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• 2003 USAMO Problems/Problem 3
  ...{} c_j = \cdots = c_{j+i} = \cdots = c_k = j </math> (<math> 0 \le i \le k-j</math>). ''Proof.'' Since the <math>j </math> terms <math>c_0, \ldots, c_{j-1} </math> are all less than <math>j </math>, no other terms that precede <m
  3 KB (636 words) - 13:39, 4 July 2013
• Pentagon
  1 KB (191 words) - 09:59, 6 June 2022
• 2004 IMO Shortlist Problems/N2
  ...e-i}) = p^e + \sum_{i=0}^{e-1}p^i(p^{e-i} - p^{e-i-1}) = p^e + e(p^e - p^{e-1}) </math>. </center> ...^{e_i} - e_i \cdot p_i^{e_i}]}{\prod p_i^{e_i}} = \prod \left(e_i \frac{p_i-1}{p_i} + 1 \right) </math>.
  6 KB (1,007 words) - 09:10, 29 August 2011
• Parallelepiped
  2 KB (248 words) - 20:08, 17 August 2023
• Chicken McNugget Theorem
  ...am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>. ...e proof is based on the fact that in each pair of the form <math>(k, mn-m-n-k)</math>, exactly one element is expressible.
  17 KB (2,748 words) - 19:22, 24 February 2024
• Dodecagon
  A '''dodecagon''' is a 12-sided [[polygon]]. The sum of its internal [[angle]]s is <math>1800^{\circ}<
  1 KB (219 words) - 13:08, 15 June 2018
• Triangular number
  <math>1 +2+3 + 4............. +(n-1)+(n)</math>. ...math>T_{n} = 1 + 2 + \ldots + (n-1) + n = (1 + 2 + \ldots + n-1) + n = T_{n-1} + n</math>.
  2 KB (275 words) - 08:39, 7 July 2021
• 2007 AMC 12A Problems/Problem 24
  ...mod{4}</math> is that <math>v_2(2j+1)=0,</math> so we must have <math>v_2(n-1)>v_2(n+1)</math> since <math>v_2(2k) \geq 1 >v_2(2j+1).</math> Therefore, {{AMC12 box|year=2007|num-b=23|num-a=25|ab=A}}
  4 KB (588 words) - 14:40, 23 August 2023
• 2006 AIME II Problems/Problem 13
  ...math>, <math>n\geq 0</math>. Then the final odd integer is <math>2n+1 + 2(j-1) = 2(n+j) - 1</math>. The odd integers form an [[arithmetic sequence]] wit ...ual <math>0</math>. We then perform casework based on the parity of <math>p-q</math>.
  4 KB (675 words) - 10:40, 14 July 2022
• 2006 AIME II Problems
  When rolling a certain unfair six-sided die with faces numbered <math>1, 2, 3, 4, 5</math>, and <math>6</math> ...h> and <math>t_2</math> intersect at <math>(x,y),</math> and that <math>x=p-q\sqrt{r},</math> where <math>p, q,</math> and <math>r</math> are positive i
  8 KB (1,350 words) - 12:00, 4 December 2022
• 1998 PMWC Problems
  for(int j = n-i; j > 0; --j){ ...ad a total of <math>30</math> balls, how many balls were there in the right-hand box?
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2001 PMWC Problems
  F=O+(G-O)+(E-O); draw((xstart,A.y)--(xstart,A.y-len));
  4 KB (641 words) - 21:24, 21 April 2014
• 2002 PMWC Problems
  label("$" + string(i) + "^\textrm{" + ord[i-1] + "}$ Row:", (-5,(-i+1)*sqrt(3)/2+correction)); draw( (1-cos(pi/3)*correction,-sin(pi/3)*correction)--(0+cos(pi/3)*correction,-sqrt(3
  5 KB (725 words) - 16:07, 23 April 2014
• 2003 PMWC Problems
  filldraw((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,black); fill((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,white);
  7 KB (918 words) - 16:15, 22 April 2014
• 1989 USAMO Problems/Problem 3
  Let <math>P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n</math> be a [[polynomial]] in the complex variable <math>z <cmath> \lvert z_i-i \rvert \cdot \lvert z_j-i \rvert < 1. </cmath>
  2 KB (340 words) - 19:11, 18 July 2016
• Cramer's Rule
  2 KB (352 words) - 18:22, 11 October 2023
• Lucas' Theorem
  ...n of <math>n</math> in [[base]] <math>p</math> and <math>(\overline{i_mi_{m-1}\cdots i_0})_p</math> is the representation of <math>i</math> in base <mat For all <math>1\leq k \leq p-1</math>, <math>\binom{p}{k}\equiv 0 \pmod{p}</math>. Then we have
  1 KB (251 words) - 15:13, 11 August 2020
• Summation
  ...r algebraic structure), <math>\sum_{i=a}^{b}c_i=c_a+c_{a+1}+c_{a+2}...+c_{b-1}+c_{b}</math>. Here <math>i</math> refers to the index of summation, <math ...i= \frac{n(n+1)}{2}</math>, and in general <math>\sum_{i=a}^{b} i= \frac{(b-a+1)(a+b)}{2}</math>
  3 KB (482 words) - 16:39, 8 October 2023
• Zorn's Lemma
  ...intermediate results, viz., Bourbaki's Theorem (also known as the Bourbaki-Witt theorem). ...h>x \in A</math> does <math>x = f(x)</math>, which contradicts the Bourbaki-Witt Theorem. {{halmos}}
  9 KB (1,669 words) - 19:02, 1 August 2018
• Ideal
  ...ls in rings are the [[kernel]]s of ring [[homomorphism]]s; in this way, two-sided ideals of rings are similar to [[normal subgroup]]s of [[group]]s. ...frak{a}</math> is both a left ideal and a right ideal, it is called a ''two-sided ideal''. In a [[commutative ring]], all three kinds of ideals are the
  8 KB (1,389 words) - 23:44, 17 February 2020

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