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• Pigeonhole Principle
  In [[combinatorics]], the '''pigeonhole principle''' states that if <math>n+1</math> or more pigeons are placed into <math>n</math> holes, o ...inciple'''. A common phrasing of the principle uses balls and boxes and is that if <math>n</math> balls are to be placed in <math>k</math> boxes and <math>
  11 KB (1,986 words) - 18:13, 19 June 2024
• 2021 JMC 10
  <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qqu <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2
  12 KB (1,784 words) - 15:49, 1 April 2021
• 2005 IMO Shortlist Problems/C2
  ...or indirectly), then that someone wins a free instructional video. Prove that if <math> \displaystyle n </math> persons bought sombreros, then at most <m ...k+2) </math>. Since this is clear for <math>m=0</math>, let us WLOG assume that <math>m</math> is positive. Under this assumption, we will now prove the st
  3 KB (538 words) - 08:12, 29 August 2011
• Chicken McNugget Theorem
  ...ritten in the form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. ...expressed in the form <math>am + bn</math>. The proof is based on the fact that in each pair of the form <math>(k, mn-m-n-k)</math>, exactly one element is
  17 KB (2,823 words) - 22:06, 15 November 2024
• 1990 APMO Problems
  ...h>, how many non-similar triangles are there in which <math>AEGF</math> is a cyclic quadrilateral? ...of <math>a_1, a_2, \dots , a_n</math> taken <math>k</math> at a time. Show that
  2 KB (364 words) - 22:07, 12 September 2016
• 2008 USAMO Problems/Problem 6
  ...mber of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integ ...e [[vector space]] of all such functions. Define the linear operator <math>A : V \to V</math> as
  13 KB (2,414 words) - 13:37, 11 July 2016
• 2001 IMO Shortlist Problems/C3
  ...iques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-cliq If there exists only one 3-clique, remove anyone in that clique. (If there are no 3-cliques, we are done!) Otherwise, consider the f
  3 KB (479 words) - 14:01, 23 November 2017
• Millennium Problems
  ...set of seven problems for which the [[Clay Mathematics Institute]] offered a US \$7 million prize fund (\$1 million per problem) to celebrate the new mi ...24, 2000 at the [[Collège de France]]. [[Timothy Gowers]] first presented a lecture titled ''The Importance of Mathematics'' as an introduction. After
  13 KB (1,969 words) - 16:57, 22 February 2024
• 2008 iTest Problems
  ...s as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fue When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.
  71 KB (11,749 words) - 11:39, 20 November 2024
• Mock AIME 2 2010 Answer Key
  ...an be easily proved through strong induction. Starting from 2010, which is a multiple of 15, we must first purge 1 lemming. We can then purge 4 lemmings ...h>0 \le b_i \le 2, 3|a_i-b_i</math> for <math>1 \le i \le 10</math>. Given that <math>f</math> can take on <math>K</math> distinct values, find the remaind
  36 KB (6,214 words) - 19:22, 13 July 2023
• 1982 USAMO Problems
  ...h <math>1982</math> persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in Let <math>S_r=x^r+y^r+z^r</math> with <math>x,y,z</math> real. It is known that if <math>S_1=0</math>,
  2 KB (348 words) - 21:36, 19 March 2020
• 2012 AIME II Problems
  ...from each of the three departments. Find the number of possible committees that can be formed subject to these requirements. ...th edge of the field. Cao arrives at point <math>D</math> at the same time that Ana and Bob arrive at <math>D</math> for the first time. The ratio of the f
  7 KB (1,228 words) - 11:16, 13 March 2020
• 2014 USAJMO Problems/Problem 5
  ...annot win in a finite number of moves, or prove that no such minimum value exists. ...or <math>k<5</math> as well) and show that <math>B</math> can thwart <math>A</math> for <math>k\geq 6</math>.
  4 KB (665 words) - 19:18, 18 April 2016
• 1986 AHSME Problems
  <math>\textbf{(A)}\ 2y \qquad <math>\textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad
  17 KB (2,512 words) - 17:30, 12 October 2023
• 2016 AIME II Problems
  ...eats <math>25</math> of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had in ...ath> and <math>b</math> are relatively prime positive integers. Find <math>a+b</math>.
  8 KB (1,312 words) - 20:16, 3 March 2021
• Gmaas
  ...9999999^{9999999999999999999999999999999999999}</math> years and counting. That's older than he is. EDIT: It is not older than he is because Gmaas is infin ...141592653589793238462643383 "2. Games can turn things into anything. Using that fact, you can use the Games theorem to solve any problem.
  69 KB (11,805 words) - 19:49, 18 December 2019
• MIE 2015/Day 1/Problem 6
  ...e an arithmetic progression, both in these order, so we can say that <math>a</math>, <math>b</math> and <math>c</math>: (a) are the sides of an obtusangle triangle.
  6 KB (962 words) - 23:39, 31 August 2021
• GMAAS
  7. All humans shalt remember that the number of commandments is infinite, but humans aren’t capable enough ...̃̀̉̋̅̃ͮ͝͡͞_͇̤̺͓͕͚̮̥̱̳̪̩̍̽̓ͪ̎̒̾ͯ̏̚͘͜͠͝ͅa̷̸̸̶̵̧̡̢̡̛͔͈͓̖̠̱͉̥̟͓̯̭͎̪̱̳̜̙̝͓̰͉̹̤̰͎͓͈
  99 KB (14,063 words) - 20:43, 13 November 2024
• 2020 CIME I Problems
  A knight begins on the point <math>(0,0)</math> in the coordinate plane. From ...es tax of <math>7.5\%</math> is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possib
  7 KB (1,188 words) - 17:00, 31 August 2020
• G285 2021 Summer Problem Set
  <math>\textbf{(A)}\ 42\qquad\textbf{(B)}\ 43\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 45\qqu ...> is a positive integer, find the sum of all values of <math>x</math> such that <math>f(x,y) \neq k</math> for some constant <math>k</math>.
  9 KB (1,577 words) - 22:28, 28 June 2021

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