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• Orthocenter
  dot((4,0)); draw((0,0)--(4,0)--(3,3)--cycle);
  5 KB (829 words) - 13:11, 20 February 2024
• 1970 Canadian MO Problems
  draw((2.5,0)--(2.5,7.5/4)--(5,0)--cycle,black); MP("C",(0,0),SW);MP("D",(16/5,12/5),N);MP("B",(5,0),SE);
  4 KB (604 words) - 04:32, 8 October 2014
• Perfect number
  A [[positive integer]] <math>n</math> is called a '''perfect number''' if it is the sum of its [[proper di * <math>28 = 14 + 7 + 4 + 2 + 1</math>
  1 KB (193 words) - 19:09, 3 March 2022
• Stirling number
  ...kind''' <math>c(n, k)</math> is the number of [[permutation]]s of an <math>n</math>-[[element]] [[set]] with exactly <math>k</math> [[cycle of a permuta ...h>\{(1)(234), (1)(243), (134)(2),(143)(2),(124)(3),(142)(3),(123)(4),(132)(4), (12)(34), (13)(24), (14)(23)\}</math>.
  2 KB (253 words) - 00:50, 15 May 2024
• 1970 IMO Problems
  ...the system with base <math>a</math> and <math>B_{n-1}</math> and <math>B_{n}</math> are numbers in the system with base <math>b</math>; these are relat <math>A_{n} = x_{n}x_{n-1}\cdots x_{0}, A_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}</math>,
  3 KB (558 words) - 00:17, 10 December 2022
• Argument
  ...est considered as an [[equivalence class]] <math>\mathbf r = \{r + 2\pi n, n \in \mathbb{Z}\}</math>. The advantages of this are several: most importan
  1 KB (188 words) - 20:59, 31 July 2020
• Proper divisor
  ...ger]] <math>n</math> is any [[divisor]] of <math>n</math> other than <math>n</math> itself. Thus <math>1</math> has no proper divisors, [[prime number] *[[2017 USAJMO Problems/Problem 4]]
  436 bytes (63 words) - 16:27, 10 May 2021
• 2007 iTest Problems
  <math>\mathrm{(A)}\, 4</math> ===Problem 4===
  30 KB (4,794 words) - 23:00, 8 May 2024
• 1981 IMO Problems/Problem 2
  ..., n \} </math>. Each of these subsets has a smallest member. Let <math>F(n,r) </math> denote the arithmetic mean of these smallest numbers; prove that F(n,r) = \frac{n+1}{r+1}.
  5 KB (879 words) - 11:18, 27 June 2020
• 1981 IMO Problems/Problem 3
  ...rs satisfying <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math>( n^2 - mn - m^2 )^2 = 1 </math>. ...m</math>, since if we had <math>n < m</math>, then <math>n^2 -nm -m^2 = n(n-m) - m^2 </math> would be the sum of two negative integers and therefore le
  1 KB (248 words) - 10:23, 13 May 2019
• 1981 IMO Problems/Problem 4
  ...t is a [[divisor]] of the [[least common multiple]] of the remaining <math>n-1</math> numbers? (b) For which values of <math>n>2</math> is there exactly one set having the stated property?
  3 KB (516 words) - 09:43, 28 March 2012
• 1987 IMO Problems/Problem 5
  ...an integer greater than or equal to 3. Prove that there is a set of <math>n </math> points in the plane such that the distance between any two points i ...the set of points <math>S = \{ (x,x^2) \mid 1 \le x \le n , x \in \mathbb{N} \}</math> in the <math>xy</math>-plane.
  2 KB (290 words) - 13:37, 26 July 2009
• Apothem
  ...</math>, the length of the apothem is <math>\frac{s}{2\tan\left(\frac{\pi}{n}\right)}</math>. ..., <math>R</math>, the length of the apothem is <math>R\cos\left(\frac{\pi}{n}\right)</math>.
  1 KB (169 words) - 18:22, 9 March 2014
• 2005 AMC 10A Problems
  <math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 20 < ...-8\qquad \mathrm{(B) \ } -4\qquad \mathrm{(C) \ } -2\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 8 </math>
  14 KB (2,026 words) - 11:45, 12 July 2021
• Euler line
  ...ath>G</math>, [[circumcenter]] <math>O</math>, [[nine-point center]] <math>N</math> and [[De Longchamps point | de Longchamps point]] <math>L</math>. I ...ath>, and <math>\triangle CH_AH_B</math> [[concurrence | concur]] at <math>N</math>, the nine-point circle of <math>\triangle ABC</math>.
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2003 AMC 10A Problems
  Members of the Rockham Soccer League buy socks and T-shirts. Socks cost &#36;4 per pair and each T-shirt costs &#36;5 more than a pair of socks. Each memb <math> \mathrm{(A) \ } 4.5\qquad \mathrm{(B) \ } 9\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 18
  13 KB (1,900 words) - 22:27, 6 January 2021
• 1997 AJHSME Problems/Problem 24
  A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0); label("$B$",B,N);
  2 KB (394 words) - 17:05, 20 October 2023
• Fermat point
  pair A=(2,4), B=(1,1), C=(6,1); .../math> minimizes <math>m\cdot AP+n\cdot BP+p\cdot CP</math>, where <math>m,n,p</math> are positive reals?
  4 KB (769 words) - 16:07, 29 December 2019
• Balkan MO Problems and Solutions, with authors
  * [[1984 BMO Problems/Problem 4 | Problem 4]] * [[1985 BMO Problems/Problem 4 | Problem 4]]
  4 KB (371 words) - 16:41, 1 January 2024
• Chinese Remainder Theorem
  ...on of each residue class mod <math>m</math> with a residue class mod <math>n</math> is a residue class mod <math>mn</math>. ...an deduce that <math>b \equiv c \pmod{m}</math> and <math>b \equiv c \pmod{n}.</math>
  6 KB (1,022 words) - 14:57, 6 May 2023

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