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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) - 07:29, 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) - 09:24, 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;

19 KB (3,221 words) - 01:05, 7 February 2023
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) - 16:42, 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) - 22:46, 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) - 21:43, 24 March 2022
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) - 04:47, 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) - 07:28, 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) - 22:44, 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) - 08:40, 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) - 07:27, 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) - 11:54, 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) - 21:48, 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) - 00:30, 18 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) - 11:30, 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) - 22:24, 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) - 04:49, 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) - 01:42, 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) - 01:15, 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) - 16:47, 2 August 2023
Radical axis
First, we tackle part (b). Suppose the circles intersect at points <math>X</math> and <math>Y</math> and point P lies on <math>XY</math>. Then by Theo ...on <math>XY</math>. In particular, line <math>PY</math> does not intersect X. Then <math>PY</math> intersects circles <math>\omega_1</math> and <math>\o

10 KB (1,797 words) - 02:05, 24 October 2023
2010 AIME II Problems
...yz) + (\log_{10}y) (\log_{10}z) = 468</math>. Find <math>\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}</math>. ...ositive integer <math>n</math> with the property that the polynomial <math>x^4 - nx + 63</math> can be written as a product of two nonconstant polynomia

8 KB (1,246 words) - 21:58, 10 August 2020
2010 AIME II Problems/Problem 15
...ircumcircles of <math>\triangle AMN</math> and <math>\triangle ADE</math>. Ray <math>AP</math> meets <math>BC</math> at <math>Q</math>. The ratio <math>\f ...>f(X)=\text{Pow}_{(AMN)}(X)-\text{Pow}_{(ADE)}(X)</cmath> for points <math>X</math> in the plane. Then <math>f</math> is linear, so <math>\frac{BQ}{CQ}=

9 KB (1,523 words) - 15:24, 21 November 2023
Telescope
...types of telescopes have been invented such as the X-Ray telescope, Gamma-ray telescope, and so forth.

625 bytes (95 words) - 16:36, 27 June 2023
2011 AMC 12A Problems
...ht|+\left|x-y\right|=2</math>. What is the maximum possible value of <math>x^2-6x+y^2</math>? Let <math>f(x)=ax^2+bx+c</math>, where <math>a</math>, <math>b</math>, and <math>c</math>

13 KB (1,994 words) - 13:52, 3 July 2021
2011 AMC 12A Problems/Problem 22
...w many points are <math>100</math>-ray partitional but not <math>60</math>-ray partitional? ...l vary. We start by finding the corner-most point that is <math>100</math>-ray partitional (let this point be the bottom-left-most point).

11 KB (1,818 words) - 17:38, 6 September 2021
2011 AMC 10A Problems
...16 + \cdots + 102. \end{eqnarray*} </cmath> What is the value of <math>Y - X</math>? <math>\textbf{(A)}\:(x+7)^2=0</math>

13 KB (1,903 words) - 18:09, 19 April 2021
1993 AHSME Problems
...circ</math>" by <math>x\circ y=4x-3y+xy</math>, for all real numbers <math>x</math> and <math>y</math>. For how many real numbers <math>y</math> does <m ...f <math>a^b</math> and <math>x^b</math> where <math>x>0</math>, then <math>x=</math>

20 KB (2,814 words) - 08:15, 27 June 2021
2011 AIME I Problems
...n the new coordinate system, point <math>A</math> is on the positive <math>x</math>-axis, and point <math>B</math> is on the positive <math>y</math>-axi ...h>X</math> and <math>Y</math> are on <math>\overline{BC}</math> with <math>X</math> on <math>\overline{CY}</math>, and points <math>Z</math> and <math>U

10 KB (1,634 words) - 22:21, 28 December 2023
2011 AIME I Problems/Problem 14
...\overline{A_7 A_8}</math>, respectively. For <math>i = 1, 3, 5, 7</math>, ray <math>R_i</math> is constructed from <math>M_i</math> towards the interior pair A,B,C,D,E,F,G,H,M,N,O,O2,P,W,X,Y,Z;

8 KB (1,344 words) - 18:39, 9 February 2023
2011 USAJMO Problems/Problem 3
...that all the points <math>P=(a,a^2)</math> belong to the parabola <math>y=x^2</math> which we will denote <math>p</math>. This parabola has a focus <ma ...th> has equation <math>y=2a(x-a)+a^2</math> and does not intersect <math>y=x^2</math> at any point besides <math>P</math>. Now for any point <math>P</ma

15 KB (2,593 words) - 13:37, 29 January 2021
2010 PUMaC Problems
...\lfloor x\rfloor</math> is the largest integer less than or equal to <math>x</math>. ...<math>S</math> be the sum of all real <math>x</math> such that <math>4^x = x^4</math>. Find the nearest integer to <math>S</math>.

22 KB (3,694 words) - 23:58, 3 June 2022
2012 AMC 10A Problems/Problem 11
.../math>, respectively. A line externally tangent to both circles intersects ray <math>AB</math> at point <math>C</math>. What is <math>BC</math>? label("$x$",(C--B),N);

2 KB (291 words) - 18:41, 22 April 2024
2012 AMC 10A Problems
.../math>, respectively. A line externally tangent to both circles intersects ray <math>AB</math> at point <math>C</math>. What is <math>BC</math>? ...some positive integer <math>n</math>. The probability that no two of <math>x</math>, <math>y</math>, and <math>z</math> are within 1 unit of each other

13 KB (1,994 words) - 01:31, 22 December 2023
2006 SMT/General Problems
...f the region defined by <math> x^2+y^2\le\pi^2 </math> and <math> y\ge\sin x </math>. ...terms of <math> a, b, x, </math> and <math> y </math>? Assume that <math> x, y, a, </math> and <math> b </math> are positive.

10 KB (1,477 words) - 16:02, 27 May 2012
2006 SMT/Team Problems
...h> is formed when <math> \triangle ABC </math> is rotated about the <math> x </math> axis, and cone <math> N </math> is formed when <math> \triangle ABC Let <math> x+y=a </math> and <math> xy=b </math>. The expression <math> x^6+y^6 </math> can be written as a polynomial in terms of <math> a </math> a

6 KB (910 words) - 17:32, 27 May 2012
2012 IMO Problems
to the ray <math>AB</math> beyond <math>B</math>, and to the ray <math>AC</math> beyond <math>C</math>.) ...cified in some previous question), and asking <math>A</math> whether <math>x</math> belongs to <math>S</math>. Player <math>B</math> may ask as many suc

5 KB (834 words) - 11:00, 24 February 2021
2013 AMC 12B Problems
...line, and Tom's car averages <math>10</math> miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rat ...satisfy the equation <math>x^2 + y^2 = 10x - 6y - 34</math>. What is <math>x + y</math>?

16 KB (2,459 words) - 02:46, 30 January 2021
2013 AMC 10B Problems
...allon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rat ...atisfy the equation <math>x^2 + y^2 = 10x - 6y - 34</math>. What is <math>x+y</math>?

12 KB (1,926 words) - 21:54, 6 October 2022
1981 AHSME Problems
If <math>\sqrt{x+2}=2</math>, then <math>(x+2)^{2}</math> equals For <math>x\neq0</math>, <math>\dfrac{1}{x}+\dfrac{1}{2x}+\dfrac{1}{3x}</math> equals

17 KB (2,633 words) - 15:44, 16 September 2023
2014 USAJMO Problems/Problem 6
...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>. (a) Prove that <math>I</math> lies on ray <math>CV</math>.

7 KB (1,273 words) - 18:17, 28 August 2021
2014 AIME II Problems
pair A,B,C,D,E,F,R,S,T,X,Y,Z; X = (12*4.472+ 22,24);

8 KB (1,410 words) - 00:04, 29 December 2021
2014 AIME I Problems
...d <math>2014</math>, respectively, and each graph has two positive integer x-intercepts. Find <math>h</math>. ...ath>, for <math>w \neq 0</math>, denotes the measure of the angle that the ray from <math>0</math> to <math>w</math> makes with the positive real axis in

9 KB (1,472 words) - 13:59, 30 November 2021
2014 AIME I Problems/Problem 7
...ath>, for <math>w \neq 0</math>, denotes the measure of the angle that the ray from <math>0</math> to <math>w</math> makes with the positive real axis in ...imaginary part of the above expression divided by the real part. Let <math>x = \alpha - \beta</math>. Then, we have that:

5 KB (782 words) - 20:25, 10 October 2023
1978 AHSME Problems
If <math>1-\frac{4}{x}+\frac{4}{x^2}=0</math>, then <math>\frac{2}{x}</math> equals ...and <math>y</math> such that <math>x = 1/y</math>, <math>\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)</math> equals

15 KB (2,432 words) - 01:06, 22 February 2024
2007 iTest Problems/Problem 16
<cmath>x^2+y^2=100</cmath> * For points on the axes, there are <math>10</math> points on each ray plus the origin, making a total of <math>41</math> points.

3 KB (420 words) - 20:28, 17 June 2018
2013 Canadian MO Problems
Determine all polynomials <math>P(x)</math> with real coefficients such that <math>(x+1)P(x-1)-(x-1)P(x)</math>

2 KB (381 words) - 13:45, 8 October 2014
2017 AMC 12A Problems/Problem 9
...ordinate plane such that two of the three quantities <math>3</math>, <math>x+2</math>, and <math>y-4</math> are equal and the third of the three quantit ...math> where <math>y\leqslant 7</math> is part of <math>S</math>. This is a ray with an endpoint of <math>(1, 7)</math>.

3 KB (446 words) - 14:58, 10 June 2023
2017 AMC 10A Problems/Problem 12
...th> in the coordinate plane such that two of the three quantities <math>3,~x+2,</math> and <math>y-4</math> are equal and the third of the three quantit ...x=1</math> where <math>y \le 7</math> is part of <math>S</math>. This is a ray with an endpoint of <math>(1, 7)</math>.

2 KB (421 words) - 00:34, 1 August 2021
2017 USAMO Problems
...riangle with circumcircle <math>\Omega</math> and incenter <math>I</math>. Ray <math>AI</math> meets <math>\overline{BC}</math> at <math>D</math> and meet ...>P_{2n}</math> be <math>2n</math> distinct points on the unit circle <math>x^2+y^2=1</math>, other than <math>(1,0)</math>. Each point is colored either

4 KB (771 words) - 13:49, 22 November 2023

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