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• Trivial Inequality
  *Find all integer solutions <math>x,y,z</math> of the equation <math>x^2+5y^2+10z^2=4xy+6yz+2z-1</math>. ...es the equality hold? ([[1969 Canadian MO Problems/Problem 3|1969 Canadian MO]])
  3 KB (560 words) - 22:51, 13 January 2024
• Discriminant
  ...possible to us to use Cardano's formula, by doing the substitution <math>x=z-\frac{a}{3}</math> on the polynomial <math>p(x)=x^3+ax^2+bx+c</math>. * [[1977_Canadian_MO_Problems/Problem_1 | 1977 Canadian MO Problem 1]]
  4 KB (734 words) - 19:19, 10 October 2023
• 1983 AIME Problems/Problem 15
  ...h> to <math>BA</math> and call its intersection with <math>BA</math> <math>Z</math>. Then, by the Pythagorean theorem, <math>OZ = \frac{7\sqrt{2}}{2}</m Let Z be (0,5).
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1970 Canadian MO Problems
  Find all number triples <math>(x,y,z)</math> such that when any of these numbers is added to the product of the [[1970 Canadian MO Problems/Problem 1 | Solution]]
  4 KB (604 words) - 04:32, 8 October 2014
• 2006 Canadian MO Problems/Problem 4
  ...math> beats <math>Y</math>, <math>Y</math> beats <math>Z</math>, and <math>Z</math> beats <math>X</math>. *[[2006 Canadian MO]]
  604 bytes (100 words) - 18:20, 28 November 2023
• 2006 Canadian MO Problems/Problem 5
  ...th> and <math>F</math>, <math>F</math> and <math>D</math> be labeled <math>Z, X,Y</math>, respectively. *[[2006 Canadian MO]]
  2 KB (347 words) - 17:02, 3 June 2011
• Spiral similarity
  ...at <math>B</math> with angle <math>\alpha</math> and coefficient <math>k, Z = t(Y), \frac {BZ}{BY} = k.</math> We use the complex plane <math>x = \vec X, y = \vec Y, z = \vec Z, a = \vec A,b = \vec B, o = \vec O.</math>
  28 KB (4,863 words) - 00:29, 16 December 2023
• 2004 AMC 12B Problems/Problem 14
  ...lar from <math>J</math> to <math>MO</math> and let them intersect at <math>Z</math>. There is a obvious pair of congruent triangles. Fill in the gap. He
  7 KB (1,083 words) - 22:41, 23 November 2020
• 1973 Canadian MO Problems/Problem 7
  Thus, by induction, the formula holds for all <math>n \in \mathbb{Z^{+}}</math> *[[1973 Canadian MO]]
  2 KB (351 words) - 18:56, 27 May 2022
• Mock AIME I 2012 Problems
  ...h>, <math>NO=21</math>, <math>MO=23</math>. The trisection points of <math>MO</math> are <math>E</math> and <math>F</math>, with <math>ME<MF</math>. Segm ...ntersect at points <math>X, Y</math> such that <math>XY=2</math>. If <math>Z</math> equals the area of <math>\triangle{PMN}</math>, <math>24Z</math> can
  7 KB (1,309 words) - 11:13, 8 April 2012
• 2005 Indonesia MO Problems/Problem 6
  Find all triples <math> (x,y,z)</math> of integers which satisfy <math> x(y + z) = y^2 + z^2 - 2</math>
  2 KB (286 words) - 11:56, 17 March 2020
• 2004 Indonesia MO Problems/Problem 4
  ...gle A, \angle B, \angle C, \angle D</math> be <math>w,x,y</math> and <math>z,</math> respectively. ...\angle ZWX = \tfrac{w+x}{2}</math> degrees and <math>\angle ZYX = \tfrac{y+z}{2}</math> degrees.
  2 KB (343 words) - 12:02, 5 August 2018
• 2002 Indonesia MO Problems
  [[2002 Indonesia MO Problems/Problem 1|Solution]] [[2002 Indonesia MO Problems/Problem 2|Solution]]
  2 KB (386 words) - 00:08, 4 August 2018
• 2002 Indonesia MO Problems/Problem 3
  <math>\left\{\begin{array}{l}x+y+z = 6\\x^2 + y^2 + z^2 = 12\\x^3 + y^3 + z^3 = 24\end{array}\right.</math> <cmath>x^2 + y^2 + z^2 + 2(xy + yz + xz) = 36</cmath>
  2 KB (353 words) - 23:38, 13 July 2022
• 2004 Indonesia MO Problems/Problem 7
  label("Z",(18,16),NE); ...</math> be the intersection of the incircle and <math>BC</math>, and <math>Z</math> be the intersection of the incircle and <math>AB</math>.
  2 KB (337 words) - 13:26, 28 February 2020
• 2006 Indonesia MO Problems
  [[2006 Indonesia MO Problems/Problem 1|Solution]] [[2006 Indonesia MO Problems/Problem 2|Solution]]
  3 KB (439 words) - 12:39, 4 September 2018
• 2005 Indonesia MO Problems
  [[2005 Indonesia MO Problems/Problem 1|Solution]] [[2005 Indonesia MO Problems/Problem 2|Solution]]
  3 KB (485 words) - 00:31, 5 September 2018
• 2019 AIME I Problems
  ...overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. ...tegers <math>m</math> and <math>n</math> such that one such value of <math>z</math> is <math>m+\sqrt{n}+11i</math>. Find <math>m+n</math>.
  8 KB (1,331 words) - 06:57, 4 January 2021
• 2006 Indonesia MO Problems/Problem 3
  ...y, \tan C = z</math>. Note that <math>A+B+C = 180^\circ</math>, so <math>z = \tan C = \tan (180^\circ - (A+B)) = -\tan(A+B)</math>. ...esults in <math>x+y = xyz - z</math>, and rearranging results in <math>x+y+z = xyz</math>. We want this equation to be satisfied by positive [[integers
  3 KB (465 words) - 12:00, 26 September 2019
• 2020 AIME I Problems/Problem 15
  ..., Op=2*H-O, Z=bisectorpoint(O,Op), X=IP(omega,L(H,Z,0,50)), Y=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, ...ion of <math>O</math> onto <math>\overline{HO'}</math>. Then <math>TO' = R-MO</math>, so the Pythagorean Theorem applied to <math>\triangle TOO'</math> y
  16 KB (2,678 words) - 22:45, 27 November 2023

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