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• 1986 AIME Problems/Problem 11
  ...>C = \{1, (x+1), (x+1)^2, ..., (x+1)^{17} \}</math> be two bases for <math>V</math>. ...e polynomial given in the problem, and it is easy to see that <math>[ \vec{v} ]_B = \langle 1, -1, 1, -1, ... , 1, -1 \rangle.</math>
  6 KB (872 words) - 16:51, 9 June 2023
• 1988 AIME Problems/Problem 13
  ...(x)</math> are also roots of <math>f(x)</math>. Let these roots be <math>u,v</math>. We get the system If we multiply the first equation by <math>v^{16}</math> and the second by <math>u^{16}</math> we get <cmath>\begin{alig
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1988 AIME Problems/Problem 10
  ...lies on exactly one vertex of a square/hexagon/octagon, we have that <math>V = 12 \cdot 4 = 8 \cdot 6 = 6 \cdot 8 = 48</math>. ...h of its endpoints, the number of edges <math>E</math> is <math>\frac{3}{2}V = 72</math>.
  5 KB (811 words) - 19:10, 25 January 2021
• 1990 AIME Problems/Problem 14
  Finally, we substitute <math>h</math> into the volume equation to find <math>V = 6\sqrt{133}\left(\frac{99}{\sqrt{133}}\right) = \boxed{594}</math>. ...ave the base area as <math>18\sqrt {133}</math>. Thus, the volume is <math>V = \frac {1}{3}\cdot18\sqrt {133}\cdot\frac {99}{\sqrt {133}} = 6\cdot99 = 5
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1993 AIME Problems/Problem 12
  ...th>(u,v)</math> and <math>(p,q)</math>, then <math>u=2r-p</math> and <math>v=2s-q</math>. So we start with the point they gave us and work backwards. We
  4 KB (611 words) - 13:59, 15 July 2023
• 1993 AIME Problems/Problem 10
  ...<math>P</math> pentagonal faces meet. What is the value of <math>100P+10T+V</math>? ...ge). Thus, <math>E=60</math>. Finally, using Euler's formula we have <math>V=E-30=30</math>.
  4 KB (716 words) - 20:50, 17 April 2022
• 1994 AIME Problems/Problem 6
  ...(-20/sqrt(3),0)-2*u+i*u--(0,20)--(20/sqrt(3),0)+2*d-i*d;draw(shift(0,-2*i)*v);} ...(-20/sqrt(3),0)-2*u+i*u--(0,20)--(20/sqrt(3),0)+2*d-i*d);draw(shift(0,2*i)*v);}
  4 KB (721 words) - 16:14, 8 March 2021
• 1995 AIME Problems/Problem 12
  ...as <math>\vec{u}\cdot \vec{v} = \parallel \vec{u}\parallel \parallel \vec{v}\parallel \cos \theta</math>, we will be able to solve for <math>\cos \thet <cmath>\vec{v} = \overrightarrow{OB}\times \overrightarrow{OC} - \left|\begin{array}{ccc}
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1997 AIME Problems/Problem 14
  ...c{m}{n}</math> be the [[probability]] that <math>\sqrt{2+\sqrt{3}}\le\left|v+w\right|</math>, where <math>m</math> and <math>n</math> are [[relatively p Now, let <math>v</math> be the root corresponding to <math>m\theta=2m\pi/1997</math>, and le
  5 KB (874 words) - 22:30, 1 April 2022
• 1999 AIME Problems/Problem 15
  ...coordinates of the vertex of the resulting pyramid. Call this point <math>V</math>. Clearly, the height of the pyramid is <math>z</math>. The desired v ...= QC</math>. We then use distance formula to find the distances from <math>V</math> to each of the vertices of the medial triangle. We thus arrive at a
  7 KB (1,107 words) - 20:34, 27 January 2023
• 2000 AIME I Problems/Problem 8
  (Computational) The volume of a cone can be found by <math>V = \frac{\pi}{3}r^2h</math>. In the second container, if we let <math>h',r'< From the formula <math>V=\frac{\pi r^2h}{3}</math>, we can find that the volume of the container is
  4 KB (677 words) - 16:33, 30 December 2023
• 2000 AIME I Problems/Problem 2
  ...rea of [[pentagon]] <math>ABCDE</math> is <math>451</math>. Find <math>u + v</math>. D(D(MP("A\ (u,v)",A,(1,0)))--D(MP("B",B,N))--D(MP("C",C,N))--D(MP("D",D))--D(MP("E",E))--cy
  3 KB (434 words) - 22:43, 16 May 2021
• 2001 AIME I Problems/Problem 12
  ...ath>P</math> perpendicular to plane <math>ABC</math> can be found as <math>V=(A-C)\times(B-C)=\langle 8, 12, 24 \rangle</math> ...r each pyramid(base times height divided by 3) we have <math>\dfrac{rF}{3}=V</math>. The surface area of the pyramid is <math>\dfrac{6\cdot{4}+6\cdot{2}
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2014 USAJMO Problems
  ...line{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</mat
  3 KB (600 words) - 16:42, 5 August 2023
• 2003 AIME II Problems/Problem 5
  ...another identical wedge and sticking it to the existing one). Thus, <math>V=\dfrac{6^2\cdot 12\pi}{2}=216\pi</math>, so <math>n=\boxed{216}</math>. Part 1: https://www.youtube.com/watch?v=EzE53aPGbrQ&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=5
  1 KB (204 words) - 17:41, 30 July 2022
• 2001 AIME II Problems/Problem 15
  triple S=(1,0,0), T=(2,0,2), U=(8,6,8), V=(8,8,6), W=(2,2,0), X=(6,8,8); ...-U--V--W--cycle); draw((0,0,1)--T--U--X--(0,2,2)--cycle); draw((0,1,0)--W--V--X--(0,2,2)--cycle);
  4 KB (518 words) - 15:01, 31 December 2021
• 2001 AIME II Problems/Problem 7
  ...ine{UV}</math> with <math>U</math> on <math>\overline{PQ}</math> and <math>V</math> on <math>\overline{QR}</math> such that <math>\overline{UV}</math> i pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(60, 40), O1 = (30,30), O2 = (15, 75), O3 = (70, 10);
  7 KB (1,112 words) - 02:15, 26 December 2022
• Integration by parts
  <math>\int u\, dv=uv-\int v\,du</math> ...math>u</math> will show up as <math>du</math> and <math>dv</math> as <math>v</math> in the integral on the RHS, <math>u</math> should be chosen such tha
  1 KB (235 words) - 17:01, 11 March 2022
• 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
• Expected value
  https://www.youtube.com/watch?v=BBD66Q3KXuI ...enter connecting the midpoints of the two sides of the small triangle with V as an endpoint. Find, with proof, the expected value of the number of full
  4 KB (719 words) - 19:41, 25 November 2020
• 1977 Canadian MO Problems
  the vertex <math>V</math> to this path? MP("P",(-1,0),W);MP("V",(-.5,2.4),N);
  3 KB (560 words) - 19:23, 10 March 2015
• Derivative/Formulas
  | <math>\left(u(x)\times v(x)\right)'=u(x)v'(x)+u'(x)v(x)</math> | <math>\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}</math>
  3 KB (506 words) - 16:23, 11 March 2022
• P versus NP
  *Given a weighted, undirected graph <math>G = (V,E)</math> and two vertices <math>s, t \in E</math>, does there exist a path
  6 KB (1,104 words) - 15:11, 25 October 2017
• 2015 IMO Problems/Problem 6
  ...for which <cmath>\left\vert\sum_{j=m+1}^n(a_j-(v+1))\right\vert\le (T-v)\,v \le \left(\frac T 2\right)^2</cmath>for all integers <math>m</math> and <ma ...is:<cmath>\sum_{i=1}^v (T-v+i) - \sum_{i=1}^v i=\sum_{i=1}^v (T-v)=(T-v)\,v\;. \quad \blacksquare</cmath>
  4 KB (833 words) - 01:33, 31 December 2019
• Zero (constant)
  ...s had to make change on a purchase of LXIV dollars with bills marked L, X, V and I when handed XC dollars.
  2 KB (371 words) - 12:55, 21 June 2023
• Free group
  ...group over a set <math>I</math> and <math>uv = vu</math>, then <math>u^m = v^n</math>, for some [[integer]]s <math>m</math> and <math>n</math>.
  2 KB (454 words) - 17:54, 16 March 2012
• 2007 iTest Problems
  \text{(V) } 2007 \quad </math> <math>\text{(V) Ying} \quad
  33 KB (5,177 words) - 21:05, 4 February 2023
• 1981 IMO Problems/Problem 2
  ...ng subset. Hence <cmath>F(n,r)=\frac{1}{\binom{n}{r}} \sum_{v \in B} \deg (v)= \frac{n+1}{r+1}.</cmath>
  5 KB (879 words) - 11:18, 27 June 2020
• Roman numerals
  5 - '''V''' ''(quinque)''
  865 bytes (140 words) - 13:58, 24 March 2019
• 2002 IMO Shortlist Problems/N3
  ..., then the [[greatest common factor]] of <math>2^u + 1 </math> and <math>2^v + 1 </math> is 3. ...ath>t </math>, contradicting our assumption that <math>u </math> and <math>v </math> are relatively prime.
  10 KB (1,739 words) - 06:38, 12 November 2019
• 2020 AMC 10A Problems
  label("V", (2, 6), NE);
  13 KB (1,968 words) - 18:32, 29 February 2024
• 2007 AIME II Problems/Problem 15
  ...ale the triangle with the circumradius by a [[line]]ar scale factor, <math>v</math>. :<math>\frac{65}{8}v=8u</math>
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2007 AMC 12A Problems/Problem 20
  ...cut off corners is a [[pyramid]], whose volume can be calculated by <math>V = \frac 13 Bh</math>. Use the base as one of the three [[congruency (geomet
  2 KB (317 words) - 10:10, 16 September 2022
• 1989 USAMO Problems/Problem 2
  Call it a vertex set <math>V</math>. <math>10</math> vertices remain outside <math>V</math> and each has to be attached to at least one edge.
  3 KB (438 words) - 01:19, 27 December 2023
• 1989 USAMO Problems
  Let <math>u</math> and <math>v</math> be real numbers such that (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.
  2 KB (326 words) - 18:52, 18 July 2016
• 2007 USAMO Problems/Problem 4
  Chose a vertex, <math>v</math>, for which <math>a</math> is minimal (i.e. out of all maximal elemen ...h>. So then in this special case, we have <math>D = 4</math>, and so <math>V = 2006 \times 4+1</math> (a possible configuration of this size that works
  10 KB (1,878 words) - 14:56, 30 June 2021
• Octahedron
  * The volume <math>V</math> of a regular octahedron with side length <math>a</math> is <math>\fr
  1 KB (155 words) - 12:19, 21 July 2023
• 1989 USAMO Problems/Problem 5
  Let <math>u</math> and <math>v</math> be real numbers such that <cmath> (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. </cmath>
  2 KB (300 words) - 19:16, 18 July 2016
• 2004 AMC 12A Problems/Problem 11
  ...bstituting yields: <math>20n+25=21(n+1),</math> so <math>n=4</math>, <math>v = 80.</math> Then, we see that the only way Paula can satisfy this rule is
  2 KB (310 words) - 17:39, 31 December 2022
• 2004 AMC 12A Problems/Problem 6
  Let <math>U=2\cdot 2004^{2005}</math>, <math>V=2004^{2005}</math>, <math>W=2003\cdot 2004^{2004}</math>, <math>X=2\cdot 20 <math>\mathrm {(A)} U-V \qquad \mathrm {(B)} V-W \qquad \mathrm {(C)} W-X \qquad \mathrm {(D)} X-Y \qquad \mathrm {(E)} Y-
  1 KB (139 words) - 02:10, 30 December 2020
• Binet's Formula
  ...d <math>p= \frac{1-\sqrt{5}}{2}.</math> We get that <cmath>F_n = \lambda_1 v^n + \lambda_2 p^n.</cmath>Plugging in our initial conditions, we get <cmath>0 = \lambda_1 + \lambda_2 </cmath> <cmath> 1= \lambda_1 v + \lambda_2 p </cmath>
  3 KB (550 words) - 16:12, 24 February 2024
• Celsius
  ...cncn vmnsnfdv jcd vndlfencm savdnm vs,czfvndm ,cxafkvsnsm dvhbfvm c,cdfemw v,dsnekmc xvzmvsdv vndnf cnvdjnf dfnwfd fwe vndkbfwenj sdbfwenj fbhkfenlkjs
  4 KB (357 words) - 22:40, 17 April 2024
• Mass
  ...math>, where <math>M</math> is momentum, <math>m</math> is mass, and <math>v</math> is [[velocity]].
  1 KB (188 words) - 22:44, 10 October 2013
• Graph (graph theory)
  ...may be defined using ordered pairs from the [[product set]] <math>V \times V</math>. ...ay the edges <math>e</math> and <math>f</math> are ''coincident'' at <math>v</math>.
  8 KB (1,428 words) - 10:26, 27 August 2020
• 2000 AMC 12 Problems/Problem 13
  https://www.youtube.com/watch?v=OT42J21ZNC8&feature=youtu.be https://www.youtube.com/watch?v=7QU8OlnljHw ~David
  6 KB (996 words) - 18:14, 29 June 2023
• 1997 PMWC Problems
  ...uilateral triangle with sides of length three units. <math>U</math>, <math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> label("$V$",(-1/3,sqrt(3)/6),NW);
  15 KB (2,057 words) - 19:13, 10 March 2015
• 1951 AHSME Problems
  ...he area <math>(A)</math> of the sail and the square of the velocity <math>(V)</math> of the wind. The pressure on a square foot is <math>1</math> pound
  23 KB (3,641 words) - 22:23, 3 November 2023
• 2002 AMC 12B Problems/Problem 5
  ...s of a [[pentagon]]. Suppose that <math>v < w < x < y < z</math> and <math>v, w, x, y, </math> and <math>z</math> form an [[arithmetic sequence]]. Find ...ath> triangles) is <math>3 \cdot 180 = 540^{\circ}</math>. If we let <math>v = x - 2d, w = x - d, y = x + d, z = x+2d</math>, it follows that
  2 KB (263 words) - 05:58, 8 August 2023
• 2003 AMC 12B Problems/Problem 17
  ...that <cmath>2 \cdot \textbf{(2)} + \textbf{(1)} = 5u + 5v = 3 \implies u + v = \frac{3}{5} \implies c = \boxed{\textbf{(D)}~\frac{3}{5}}</cmath>.
  2 KB (329 words) - 13:49, 4 April 2024
• Isogonal conjugate
  ...nes. Let coordinates of images of points be <cmath>X(-a, 0), X'(a,u), Y(-b,v), Y'(b,w).</cmath> ...f a straight line <math>XY</math> is <math>\frac{x + a}{a - b} = \frac {y}{v}.</math>
  54 KB (9,416 words) - 08:40, 18 April 2024

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