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Physics
<math>\delta x'(t)=v(t)</math> where <math>\delta x(t)</math> is displacement at time <math>t</math>, <math>v</math> is velocit

9 KB (1,355 words) - 06:29, 29 September 2021
2006 USAMO Problems
...ace x\rbrace=x-\lfloor x\rfloor</math> denote the fractional part of <math>x</math>. ...h> and <math>BC</math>, respectively, such that <math>AE/ED=BF/FC</math>. Ray <math>FE</math> meets rays <math>BA</math> and <math>CD</math> at <math>S</

3 KB (520 words) - 08:24, 14 May 2021
1983 AIME Problems/Problem 15
...ath>P</math> and a line <math>\ell,</math> what is the set of points <math>X</math> such that the midpoint of <math>PX</math> lies on line <math>\ell</m pair X = .6*M1 + .4*M2, M = (P+X)/2;

20 KB (3,497 words) - 14:37, 27 May 2024
2014 USAJMO Problems
...2}{x}+f(yf(y))</cmath> for all <math>x, y \in \mathbb{Z}</math> with <math>x \neq 0</math>. ...line <math>MN</math> and line <math>MP</math>, respectively, and let <math>X</math> be the midpoint of arc <math>BAC</math> of <math>\Gamma</math>.

3 KB (600 words) - 15:42, 5 August 2023
2006 USAMO Problems/Problem 6
...and <math>BC</math>, respectively, such that <math>AE/ED = BF/FC</math>. Ray <math>FE</math> meets rays <math>BA</math> and <math>CD</math> at <math>S</ ...ion of the circumcircles of <math>SAE</math> and <math>SBF</math> be <math>X</math>, and let the intersection of the circumcircles of <math>TCF</math> a

5 KB (986 words) - 21:46, 18 May 2015
Congruent (geometry)
...ine. Then in the plane <math>\alpha'</math> there is one and only one half-ray <math>k'</math> such that the angle <math>(h,k)</math>, or <math>(k,h)</mat pair X,Y,Z;

10 KB (1,655 words) - 15:56, 17 September 2024
Euler line
Point <math>F</math> lyes on ray <math>BC, \frac {\vec {BF}}{\vec {CF}} = \frac {\tan \alpha – \tan \gamma ...BD}{\vec DA}, m = \frac {\vec CE}{\vec EA}, p = \frac {\vec CX}{\vec XB}, X \in BC, q = \frac {AY}{YX}, Y \in AX.</math>

59 KB (10,203 words) - 03:47, 30 August 2023
1985 IMO Problems/Problem 1
...notation of the previous solution. Let <math>X</math> be the point on the ray <math>AD</math> such that <math>AX = AO</math>. We note that <math>OF = OG

4 KB (684 words) - 06:28, 3 October 2021
1995 AHSME Problems
If <math>\sqrt {2 + \sqrt {x}} = 3</math>, then <math>x =</math> ...ulting cube, which of the lettered faces is opposite the face marked <math>x</math>?

17 KB (2,387 words) - 21:44, 26 May 2021
Isogonal conjugate
...h>(x,y)</math> into a point with coordinates <math>(\frac{1}{x}, \frac {y}{x}),</math> is projective. ...efine the equations <math>y = \pm kx,</math> and the lines <math>(\frac{1}{x}, \pm k)</math> symmetrical with respect to the line <math>\ell</math> beco

54 KB (9,416 words) - 07:40, 18 April 2024
Mock AIME 5 Pre 2005 Problems
...eronian'' triangle <math>COP</math> has sides <math>29,6,</math> and <math>x</math>. If the distance from its incenter to the circumcenter is exprssed a ...\overline {XY}</math>, dividing <math>XV:VY = 1:2</math>. It is found that ray VT intersects <math>\overline {RS}</math> at Z. Find

6 KB (909 words) - 06:27, 12 October 2022
2008 USAMO Problems/Problem 2
...ors of <math>\overline{AB}</math> and <math>\overline{AC}</math> intersect ray <math>AM</math> in points <math>D</math> and <math>E</math> respectively, a pair B=(0,0),C=(5,0),A=(1,4); /* A.x > C.x/2 */

20 KB (3,565 words) - 10:54, 1 May 2024
2008 USAMO Problems
...ors of <math>\overline{AB}</math> and <math>\overline{AC}</math> intersect ray <math>AM</math> in points <math>D</math> and <math>E</math> respectively, a ...be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y)</math> with integer coordinates such that

4 KB (674 words) - 20:48, 12 August 2014
2008 IMO Problems
Let <math>x, y, z\neq 1</math> be three real numbers, such that <math>xyz = 1</math> <math>\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\le

4 KB (613 words) - 23:30, 17 February 2021
1993 IMO Problems/Problem 4
...math>C</math> be given points in the plane. Prove that for any point <math>X</math> in the plane, ...math>ABC</math> have distance exceeding <math>a</math>. Thus, since <math>X</math> is within <math>ABC</math>, the longest side of any of the triangles

4 KB (854 words) - 10:30, 21 November 2023
2001 USAMO Problems/Problem 4
Set <math>A(0,0)</math>, <math>B(1,0)</math>, <math>C(x,y)</math>, <math>P(p,q)</math>. <cmath>(p-1)^2 + q^2 + (p-x)^2 + (q-y)^2 + 1 + x^2 + y^2 \geq p^2 + q^2 + (x-1)^2 + y^2 </cmath>

8 KB (1,470 words) - 21:24, 18 June 2022
1963 IMO Problems/Problem 2
...mferences of all circles that have a diameter <math>AX</math>, where <math>X</math> is some point on <math>BC</math>. As <math>X</math> moves from <math>B</math> to <math>C</math>, the motion of the circl

2 KB (306 words) - 03:49, 19 February 2019
1995 USAMO Problems/Problem 3
...verline{AB}</math> respectively. Point <math>A_2</math> is located on the ray <math>OA_1</math> so that <math>\triangle OAA_1</math> is similar to <math> ...e OAC</math>, by lemma 1. Therefore, <math>\angle A_1AC = 90 - \angle B + x = \angle BAA_2</math>. Since <math>A_1</math> is the midpoint of <math>BC<

2 KB (364 words) - 00:42, 19 April 2024
2008 IMO Problems/Problem 6
...ega</math> tangent to ray <math>BA</math> beyond <math>A</math> and to the ray <math>BC</math> beyond <math>C</math>, which is also tangent to the lines < Use sin (90-x) = cos x and cos(90-x) = sinx

1 KB (283 words) - 00:15, 19 November 2023
2009 AIME II Problems/Problem 3
...h> is <math>x</math> and the slope of <math>\overline{BE}</math> is <math>-x/2</math>. They are perpendicular, so they multiply to <math>-1</math>, that <cmath>x\cdot-\frac{x}{2}=-1,</cmath>

4 KB (572 words) - 15:47, 2 August 2023

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