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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
Symmetry
Let <math>L</math> be the point in the ray <math>FE,</math> such that <math>\frac {KE} {FL} = \frac {AB}{AC}.</math> ...X_0</math> the result of transformation <math>T</math> of point <math>X: T(X) = X_0.</math>

19 KB (3,292 words) - 13:04, 13 May 2024
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
Circular Inversion
...e{OA'} = k^2</math>, where <math>A'</math> is the transformed point on the ray extending from <math>O</math> through <math>A</math>. pair[] x = intersectionpoints(circle, line);

16 KB (2,516 words) - 23:48, 15 January 2024
2017 USAJMO Problems
<cmath>\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.</cmath> (a) Prove that there are infinitely many pairs <math>(x,y)</math> of positive integers satisfying the equation.

3 KB (570 words) - 16:44, 5 August 2023
2017 USAMO Problems/Problem 3
...riangle with circumcircle <math>\Omega</math> and incenter <math>I.</math> Ray <math>AI</math> meets <math>BC</math> at <math>D</math> and <math>\Omega</m Let <math>X</math> be the point on circle <math>\Omega</math> opposite <math>M</math>.

4 KB (711 words) - 18:24, 8 May 2023
2000 IMO Problems/Problem 1
...<math>\triangle{ABP}</math> are tangent to <math>BC</math>. Prove that ray <math>AP</math> bisects <math>BC</math>. ...ur lemma, <math>\textit{(the two circles are tangent to AB)}</math>, <math>X</math> bisects <math>AB</math>. Since <math>\triangle{NAX}</math> and <math

3 KB (451 words) - 00:04, 19 November 2023
Optics
*[[X-Rays]] ...elongs to classical physics. Wave optics are a topic of modern physics. In ray, optics covers topics such as reflection and refraction and the dispersion

994 bytes (152 words) - 10:07, 8 July 2019
2019 AIME I Problems
There are positive integers <math>x</math> and <math>y</math> that satisfy the system of equations ...{10} (\gcd(x,y)) = 60</cmath> <cmath>\log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) = 570.</cmath>

8 KB (1,331 words) - 06:57, 4 January 2021
2019 AIME I Problems/Problem 13
...h>, and <math>CA=6</math>. Points <math>D</math> and <math>E</math> are on ray <math>AB</math> with <math>AB<AD<AE</math>. The point <math>F \neq C</math> pair A, B, C, D, E, F, X, O1, O2;

10 KB (1,620 words) - 20:44, 20 December 2023
What is the definition of Pure Mathematics?
...of the tie represented by a letter. This letter is most of the time <math>x</math>,<math>y</math> or <math>z</math> however it can be really any non-us ...or expression involving one or more variables. For example, if <math>f(x)=x+3</math> then

35 KB (5,882 words) - 18:08, 28 June 2021
2006 SMT/Team Problems/Problem 13
...This ray makes an angle of <math> \theta </math> with the positive <math> x </math> axis. Compute <math> \cos\theta </math>. ...olution. This means that the discriminant of the equation <math>x^2(m^2-1)+x-1=0</math> will be equal to zero. Solving:

1 KB (261 words) - 18:38, 14 January 2020
2020 AIME I Problems/Problem 15
...BC</math> at <math>H</math> intersects <math>\omega</math> at points <math>X</math> and <math>Y</math> with <math>HA=3,HX=2,</math> and <math>HY=6.</mat ...IP(omega,L(H,Z,50,0)), D=2*H-A, K=extension(B,C,X,Y), E=extension(A,origin,X,Y), L=foot(H,B,C);

16 KB (2,678 words) - 22:45, 27 November 2023
Light
* [[X-Rays]] ...elongs to classical physics. Wave optics are a topic of modern physics. In ray, optics covers topics such as reflection and refraction and the dispersion

2 KB (198 words) - 12:52, 8 July 2019
2000 SMT/Calculus Problems
Find the slope of the tangent at the point of inflection of <math>y = x^3 - 3x^2 + 6x + 2000</math>. ...he pulls herself up <math>x</math> feet at once, then the rope slips <math>x^3</math> feet down. How many feet at a time must she pull herself up to cli

3 KB (413 words) - 13:10, 21 January 2020
2020 AIME II Problems/Problem 13
...ath>. Extend <math>CD</math>, cross ray <math>AB</math> at <math>M</math>, ray <math>AE</math> at <math>N</math>, then by AAS we have triangle <math>END</ Substituting <math>x=r^2</math> gives the quadratic equation <math>5x^2-84x+64=0</math>, with so

13 KB (2,197 words) - 23:00, 8 January 2024
2022 AIME I Problems/Problem 14
Consider the splitting line through <math>M</math>. Extend <math>D</math> on ray <math>BC</math> such that <math>CD=CA</math>. Then the splitting line bisec ...id a</math> and <math>3\mid b</math>. Thus, it's equivalent to solve <math>x^2+xy+y^2 = 73^2</math>.

16 KB (2,730 words) - 02:56, 4 January 2023
2021 AMC 12A Problems/Problem 11
...and bounce off the <math>y</math>-axis, then hit and bounce off the <math>x</math>-axis, then hit the point <math>(7,5)</math>. What is the total dista label("$x$",(xMax,0),(2,0));

10 KB (1,514 words) - 01:28, 1 November 2023
2016 APMO Problems
...math>O</math> that is tangent to ray <math>AC</math> at <math>E</math> and ray <math>AB</math> at <math>F</math>. Let <math>R</math> be a point on segment ...= f(xf(z) + y) + f(yf(z) + x),</cmath>for all positive real numbers <math>x, y, z</math>.

3 KB (454 words) - 22:09, 11 July 2021
2018 TSTST Problems
...{\mathbb Z}</math> such that for any polynomials <math>p,q \in {\mathbb Z}[x]</math>, ...th>, respectively. Line <math>MO</math> meets <math>\omega</math> at <math>X</math> and <math>Y</math>, while line <math>CO</math> meets <math>\omega</m

5 KB (930 words) - 08:38, 22 February 2023
2023 USAJMO Problems
Find all triples of positive integers <math>(x,y,z)</math> that satisfy the equation 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023

4 KB (681 words) - 21:38, 19 March 2024
1992 IMO Problems/Problem 4
...th the circle center at <math>(0,0)</math> and circle equation to be <math>x^{2}+y{2}=r^{2}</math> Therefore, to get the coordinates of point <math>T=(T_{x},T_{y})</math>, we solve the following equations:

6 KB (1,052 words) - 00:42, 17 November 2023
2023 USAJMO Problems/Problem 6
.../math> intersects rays <math>AB</math> and <math>AC</math> at points <math>X</math> and <math>Y</math>, respectively. Prove that <math>\angle XDE = \ang

4 KB (701 words) - 21:54, 7 April 2024
Problems Collection
<cmath>x \equiv 2^4 \cdot 3^4 \cdot 7^4+2^7 \cdot 3^7 \cdot 5^6 \pmod{7!}</cmath> Find the remainder when <math>\min{x}</math> is divided by <math>1000</math>.

43 KB (7,006 words) - 14:24, 19 February 2024
2024 USAJMO Problems
...>S</math> be the set of integer points <math>(x,y)</math> with <math>1\leq x\leq2m</math> and <math>1\leq y\leq2n</math>. A configuration of <math>mn</m <cmath>f(x^2-y)+2yf(x)=f(f(x))+f(y)</cmath>

3 KB (482 words) - 13:58, 26 March 2024
Sharygin Olympiads, the best
...the circumcircle at <math>A</math> and <math>B</math> meet at point <math>X,</math> and the tangents at points <math>P</math> and <math>Q</math> meets Let <math>K</math> be the point on ray <math>EF</math> such that <math>EK = AB.</math>

29 KB (4,997 words) - 18:06, 16 May 2024

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