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• 2006 AIME I Problems/Problem 2
  == Problem == ...]] <math> \mathcal{A} </math> be a 90-[[element]] [[subset]] of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements
  1 KB (189 words) - 20:05, 4 July 2013
• 2006 AIME II Problems/Problem 2
  == Problem == {{AIME box|year=2006|n=II|num-b=1|num-a=3}}
  1 KB (164 words) - 14:58, 14 April 2020
• 2005 AIME II Problems/Problem 2
  == Problem == *Person 2: <math>\frac{6 \cdot 4 \cdot 2}{6 \cdot 5 \cdot 4} = \frac 25</math>
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME I Problems/Problem 2
  == Problem == ...th>(334,6)</math>, <math>(501,4)</math>, <math>(668,3)</math>, <math>(1002,2)</math> and <math>(2004,1)</math>, and each of these gives a possible value
  2 KB (303 words) - 01:31, 5 December 2022
• 2004 AIME I Problems/Problem 2
  == Problem == ...Therefore, the largest element in <math>A</math> is <math>2 + \frac{m-1}{2}</math>.
  8 KB (1,431 words) - 17:50, 29 December 2024
• 2004 AIME II Problems/Problem 2
  == Problem == <cmath>2 \cdot \frac{14}{323} + \frac{90}{323} = \frac{118}{323}</cmath>
  2 KB (330 words) - 13:42, 1 January 2015
• 2019 AIME I Problems/Problem 2
  ==Problem== ...nct from <math>J</math>. The value of <math>B - J</math> is at least <math>2</math> with a probability that can be expressed in the form <math>\frac{m}{
  5 KB (831 words) - 18:47, 29 January 2025
• 1983 AIME Problems/Problem 2
  == Problem == == Solution 2 ==
  1 KB (184 words) - 22:11, 4 February 2025
• 1984 AIME Problems/Problem 2
  == Problem == == Solution 2==
  2 KB (260 words) - 01:24, 18 July 2024
• 1986 AIME Problems/Problem 2
  == Problem == ...,z)=\left(\sqrt5,\sqrt6,\sqrt7\right)</math> so that <math>\left(x^2,y^2,z^2\right)=(5,6,7).</math>
  3 KB (460 words) - 00:44, 5 February 2022
• 1987 AIME Problems/Problem 2
  == Problem == ...[[point]]s, one on the [[sphere]] of [[radius]] 19 with [[center]] <math>(-2,-10,5)</math> and the other on the sphere of radius 87 with center <math>(1
  1 KB (170 words) - 22:53, 3 February 2025
• 1988 AIME Problems/Problem 2
  == Problem == ...te the square of the sum of the digits of <math>k</math>. For <math>n \ge 2</math>, let <math>f_n(k) = f_1(f_{n - 1}(k))</math>. Find <math>f_{1988}(1
  696 bytes (103 words) - 19:16, 27 February 2018
• 1989 AIME Problems/Problem 2
  == Problem == ...ose 1} = 10</math> have 1 member and <math>{10 \choose 2} = 45</math> have 2 members. Thus the answer is <math>1024 - 1 - 10 - 45 = \boxed{968}</math>.
  911 bytes (135 words) - 08:30, 27 October 2018
• 1990 AIME Problems/Problem 2
  == Problem == Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>.
  5 KB (765 words) - 23:00, 26 August 2023
• 1991 AIME Problems/Problem 2
  == Problem == MP("4",(A+B)/2,N,f);MP("\cdots",(A+B)/2,f);
  4 KB (595 words) - 12:51, 17 June 2021
• 1992 AIME Problems/Problem 2
  == Problem == ...</math> potential ascending numbers, one for each [[subset]] of <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9\}</math>.
  2 KB (336 words) - 05:18, 4 November 2022
• 1993 AIME Problems/Problem 2
  == Problem == ...ay south, on the fifth day east, etc. If the candidate went <math>n^{2}_{}/2</math> miles on the <math>n^{\mbox{th}}_{}</math> day of this tour, how man
  2 KB (241 words) - 11:56, 13 March 2015
• 1994 AIME Problems/Problem 2
  == Problem == [[Image:1994 AIME Problem 2.png]]
  2 KB (272 words) - 03:53, 23 January 2023
• 1995 AIME Problems/Problem 2
  == Problem == <math>\sqrt{1995}x^{\log_{1995}x}=x^2</math>.
  2 KB (362 words) - 00:40, 29 January 2021
• 1996 AIME Problems/Problem 2
  == Problem == ...>n</math> is it true that <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
  1 KB (163 words) - 19:31, 4 July 2013

Page text matches

• 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 ...C^2=(AD)(AB)+(BD)(AB) \implies AC^2+BC^2=(AB)(AD+BD) \implies AC^2+BC^2=AB^2</math>
  6 KB (978 words) - 12:02, 6 March 2025
• Resources for mathematics competitions
  ...idual articles often have sample problems and solutions for many levels of problem solvers. Many also have links to books, websites, and other resources rele * CompetifyHub's Problem Sets [//competifyhub.com/resources/ Free Competition Resources for Grades 1
  14 KB (1,913 words) - 23:52, 6 March 2025
• Inequality
  ...ticular, notice that although <math>3 > 2</math>, we must have <math>-3 < -2</math>. In particular, when multiplying or dividing by negative quantities ...n the example <math>x \ge \frac{3}{2}</math>, the value <math>x = \frac{3}{2}</math> satisfies the inequality because the inequality is nonstrict.
  12 KB (1,806 words) - 06:07, 19 June 2024
• AMC 10
  ...are invited to take the [[American Invitational Mathematics Examination]] (AIME). ...administered by the [[American Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC.
  4 KB (596 words) - 04:53, 3 February 2025
• AMC 12
  ...ke the more challenging [[American Invitational Mathematics Examination]] (AIME). ...administered by the [[American Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC.
  5 KB (646 words) - 04:52, 3 February 2025
• American Invitational Mathematics Examination
  ...ed States at the [[International Mathematics Olympiad]] (IMO). While most AIME participants are high school students, some bright middle school students a High scoring AIME students are invited to take the prestigious [[United States of America Mat
  5 KB (669 words) - 17:19, 11 March 2025
• American Mathematics Competitions
  * [[American Invitational Mathematics Examination]] (AIME) &mdash; high scorers from the AMC 10/12 exams. * [[United States of America Mathematics Olympiad]] (USAMO) &mdash; high AIME and AMC scorers.
  5 KB (696 words) - 03:47, 24 December 2019
• United States of America Mathematical Olympiad
  ...dministered by the [[Mathematical Association of America]] (MAA). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the MAA. ...ficulty=7-9|breakdown=<u>Problem 1/4</u>: 7<br><u>Problem 2/5</u>: 8<br><u>Problem 3/6</u>: 9}}
  6 KB (836 words) - 04:57, 3 February 2025
• Joining an ARML team
  * Performance on problems at practice sessions. * Performance on AMC and AIME, including current and previous years.
  22 KB (3,532 words) - 11:25, 27 September 2024
• Trivial Inequality
  For all [[real number]]s <math>x</math>, <math>x^2 \ge 0</math>. ...e numbers under multiplication. Finally, if <math>x<0</math>, then <math>x^2 = (-x)(-x) > 0,</math> again by the closure of the set of positive numbers
  3 KB (583 words) - 21:20, 2 August 2024
• Completing the square
  ...as added to this, then we would have a [[perfect square]], <math>(x-3)^2=x^2-6x+9</math>. To do this, add <math>7</math> to each side of the equation t <math>x^2-6x+9=7\Rightarrow(x-3)^2=7\Rightarrow x-3=\pm{\sqrt7}\Rightarrow x=3\pm\sqrt{7}</math>
  2 KB (422 words) - 16:20, 5 March 2023
• Simon's Favorite Factoring Trick
  ...nd make the equation factorable. It can be used to solve more than algebra problems, sometimes going into other topics such as number theory. == Fun Practice Problems ==
  4 KB (682 words) - 10:25, 18 February 2025
• Principle of Inclusion-Exclusion
  ...d only once. In particular, memorizing a formula for PIE is a bad idea for problem solving. ==== Problem ====
  9 KB (1,703 words) - 01:20, 7 December 2024
• Combination
  ...as a great introductory video to combinations, permutations, and counting problems in general! [https://bit.ly/CombinationsAndPermutations Permutations & Comb * <math>\sum_{x=0}^{n} \binom{n}{x} = 2^n</math>
  4 KB (638 words) - 21:55, 5 January 2025
• Factorial
  ...rial is defined for [[positive integer]]s as <math>n!=n \cdot (n-1) \cdots 2 \cdot 1 = \prod_{i=1}^n i</math>. Alternatively, a [[recursion|recursive d * <math>2! = 2</math>
  10 KB (809 words) - 16:40, 17 March 2024
• 2006 AIME II
  ...contains the full set of test problems. The rest contain each individual problem and its solution. * [[2006 AIME II Problems]]
  1 KB (133 words) - 12:32, 22 March 2011
• Fermat's Little Theorem
  In contest problems, Fermat's Little Theorem is often used in conjunction with the [[Chinese Re <cmath>(a+1)^p = a^p + {p \choose 1} a^{p-1} + {p \choose 2} a^{p-2} + \cdots + {p \choose p-1} a + 1.</cmath>
  15 KB (2,618 words) - 12:03, 19 February 2025
• Parabola
  ...equation]] of a parabola. The first is [[polynomial]] form: <math>y = a{x}^2+b{x}+c</math> where a, b, and c are [[constant]]s. This is useful for manip ...nd is [[completing the square | completed square]] form, or <math>y=a(x-h)^2+k</math> where a, h, and k are constants and the [[vertex]] is (h,k). This
  3 KB (551 words) - 16:22, 13 September 2023
• Euler's Totient Theorem
  ...r, <math>\phi{(n)}</math> is the number of integers in the range <math>\{1,2,3\cdots{,n}\}</math> which are relatively prime to <math>n</math>. If <math ==Problems==
  4 KB (569 words) - 22:34, 30 December 2024
• Brahmagupta's Formula
  ...th>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. .../math>. Hence, <math>[ABCD]=\frac{\sin B(ab+cd)}{2}</math>. Multiplying by 2 and squaring, we get:
  3 KB (543 words) - 19:35, 29 October 2024

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