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- 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>\o10 KB (1,797 words) - 02:05, 24 October 2023
- ...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 polynomia8 KB (1,246 words) - 21:58, 10 August 2020
- ...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
- ...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
- ...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
- ...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
- ...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
- ...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
- ...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>U10 KB (1,634 words) - 22:21, 28 December 2023
- ...\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
- ...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</ma15 KB (2,593 words) - 13:37, 29 January 2021
- ...\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
- .../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
- .../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 other13 KB (1,994 words) - 01:31, 22 December 2023
- ...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
- ...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> a6 KB (910 words) - 17:32, 27 May 2012
- 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 suc5 KB (834 words) - 11:00, 24 February 2021
- ...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
- ...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
- 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> equals17 KB (2,633 words) - 15:44, 16 September 2023
