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• Lagrange's Identity
  ...\biggr )}-{\biggl (}\sum _{k=1}^{n}a_{k}b_{k}{\biggr )}^{2}&=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\\&{\biggl (}={\frac {1}{2}}\s The vector form follows from the Binet-Cauchy identity by setting <math>ci = ai</math> and <math>di = bi</math>. Th
  4 KB (914 words) - 17:27, 31 August 2022
• 1960 AHSME Problems/Problem 34
  ...answer. From the graph (where the x-axis is the time in seconds and the y-axis is distance from one side of the pool), there are five meeting points, {{AHSME 40p box|year=1960|num-b=33|num-a=35}}
  2 KB (262 words) - 19:18, 17 May 2018
• 2018 AIME I Problems
  Find the number of four-element subsets of <math>\{1,2,3,4,\dots, 20\}</math> with the property that ...at when <math>3^n</math> is written in base <math>143</math>, its two right-most digits in base <math>143</math> are <math>01</math>.
  8 KB (1,284 words) - 14:35, 9 August 2021
• 2018 AIME II Problems
  ...math> recursively to be the remainder when <math>4(a_{n-1} + a_{n-2} + a_{n-3})</math> is divided by <math>11</math>. Find <math>a_{2018} \cdot a_{2020} ...math>. The self-intersecting octagon <math>CORNELIA</math> encloses six non-overlapping triangular regions. Let <math>K</math> be the area enclosed by <
  9 KB (1,385 words) - 00:26, 21 January 2024
• 2018 AIME I Problems/Problem 10
  label("$A$", (1-r) * dir( 90), -dir( 90)); label("$B$", (1-r) * dir(162), -dir(162));
  11 KB (1,934 words) - 12:18, 29 March 2024
• 2018 AIME I Problems/Problem 11
  ...when <math>3^n</math> is written in base <math>143^2</math>, its two right-most digits in base <math>143</math> are <math>01</math>. ...iv \binom{p}{0}26^p+\binom{p}{1}26^{p-1}....+\binom{p}{p-2}26^2+\binom{p}{p-1}26+\binom{p}{p}\equiv 26p+1\equiv 1\pmod{169}</math>, so <math>26p\equiv 0
  10 KB (1,448 words) - 06:30, 21 April 2024
• 2018 AIME II Problems/Problem 9
  ...10</math> and <math>BC = DE = FG = HA = 11</math> is formed by removing 6-8-10 triangles from the corners of a <math>23</math> <math>\times</math> <math <math>=\left|\frac{297}{9}-\frac{690}{6}-102\right|=\left| 33-115-102\right|=\left|-184\right|=\boxed{184}</math>
  5 KB (782 words) - 02:30, 5 January 2024
• 2017 IMO Problems
  ...bbit is at point <math>A_{n-1}</math> and the hunter is at point <math>B_{n-1}</math>. In the nth round of the game, three things occur in order. ...sibly to a point <math>A_n</math> such that the distance between <math>A_{n-1}</math> and <math>A_n</math> is exactly 1.
  4 KB (720 words) - 12:12, 5 August 2021
• 1964 AHSME Problems/Problem 13
  ...implies <math>AJ=x</math>. Thus <math>KB=BL=17-x</math> and <math>JC=LC=13-x</math>. ...+LC=(17-x)+(13-x)=8</math>, <math>x=11</math>. Thus <math>r:s=CL:BL=13-x:17-x=2:6=1:3</math>, hence our answer is <math>\fbox{A}</math>.
  2 KB (306 words) - 18:26, 23 July 2019
• 2018 IMO Problems/Problem 3
  ...the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from <math>1</ma Does there exist an anti-Pascal triangle with <math>2018</math> rows which contains every integer fro
  4 KB (626 words) - 01:45, 19 November 2023
• 2003 Indonesia MO Problems
  ...o <math>3</math> congruent triangles (refer to the figure). Every triangle-region is colored with a certain color so that each tile has <math>3</math> ...if <math>k-n</math> divides <math>k^m - n^{m-1}</math>, then <math>k \le 2n-1</math>.
  3 KB (497 words) - 02:20, 19 May 2024
• 2003 Indonesia MO Problems/Problem 4
  |num-b=3 |num-a=5
  2 KB (296 words) - 18:00, 11 August 2018
• 2012 Indonesia MO Problems
  ...ber <cmath>n=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b</cmath> is an even non-negative integer. (i) On each row of <math>P</math>, from left to right, the numbers are non-increasing,
  5 KB (774 words) - 00:15, 13 August 2018
• 2003 JBMO Problems
  2 KB (330 words) - 13:59, 25 August 2018
• 2006 Indonesia MO Problems/Problem 8
  Find the largest 85-digit integer which has property: the sum of its digits equals to the produc ...we'll find the largest value of <math>a_1</math> and use it to find the 85-digit integer.
  6 KB (906 words) - 22:48, 23 September 2018
• 2018 AMC 8 Problems
  The twelve-sided figure shown has been drawn on <math>1 \text{ cm}\times 1 \text{ cm}</ What is the value of <math>1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018</math>?
  14 KB (2,191 words) - 02:31, 7 June 2024
• 2018 AMC 8 Problems/Problem 24
  ...ath> respectively. Let <math>R</math> be the ratio of the area of the cross-section <math>EJCI</math> to the area of one of the faces of the cube. What ...with sides <math>a,b,c,d</math> then <cmath>[ABCD]=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left({\frac{B+D}{2}}\right)}</cmath>where <math>s=\frac{a+b+c+
  10 KB (1,684 words) - 13:25, 14 January 2024
• 1991 IMO Problems/Problem 6
  ...sequence <math>x_0,x_1,x_2,\ldots</math> such that <math>|x_i-x_j|\cdot |i-j|^a\geq 1</math> for every pair of distinct nonnegative integers <math>i,j< Suppose that we have chosen points <math>x_0,x_1,\ldots,x_{m-1}</math> satisfying
  6 KB (1,068 words) - 03:20, 24 January 2024
• 2006 iTest Problems/Problem 18
  {{iTest box|year=2006|num-b=17|num-a=19|ver=[[2006 iTest Problems/Problem U1|U1]] '''•''' [[2006 iTest Proble
  2 KB (252 words) - 12:04, 27 November 2018
• 2006 iTest Problems/Problem 15
  {{iTest box|year=2006|num-b=14|num-a=16|ver=[[2006 iTest Problems/Problem U1|U1]] '''•''' [[2006 iTest Proble
  1 KB (197 words) - 23:31, 27 November 2018

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