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• 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

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