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• Cauchy-Schwarz Inequality
  ...bf{u}+\gamma \mathbf{v})</math> and <math>\mathbf{q} = \frac{1}{2}(\mathbf{u}-\gamma \mathbf{v})</math>. Importantly, we have ...le \mathbf{u},\gamma \mathbf{v}\rangle + \langle \gamma \mathbf{v},\mathbf{u}\rangle - \gamma \bar{\gamma}||\mathbf{v}||^2)
  13 KB (2,048 words) - 15:28, 22 February 2024
• Constructive counting
  ...that each entry is between <math>2</math> and <math>9</math> inclusive and no two consecutive entries are equal?'' ...luding <math>2</math> and <math>9</math>; the key restriction here is that no two boxes right next to each other can have the same number.
  12 KB (1,896 words) - 23:55, 27 December 2023
• Fundamental Theorem of Calculus
  ...me <math>{t}</math> is <math>e^{t^2}</math> meters/second. (Yes, probably no object really moves this way, but just pretend.) ''Approximately'' how far <cmath>L(f, P) \le g(b) - g(a) \le U (f, P)</cmath>
  11 KB (2,082 words) - 15:23, 2 January 2022
• Mock AMC
  ...ular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year, but ! scope="row" | '''Mock AMC U'''
  51 KB (6,175 words) - 20:58, 6 December 2023
• 2004 AMC 12A Problems
  ...at-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters? Let <math>U=2\cdot 2004^{2005}</math>, <math>V=2004^{2005}</math>, <math>W=2003\cdot 20
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2005 AIME I Problems/Problem 13
  # There are no right angle turns in the particle's path. ...mber of diagonals. Let <math>R</math> represent a move to the right, <math>U</math> represent a move upwards, and <math>D</math> to be a move that is di
  5 KB (897 words) - 00:21, 29 July 2022
• 2000 AIME I Problems
  Find the least positive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two posit ...s. The area of pentagon <math>ABCDE</math> is <math>451</math>. Find <math>u + v</math>.
  7 KB (1,204 words) - 03:40, 4 January 2023
• 1983 AIME Problems/Problem 15
  ...ath>3</math>. By considering the graph of <math>y=f(x)</math>, which is a "U-shaped" parabola, it is now evident that <math>f(-3) > 0</math> and <math>f === Solution 7 (No Trig) ===
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1988 AIME Problems/Problem 15
  ...ion, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.) <math>\textbf{Case 1:}</math> Since letter 9 arrived before lunch, no further letters will arrive, and the number of possible orders is simply th
  7 KB (1,186 words) - 10:16, 4 June 2023
• 1988 AIME Problems/Problem 13
  ...ath>c_{i}</math> is some coefficient. However, since <math>F(x)</math> has no <math>x</math> term, it must be true that <math>c_{15} = 1</math>. ...>g(x)</math> are also roots of <math>f(x)</math>. Let these roots be <math>u,v</math>. We get the system
  10 KB (1,585 words) - 03:58, 1 May 2023
• 2014 USAJMO Problems
  ...which <math>A</math> cannot win in a finite number of moves, or prove that no such minimum value exists. ...erline{CA}</math> and <math>\overline{AB}</math>, respectively. Let <math>U,V</math> be the intersections of line <math>EF</math> with line <math>MN</m
  3 KB (600 words) - 16:42, 5 August 2023
• 2002 AIME I Problems/Problem 9
  *Ulysses starts with the third picket and paints every <math>u</math> th picket. ...itive integer <math>100h+10t+u</math> paintable when the triple <math>(h,t,u)</math> of positive integers results in every picket being painted exactly
  4 KB (750 words) - 21:09, 23 December 2021
• Holomorphic function
  Specifically, let <math>u, v : \mathbb{R \times R \to R}</math> be definted <cmath> u(x,y) = \text{Re}\,f(x+iy), \qquad v(x,y) = \text{Im}\,f(x+iy) . </cmath>
  9 KB (1,537 words) - 21:04, 26 July 2017
• Mock AIME 1 2006-2007 Problems/Problem 13
  ...se variables, which does uniquely determine these variables - but there is no obvious way of computing them. We will show a different solution. ...th>d_2</math> one can always uniquely determine the coefficients <math>s,t,u</math>.
  3 KB (568 words) - 15:50, 3 April 2012
• 1977 Canadian MO Problems
  ...ath>f(x)=x^2+x,</math> prove that the equation <math>4f(a)=f(b)</math> has no solutions in positive integers <math>a</math> and <math>b.</math> Let <math>0<u<1</math> and define
  3 KB (560 words) - 19:23, 10 March 2015
• 2007 iTest Problems
  \text{(L) There is no such value of } r\qquad</math> \text{(U) }\frac{-2007}{2^{14}}\qquad</math>
  33 KB (5,177 words) - 21:05, 4 February 2023
• 2007 AMC 12A Problems/Problem 22
  Case 1: <math>196u \Longrightarrow u = 9</math>. Easy to directly disprove. ...S(n)) = 8+u</math> if <math>u \le 2</math> and <math>S(S(n)) = 2 + (u-3) = u-1</math> otherwise.
  15 KB (2,558 words) - 19:33, 4 February 2024
• 1960 IMO Problems/Problem 6
  ...h>u = \sqrt{x^2+1}</math>. Since <math>x > 0</math>, it follows that <math>u > 1</math>. We now have: <cmath>W_1 = (u + 1)^3</cmath>
  7 KB (1,214 words) - 18:49, 29 January 2018
• Graph (graph theory)
  ...simple, loopless'' graphs: there is at most one edge joining two vertices, no edge may join a vertex to itself, and the edges are not directed. For grap ...<math>v</math> is ''isolated'' if <math>d(v) = 0</math>, i.e. if there are no edges incident to <math>v</math>.
  8 KB (1,428 words) - 10:26, 27 August 2020
• 1997 PMWC Problems
  ...</math> be an equilateral triangle with sides of length three units. <math>U</math>, <math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and label("$U$",(-1/6,sqrt(3)/3),NW);
  15 KB (2,057 words) - 19:13, 10 March 2015

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