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  • ...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>.
    5 KB (885 words) - 22:03, 5 October 2024
  • ...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 ==
    16 KB (2,192 words) - 22:06, 19 July 2024
  • 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) - 05:07, 19 June 2024
  • ...correct answers receive one point of credit, making the maximum score 15. Problems generally increase in difficulty as the exam progresses - the first few que ...ber theory]], and [[probability]] and other secondary school math topics. Problems usually require either very creative use of secondary school curriculum, or
    8 KB (1,067 words) - 19:15, 24 October 2024
  • <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) - 14:00, 1 June 2024
  • ...contain two or more pigeons. This seemingly trivial statement may be used with remarkable creativity to generate striking counting arguments, especially i '''Solution''': Intuitively, you might realize that after we select four socks of diffe
    11 KB (1,986 words) - 18:13, 19 June 2024
  • 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) - 00:20, 7 December 2024
  • ...es of the theorem by <math>a</math>. The restated form is nice because we no longer need to restrict ourselves to integers <math>{a}</math> not divisibl In contest problems, Fermat's Little Theorem is often used in conjunction with the [[Chinese Remainder Theorem]] to simplify tedious calculations.
    16 KB (2,660 words) - 22:42, 28 August 2024
  • == Solution 1 == This tells there that there is no solution for (b), since we must have <math>A^2 \ge 2</math>
    3 KB (464 words) - 00:29, 5 November 2024
  • ...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) - 17:09, 11 December 2024
  • ...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
    12 KB (1,898 words) - 07:42, 19 August 2024
  • ...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) - 16:40, 24 September 2024
  • ...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) - 16:20, 16 June 2023
  • .../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) - 13:57, 24 June 2024
  • ...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) - 10:13, 9 June 2023
  • 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) - 00:15, 4 July 2024
  • 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 (490 words) - 02:32, 23 July 2023
  • ...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 (2,406 words) - 07:56, 10 July 2024
  • 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) - 18:01, 29 October 2021
  • == Solution 1 == == Solution 2 ==
    10 KB (1,702 words) - 21:23, 25 July 2024

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