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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 (749 words) - 19:44, 25 April 2024
• 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
• 1994 USAMO Problems
  Let <math>\, k_1 < k_2 < k_3 < \cdots \,</math> be positive integers, no two consecutive, and let <math>\, s_m = k_1 + k_2 + \cdots + k_m \,</math> ...e of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modi
  2 KB (391 words) - 07:58, 19 July 2016
• 2007 AMC 12B Problems/Problem 24
  Let <math>u = \frac{a}{b}</math>. Then the given equation becomes <math>u + \frac{14}{9u} = \frac{9u^2 + 14}{9u}</math>. ...ring the denominator and simplifying, we get a quadratic in terms of <math>u</math>:
  9 KB (1,522 words) - 22:46, 12 May 2022
• 1984 IMO Problems/Problem 4
  ...ath>, and with all conditions satisfied. We shall prove that there exists no point <math>T</math> on <math>CD</math> such that <math>T</math> is a midpo ...the circle with diameter <math>CD</math> touching <math>AB</math> at <math>U</math>(as well as the circle with diameter <math>AB</math> touching <math>C
  4 KB (750 words) - 23:49, 29 January 2021
• 1991 IMO Problems/Problem 4
  ...ge joining <math>v_i</math> and <math>v_{i+1}</math> with the number <math>u+i-1</math>. ...that vertex will be sorrounded by edges whose greates common factor is 1, no matter how the other edges sorrounding it are labelled. Notice that from o
  4 KB (668 words) - 17:45, 30 January 2021
• 2002 AMC 10B Problems
  ...<math>A</math>, <math>M</math>, <math>O</math>, <math>S</math>, and <math>U</math>, we can form five-letter "words". If these "words" are arranged in a ...ath> does the equation <math>\frac{x-1}{x-2} = \frac{x-k}{x-6}</math> have no solution for <math>x</math>?
  10 KB (1,540 words) - 22:53, 19 December 2023
• 1974 USAMO Problems
  ...her and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: ...ngle ADB = \angle BDC = \angle CDA = 120^\circ</math>. Prove that <math>x=u+v+w</math>.
  3 KB (427 words) - 18:55, 3 July 2013
• 2007 AMC 10B Problems/Problem 25
  ...th>(a,b)</math> are there such that <math>a</math> and <math>b</math> have no common factors greater than <math>1</math> and: Let <math>u = \frac{a}{b}</math>. Then the given equation becomes <math>u + \frac{14}{9u} = \frac{9u^2 + 14}{9u}</math>.
  8 KB (1,425 words) - 23:05, 21 June 2023
• 2010 AIME I Problems/Problem 13
  ...ath>, and segment <math>CD</math> at distinct points <math>N</math>, <math>U</math>, and <math>T</math>, respectively. Line <math>\ell</math> divides re label("$U$",(87.13819,5.00321),NE/2);
  10 KB (1,418 words) - 23:05, 20 October 2021
• 2011 AIME I Problems/Problem 8
  ...math> on <math> \overline{CY} </math>, and points <math>Z</math> and <math>U</math> are on <math> \overline{AB} </math> with <math>Z</math> on <math> \o pair[] A, B, C, U, V, W, X, Y, Z;
  6 KB (1,077 words) - 21:47, 12 April 2022
• 1999 AMC 8 Problems
  label("U",(10,-2),S); The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is <math>11/4</math>. To the nearest who
  17 KB (2,394 words) - 19:51, 8 May 2023
• 2011 USAMO Problems/Problem 3
  In hexagon <math>ABCDEF</math>, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy <math>\ang ...}</cmath> <cmath>\vec{w} = \vec{CD} + \vec{FA}</cmath> Clearly, <math>\vec{u}+\vec{v}+\vec{w}=\textbf{0}</math>.
  11 KB (1,925 words) - 12:07, 31 August 2023
• Mock Amc 12
  ...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'''
  18 KB (2,206 words) - 19:41, 24 December 2020
• 2000 AMC 8 Problems/Problem 11
  Casework by the units digit <math>u</math> will help organize the answer. <math>u=0</math> gives no solutions, since no real numbers are divisible by <math>0</math>
  2 KB (382 words) - 15:59, 1 January 2024
• 2011 IMO Problems/Problem 2
  ...S}</math> be a finite set of at least two points in the plane. Assume that no three points of <math>\mathcal S</math> are collinear. A ''windmill'' is a (By u/basuboss on reddit)
  3 KB (577 words) - 23:19, 26 November 2023
• 1950 AHSME Problems
  ...igit number whose tens' digit is <math>t</math>, and units' digit is <math>u</math>, the new number is: ...extbf{(B)}\ 100t+10u+1\qquad\textbf{(C)}\ 1000t+10u+1\qquad\textbf{(D)}\ t+u+1\qquad\\ \textbf{(E)}\ \text{None of these answers} </math>
  22 KB (3,306 words) - 19:50, 3 May 2023
• BBCode:Tutorial
  ...utton is for underlining text. When you click on it, the symbols [u] and [/u] appear. Then you can type in whatever you want to be underlined: ...tags (not implemented on the top BBCode bar in the forums). There will be no grey box that there is when using [code][/code], and it will not automatica
  10 KB (1,680 words) - 19:26, 5 April 2023
• 2002 Romanian NMO Problems
  has no positive integer solutions. ...tarrow\mathbb{R}</math> be a function that has limits at any point and has no local extrema. Show that:
  10 KB (1,695 words) - 10:03, 10 May 2012
• 1955 AHSME Problems
  <math> \textbf{(A)}\ \text{no real values of }x\qquad\textbf{(B)}\ \text{either }x=1\text{ or }x=2\qquad\ The negation of the statement "No slow learners attend this school" is:
  22 KB (3,509 words) - 21:29, 31 December 2023
• 2020 Mock Combo AMC 10
  How many permutations of the string <math>AABBCC</math> are there such that no two identical letters are adjacent to each other? ...positive integers <math>s, t, u, v, w, x, y, z</math> where <math>s > t > u > v > w > x > y > z</math>.
  15 KB (2,452 words) - 03:03, 4 July 2020
• 2013 AMC 12A Problems/Problem 25
  ...part of a radical of a complex number: <math>\sqrt{u}</math>, where <math>u = v+wi = r e^{i\theta}</math>. <math>\operatorname{Im}(\sqrt{u}) = \operatorname{Im}(\pm \sqrt{r} e^{i\theta/2}) = \pm \sqrt{r} \sin(\thet
  4 KB (766 words) - 14:22, 8 October 2023
• 2013 AMC 12B Problems
  ...rn a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the e for (real u = 0; u < len; u += step){
  16 KB (2,459 words) - 02:46, 30 January 2021
• 2013 USAJMO Problems/Problem 6
  ...x - 9</math>. It is easily verified that <math>x^3 - 8x + 9 = 0</math> has no solution in positive numbers greater than 1. Thus, <math>\sqrt{x + xyz} \ge ...e can let <math>u = \sqrt{(y-1)(z-1)}</math>! Thus, <cmath>u^2 - 2u + 2 = (u-1)^2 + 1 > 0</cmath> and the claim holds for x = 1.
  3 KB (517 words) - 20:02, 30 April 2014
• Northeastern WOOTers Mock AIME I Problems
  ...e{uv.xy}+\overline{xy.uv} </cmath> is an integer? As an example, if <math>(u,v,x,y)=(0,1,2,3)</math>, then we have <math> N = 1.23 + 23.01 = 24.24 </mat ...re is no overlap. He continues to place T-Polyominos randomly until he can no longer do so. Let the probability that he will cover the entire board be <m
  8 KB (1,336 words) - 09:10, 30 May 2020
• 1983 IMO Problems/Problem 4
  ...olor on the respective side must have the opposite color if we are to have no monochromatic right-angled triangles. ...the points <math>U,V,T</math> must have different colors if we are to have no monochromatic right-angled triangle, and this is impossible (we only have t
  2 KB (373 words) - 23:40, 29 January 2021
• 2013 IMO Problems/Problem 2
  ...sts of <math>2013</math> red points and <math>2014</math> blue points, and no three of the points of the configuration are collinear. By drawing some lin *no line passes through any point of the configuration;
  3 KB (643 words) - 01:31, 19 November 2023
• 1973 AHSME Problems
  has no solution if and only if <math>n</math> is equal to ...alf at <math>v</math> miles per hour. Car B travels half the time at <math>u</math> miles per hour and half at <math>v</math> miles per hour. The averag
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1988 AHSME Problems
  ...=(4,-sqrt(5)), Z=(4+2/sqrt(5),-sqrt(5)), W=(4+2/sqrt(5),sqrt(5)), T=(4,0), U=(4+2/sqrt(5),-4/sqrt(5)), V=(4+2/sqrt(5),1/sqrt(5)); draw(X--Y--Z--W--X^^T--V--X^^Y--U);
  17 KB (2,535 words) - 13:45, 19 February 2020
• 1972 AHSME Problems
  \textbf{(D) }\text{no }y\qquad real t=pi/8;real u=7*pi/12;real v=13*pi/12;
  17 KB (2,732 words) - 13:54, 20 February 2020
• 2007 iTest Problems/Problem 24
  \text{(U) }20 \quad ...2</math> in <math>N</math> is at least <math>1</math>, but since there are no powers of <math>2</math> in <math>2007</math>, the exponent of <math>2</mat
  2 KB (340 words) - 19:49, 30 June 2018
• 1964 AHSME Problems
  ...uotient is <math>u</math> and the remainder is <math>v</math>, where <math>u</math> and <math>v</math> are integers. \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\
  19 KB (2,907 words) - 14:16, 20 February 2020

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