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• Inequality Introductory Problem 2
  <math>2\sum_{k=1}^{n}a_{k}^{2}\ge 2(a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}+a_{n}a_{1})</math> Which is always true.
  464 bytes (82 words) - 15:18, 23 May 2009
• Binary
  ...of computation and the ease of using ON - OFF states in computers. Binary is written as a number with two different possible digits: 1 and 0. | <math>2^3</math>|| 8|| 1000
  2 KB (294 words) - 12:55, 3 September 2019
• String
  ...[[Python]] strings can be delimited by double or single quotes (<tt>'this is also a string in Python'</tt>). ...s a string a (positive integral) number of times. For example, <tt>'spam'*3</tt> will evaluate to <tt>'spamspamspam'</tt>.
  4 KB (622 words) - 01:24, 8 April 2011
• 2004 IMO Problems/Problem 2
  ...(a) + f(-a) = 2f(a) \Longrightarrow f(a) = f(-a)</math>, so <math>f</math> is even, and all the degrees all of its terms are even. Let <math>\text{deg}\, ...f(5x) + f(8x) = 2f(7x)</math>. Comparing lead coefficients, we have <math>3^n + 5^n + 8^n = 2 \cdot 7^n</math>, which cannot be true for <math>n \ge 5<
  2 KB (373 words) - 00:52, 19 November 2023
• 2002 USAMO Problems/Problem 6
  ...+ 1)\times (m + 1)</math> square, then the <math>n\times n</math> section is covered by tiles contained in a concentric <math>(n + 2m)\times (n + 2m)</m ...th> blocks lying along the edges, hence there are at least <math>b(n) - 4n/3</math> crosses containing three used stamps.
  5 KB (940 words) - 17:33, 16 July 2014
• 2002 USAMO Problems
  ...th> be an integer with <math> 0 \le N \le 2^{2002} </math>. Prove that it is possible to color every subset of <math>S </math> either blue or red so tha (a) the union of any two red subsets is red;
  3 KB (486 words) - 09:21, 14 May 2021
• 2001 USAMO Problems/Problem 3
  The proof for the lower bound is the same as in the first solution. ...h>. Equality for the lower bound holds if and only if <math>(a,b,c)</math> is one of the triples <math>(2,0,0)</math>, <math>(0,2,0)</math> and <math>(0,
  4 KB (807 words) - 10:45, 9 April 2023
• 2001 USAMO Problems/Problem 4
  ...the side congruent to <math>PA</math>. Prove that <math>\angle BAC</math> is acute. which is true by the trivial inequality.
  8 KB (1,470 words) - 22:24, 18 June 2022
• 2001 USAMO Problems
  ...ne, with justification, the smallest integer <math>n</math> for which this is possible. ...D_2</math> at two points, the closer of which to the vertex <math>A</math> is denoted by <math>Q</math>. Prove that <math>AQ = D_2P</math>.
  3 KB (446 words) - 09:56, 20 July 2016
• 1973 USAMO Problems/Problem 5
  <cmath>q^{\dfrac{1}{3}}=p^{\dfrac{1}{3}}+md</cmath> <cmath>r^{\dfrac{1}{3}}=p^{\dfrac{1}{3}}+nd</cmath>
  4 KB (683 words) - 20:18, 29 December 2019
• 2002 AMC 10A Problems
  The ratio <math>\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}</math> is closest to which of the following numbers? == Problem 3 ==
  11 KB (1,733 words) - 11:04, 12 October 2021
• 1992 USAMO Problems/Problem 4
  ...through <math>A</math>, <math>B</math>, <math>C</math>, and <math>P</math> is tangent to the sphere through <math>A'</math>, <math>B'</math>, <math>C'</m ...contains <math>P</math>. We can easily find the <math>\triangle APB</math> is isosceles because the base angles are equal. Thus, <math>AP=BP</math>. Simi
  5 KB (807 words) - 18:37, 25 June 2021
• 2002 AMC 10A Problems/Problem 2
  ...{2}=\frac{1}{6}+\frac{8}{6}+\frac{27}{6}=\frac{36}{6}=6</math>. Our answer is then <math>\boxed{\textbf{(C) }6}</math>. ...xt{a little})</math>, and as all the options are integers, the correct one is <math>\boxed{\textbf{(C) }6}</math>.
  1,017 bytes (138 words) - 12:26, 8 November 2021
• 2002 AMC 12A Problems/Problem 8
  ...res, and <math>P</math> the area of the red square. Which of the following is correct? ...-3,3)--(-1,3)--(0,4)--(1,3)--(0,2)--(-1,3)--(1,3)--(2,4)--(3,3)--(2,2)--(1,3)--cycle;
  2 KB (345 words) - 12:50, 8 November 2021
• 2002 AMC 10B Problems
  The ratio <math>\frac{2^{2001}\cdot3^{2003}}{6^{2002}}</math> is: ...\frac{1}{2}\qquad \mathrm{(D) \ } \frac{2}{3}\qquad \mathrm{(E) \ } \frac{3}{2} </math>
  10 KB (1,540 words) - 22:53, 19 December 2023
• 2002 AMC 12A Problems/Problem 13
  ...<math>b</math> each differ from their reciprocals by <math>1</math>. What is <math>a+b</math>? \text{(E) }3
  1 KB (189 words) - 20:43, 25 August 2022
• 2002 AMC 10A Problems/Problem 9
  ...h>1001C - 2002A = 4004</math>, and <math>1001B + 3003A = 5005</math>. What is the average of A, B, and C? <math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E) }\text{Not
  2 KB (283 words) - 17:48, 25 February 2023
• 2002 AMC 12A Problems/Problem 7
  ...rcle A is equal in length to a <math>30^\circ</math> arc of circle B. What is the ratio of circle A's area and circle B's area? ...4/9 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 5/6 \qquad \textbf{(D)}\ 3/2 \qquad \textbf{(E)}\ 9/4</math>
  3 KB (492 words) - 14:46, 31 January 2024
• 2002 AMC 12A Problems/Problem 11
  ...If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work ...2</math>. From <math>d=rt</math>, we find that <math>r</math>, our answer, is <math>\boxed{\textbf{(B) }48 }</math>.
  2 KB (396 words) - 00:19, 1 February 2024
• 2002 AMC 12A Problems/Problem 9
  ...up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? ...early it is better to pick the second possibility. Thus we will have <math>3</math> disks, each with one <math>0.8</math> MB file and one <math>0.4</mat
  2 KB (355 words) - 20:42, 2 October 2023

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