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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
2007 AMC 12B Problems/Problem 19
<math>V = \pi \left(\frac{3}{\pi}\right)^2\cdot 6\sin\theta</math>

1 KB (166 words) - 16:35, 15 February 2021
1985 AJHSME Problems
label("V",(1.5,.3),N); ...bf{(A)}\ \text{Z} \qquad \textbf{(B)}\ \text{U} \qquad \textbf{(C)}\ \text{V} \qquad \textbf{(D)}\ \ \text{W} \qquad \textbf{(E)}\ \text{Y}</math>

12 KB (1,670 words) - 17:42, 24 November 2021
Mock AIME 5 Pre 2005 Problems
...>. Y is on <math>\overline {ST}</math>, dividing <math>SY:YT = 2:1</math>. V is on <math>\overline {XY}</math>, dividing <math>XV:VY = 1:2</math>. It is

6 KB (909 words) - 07:27, 12 October 2022
1974 USAMO Problems/Problem 5
...le ADB = \angle BDC = \angle CDA = 120^\circ</math>. Prove that <math>x=u+v+w</math>. label("$v$",(B+D)/2,N);

7 KB (1,221 words) - 18:57, 3 July 2013
2005 USAMO Problems/Problem 3
...math> Substituting in <math>P</math> yields <cmath>-a^2p(1-p)+vp=0\implies v = a^2(1-p).</cmath> Substituting in <math>B_1</math> yields

6 KB (1,117 words) - 01:17, 11 October 2021
Cone
...me of a pyramid, <math>V = \frac{1}{3} \cdot B \cdot h</math>, where <math>V</math> is the volume, <math>B</math> is the area of the base and <math>h</m

7 KB (1,128 words) - 20:12, 27 September 2022
2008 USAMO Problems/Problem 6
...space]] of all such functions. Define the linear operator <math>A : V \to V</math> as <cmath>(Af)(v) = \sum_{v \sim w} f(v) - f(w)</cmath>

13 KB (2,414 words) - 14:37, 11 July 2016
2008 Mock ARML 1 Problems/Problem 3
pair v(int n){ return dir(n * 60); } ...0))--MP("B",v(1),N)--MP("C",v(2),N)--MP("D",v(3),SW)--MP("E",v(4))--MP("F",v(5))--cycle);

3 KB (425 words) - 22:32, 5 December 2020
Schrodinger Equation
...ing the Hamiltonian, usually of the form <math>-\frac{\hbar^2}{2m}\Delta + V</math> where <math>\Delta</math> is the relevant Laplace(-Beltrami) operato

417 bytes (69 words) - 14:32, 21 April 2018
Jean-Victor Poncelet
*Didion 1870. ''Notice sur la vie et les ouvrages du général J. V. Poncelet''

2 KB (253 words) - 11:41, 19 December 2018
1988 IMO Problems/Problem 1
...ble positions of the midpoints <math>U</math> of <math>AB</math> and <math>V</math> of <math>AC</math> as <math>A</math> varies?</li> We claim that all points <math>U,V</math> lie on a circle centered at the midpoint of <math>OP</math>, <math>M

3 KB (545 words) - 11:32, 30 January 2021
1997 PMWC Problems/Problem T1
...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);

1 KB (188 words) - 14:38, 20 April 2014
Category (category theory)
...nd there is a morphism <math>U \to V</math> if and only if <math>U \subset V</math>.

5 KB (792 words) - 19:01, 7 April 2012
1991 IMO Problems/Problem 4
...each step in the process we remove a number of vertices from the set <math>V</math> if their exists an edge sorrounding it which is labelled. Additiona ...of the vertices not in <math>V</math>. Notice that since <math>v_1,v_n\in V</math> the edge connecting <math>v_0</math> and <math>v_1</math> and the ed

4 KB (668 words) - 17:45, 30 January 2021
2001 IMO Problems/Problem 4
..., there exists a <math>v\in\{1,2,...,m!\}</math> such that <math>t\equiv f(v)\pmod{m!}</math>. So, let <math>t\equiv f(v_t)\pmod{m!}</math>. Consider

2 KB (416 words) - 20:01, 19 May 2023
Acceleration
...known to be constant, then <cmath>\textbf{a} = \frac{\textbf{v}_2 -\textbf{v}_1 }{t_2 - t_1}</cmath> Note that velocity is a vector, so the magnitudes c ...instance, <cmath>\textbf{a} = \lim_{h\to 0} \frac{v(t+h)-v(t)}{(t+h)-t} = v'(t)</cmath>

2 KB (277 words) - 14:04, 24 April 2022
2001 USAMO Problems/Problem 3
...h>b = 2\sin u</math> and <math>c = 2\sin v</math>, where <math>0^\circ < u,v < 90^\circ</math>. Then <cmath>a = 2(-\sin u\sin v + \cos u\cos v) = 2\cos (u + v),</cmath>

4 KB (807 words) - 10:45, 9 April 2023
Time dilation
<math>\dfrac{t_1}{\sqrt{1-v^2/c^2}}=t_2</math> <math>v</math> is the relative velocity the ovject is moving to the observer.

1 KB (261 words) - 00:19, 31 January 2021
Lorentz Factor
It is equivalent to <math>\gamma=\dfrac{1}{\sqrt{1-v^2/c^2}}</math>

226 bytes (34 words) - 11:23, 5 October 2012
1992 USAMO Problems/Problem 4
By another person ^v^

5 KB (807 words) - 18:37, 25 June 2021
1974 USAMO Problems
...le ADB = \angle BDC = \angle CDA = 120^\circ</math>. Prove that <math>x=u+v+w</math>. label("$v$",(B+D)/2,N);

3 KB (427 words) - 18:55, 3 July 2013
1962 IMO Problems/Problem 3
Answer: the rhombus CUVW, where U is the center of ABCD, V is the center of ABB'A, and W is the center of BCC'B'. ...x/2 + 1/2, x/2, 1/2) = x (1, 1/2, 1/2) + (1-x) (1/2, 0, 1/2) = x W + (1-x) V, so M traces out the line VW.

3 KB (581 words) - 02:00, 6 May 2023
1962 IMO Problems/Problem 7
...n the direction of <math>V{i}</math> to <math>V{j}</math> and beyond <math>V{j}</math> for <math>i=1,2,3,4</math>; <math>j=1,2,3,4</math>; and <math>i \ ...trahedron opposite of vertex <math>V_{4}</math> below <math>\Delta V{1}V{2}V{3}</math>

11 KB (1,928 words) - 20:52, 21 November 2023
2001 AMC 10 Problems
...and diagonal are the same. Five of these numbers are represented by <math> v </math>, <math> w </math>, <math> x </math>, <math> y </math>, and <math> z label("$v$",(0.5,2.5));

14 KB (1,983 words) - 16:25, 2 June 2022
1976 USAMO Problems/Problem 4
...ath>. Let the volume of the tetrahedron be <math>V</math>. Therefore <math>V=\frac{abc}{6}</math>. ...{6V}</math>, or <math>6V\leq \frac{S^3(\sqrt{2}-1)^3}{27}</math>, or <math>V\leq \frac{S^3(\sqrt{2}-1)^3}{162}</math>, with equality only when <math>a=b

2 KB (358 words) - 23:15, 18 July 2016
1985 AJHSME Problems/Problem 11
label("V",(1.5,.3),N); ...>\text{(A)}\ \text{Z} \qquad \text{(B)}\ \text{U} \qquad \text{(C)}\ \text{V} \qquad \text{(D)}\ \ \text{W} \qquad \text{(E)}\ \text{Y}</math>

2 KB (279 words) - 09:25, 10 January 2023
2009 AMC 12B Problems/Problem 20
<math>v = \frac{200}{b}</math> ...gain <math>b</math> more edges. So the total number of new edges is <math>v*b \Rightarrow \frac{200}{b}*b \Rightarrow 200</math>. It doesn't matter how

9 KB (1,567 words) - 13:43, 19 August 2023
2009 AIME I Problems/Problem 10
...of sittings is in the form <math>N*(5!)^3</math> because for each <math>M, V, E</math> sequence we have <math>5!</math> arrangements within the Ms, Vs, ...members must sit in cycles of M, V, E, but not necessarily with one M, one V, and one E in each cycle(for example, MMVVVE, MVVVEEE, MMMVVVEE all count a

4 KB (590 words) - 16:49, 27 September 2023
2009 AIME II Problems/Problem 9
...en <math>1</math> and <math>167</math>, inclusive, and for each such <math>v</math> we have exactly one valid <math>x</math>. Hence in this case there a

5 KB (845 words) - 15:45, 28 December 2020
2008 IMO Problems/Problem 6
V be the foot of the projection of K to AC.

1 KB (283 words) - 01:15, 19 November 2023

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