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• Pythagorean Theorem
  ...e many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theo In these proofs, we will let <math>ABC</math> be any right triangle with a right angle at <math>\angle ACB</math>, and we use <math>[ABC]</math> to
  6 KB (978 words) - 12:02, 6 March 2025
• Resources for mathematics competitions
  ...tions]]. Look around the AoPSWiki. Individual articles often have sample problems and solutions for many levels of problem solvers. Many also have links to == Math Competition Problems ==
  14 KB (1,913 words) - 23:52, 6 March 2025
• Inequality
  The subject of mathematical '''inequalities''' is tied closely with [[optimization]] methods. While most of the subject of inequalities is oft ...[elementary algebra]], and relate slightly to [[number theory]]. They deal with [[relations]] of [[variable]]s denoted by four signs: <math>>,<,\ge,\le</ma
  12 KB (1,806 words) - 06:07, 19 June 2024
• 2015 IMO Problems
  <math>\mathcal{S}</math>, there is no point <math>P</math> in <math>\mathcal{S}</math> such that [[2015 IMO Problems/Problem 1|Solution]]
  4 KB (709 words) - 15:00, 1 June 2024
• Pigeonhole Principle
  ...contain two or more pigeons. This seemingly trivial statement may be used with remarkable creativity to generate striking counting arguments, especially i == Problems ==
  11 KB (1,980 words) - 22:39, 19 February 2025
• Principle of Inclusion-Exclusion
  Here, we will illustrate how PIE is applied with various numbers of sets. Just like in the Two Set Example, we start with the sum of the sizes of the individual sets <math>|A_1|+|A_2|+|A_3|</math>.
  9 KB (1,703 words) - 01:20, 7 December 2024
• Fermat's Little Theorem
  ...th sides of the theorem by <math>a</math>. This form is useful because we no longer need to restrict ourselves to integers <math>a</math> not divisible In contest problems, Fermat's Little Theorem is often used in conjunction with the [[Chinese Remainder Theorem]] to simplify tedious calculations.
  15 KB (2,618 words) - 12:03, 19 February 2025
• 1959 IMO Problems/Problem 2
  == Solution 1 == This tells there that there is no solution for (b), since we must have <math>A^2 \ge 2</math>
  4 KB (612 words) - 00:19, 7 March 2025
• 1970 IMO Problems/Problem 6
  ...onsider all possible triangles having these point as vertices. Prove that no more than <math>70 \%</math> of these triangles are acute-angled. ==Solution==
  6 KB (1,054 words) - 18:09, 11 December 2024
• Constructive counting
  ...g]] technique that involves constructing an item belonging to a set. Along with the construction, one counts the total possibilities of each step and assem ...counting is among the most fundamental techniques in counting. Familiarity with constructive counting is essential in combinatorics, especially in intermed
  13 KB (2,018 words) - 15:31, 10 January 2025
• Casework
  ...emonstrate casework in action. Unlike the selections in this article, most problems cannot be completely solved through casework. However, it is crucial as an ...be considered [[brute force]]. This is especially true if that alternative solution uses [[complementary counting]].
  5 KB (709 words) - 17:40, 24 September 2024
• Complementary counting
  ...ler approach. A large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. ...bility is not typically an intermediate step, but a framework upon which a solution is built.
  8 KB (1,192 words) - 17:20, 16 June 2023
• Fibonacci sequence
  .../math>. This is the simplest nontrivial example of a [[linear recursion]] with constant coefficients. There is also an explicit formula [[#Binet's formul Readers should be wary: some authors give the Fibonacci sequence with the [[initial condition]]s <math>F_0 = F_1 = 1</math> (or equivalently <mat
  7 KB (1,111 words) - 14:57, 24 June 2024
• Fermat's Last Theorem
  ...ve [[integers]] <math>a,b,c,n</math> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>. ...e would have circulated a proof for the special case when he had a general solution. Some think that Fermat's proof was flawed, and that he saw the flaw after
  3 KB (453 words) - 11:13, 9 June 2023
• Diophantine equation
  Finding the solution or solutions to a Diophantine equation is closely tied to [[modular arithme ...owever, <math>17</math> will never be a multiple of <math>3</math>, hence, no solutions exist.
  9 KB (1,434 words) - 01:15, 4 July 2024
• International Mathematical Olympiad
  The competition takes place over 2 consecutive days. Each day 3 problems are given to the students to work on for 4.5 hours. Following the general f ...in the form of a [[proof writing|mathematical proof]]. Since there are 6 problems, a perfect score is 42 points.
  3 KB (485 words) - 03:31, 3 February 2025
• Modular arithmetic/Introduction
  ...etic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic. ...example, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.
  16 KB (2,410 words) - 14:05, 3 January 2025
• Modular arithmetic/Intermediate
  Given integers <math>a</math>, <math>b</math>, and <math>n</math>, with <math>n > 0</math>, we say that <math>a</math> is ''congruent to'' <math>b< ...quations]], testing whether certain large numbers are prime, and even some problems in cryptology.
  14 KB (2,317 words) - 19:01, 29 October 2021
• 2006 AIME I Problems/Problem 13
  == Solution 1 == == Solution 2 ==
  10 KB (1,702 words) - 22:23, 25 July 2024
• 2006 AIME I Problems/Problem 10
  ...</math> be the union of the eight circular regions. Line <math> l, </math> with slope 3, divides <math> \mathcal{R} </math> into two regions of equal area. === Solution 1 ===
  4 KB (731 words) - 17:59, 4 January 2022

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