO Problems/Problem 5
  ...ne or only zeroes, and one where the additional digits has at least one non-zero digit. '''Case 2: Extra digits include at least one non-zero digit'''
  3 KB (405 words) - 20:32, 25 March 2020
• 2022 AMC 10A Problems
  Which expression is equal to <cmath>\left|a-2-\sqrt{(a-1)^2}\right|</cmath> for <math>a<0?</math> <math>\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3</ma
  18 KB (2,662 words) - 02:08, 9 March 2024
• Online Physics Olympiad
  ...world. This contest is a team based contest with each team consisting of 1-3 high school students. Undergraduate students in university may also partic ...</math> for more details. Competitors will submit their solutions via an on-line submission portal. Difficulty of the contest will be ranging from AP Ph
  4 KB (581 words) - 19:20, 26 May 2020
• 1991 IMO Problems
  3 KB (416 words) - 17:35, 30 January 2021
• 2023 AIME I Problems/Problem 11
  ...because <math>n-1</math> cannot be used), and otherwise there are <math>f(n-1)</math> possibilities. Thus, by induction, <math>f(n)</math> is the <math> If we have (1,2) as our pair, we are left with the numbers from 3-10 as elements that can be added to our subset. So, we must compute how many
  15 KB (2,414 words) - 06:57, 26 November 2023
• 2020 AIME II Problems/Problem 2
  {{AIME box|year=2020|n=II|num-b=1|num-a=3}}
  3 KB (403 words) - 17:24, 24 June 2020
• 2020 FMC 10A Problems
  Real numbers <math>a</math> and <math>b</math> exist such that <math>|a-3| + a^2 - 6ab + 9b^2 = 0</math>. Find the value of <math>a+b</math>. ...<math>1=d_1<d_2<d_3<\dots<d_k=n</math> is equal to the value of <math>d_{k-1} - d_2</math>. Find the sum of the first three positive numbers with zero
  14 KB (2,267 words) - 12:49, 9 June 2020
• 2020 FMC 12A Problems
  ...s ball into basket <math>b+1</math>. In general the ball has <math>\tfrac{i-1}{i}</math> chance of missing its target for basket <math>i</math>. Jensen ...(x^2-2\cos(1 ^ { \circ })x+1)\cdot (x^2-2\cos(2 ^ {\circ})x+1) \cdots (x^2-2\cos(179 ^ {\circ})x+1).</cmath>Find <math>f(1)</math>.
  15 KB (2,388 words) - 13:24, 9 June 2020
• 1982 AHSME Problems/Problem 25
  {{AHSME box|year=1982|num-b=24|num-a=26}}
  3 KB (453 words) - 19:05, 11 September 2023

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