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• 2021 Fall AMC 10B Problems/Problem 25
  ...t{10}} , \sqrt{10} \right)</math>, <math>Q = \left( \sqrt{10} , \frac{10 - v}{\sqrt{10}} \right)</math>. We have <math>m_{HP} = \frac{3}{6 - u}</math> and <math>m_{PQ} = - \frac{v}{u}</math>.
  16 KB (2,274 words) - 09:02, 10 December 2022
• Cubic Equation
  ...\left(\frac{b^3}{27a^3}\right) + \left(\frac{b}{a}\right)y^2 - \left(\frac{2b^2}{3a^2}\right)y + \left(\frac{b^3}{9a^3}\right) + \left(\frac{c}{a}\right) <math>y^3 + \left(\frac{3ac - b^2}{3a^2}\right)y + \left(\frac{2b^3 - 9abc + 27a^2d}{27a^3}\right) = 0</math>.
  6 KB (1,059 words) - 14:42, 11 December 2020
• 2021 Fall AMC 12A Problems/Problem 13
  .../math> bisects the angle between <math>\mathbf{u}</math> and <math>\mathbf{v}</math>. ...=x.</math> Therefore, we have <math>\frac{3b-a}{b-a} = -1,</math> so <math>2b=a.</math>
  16 KB (2,526 words) - 00:53, 6 May 2023
• Newton's method
  <center><cmath>p(\hat{x})=p(a_0+a_1p)=p(a_0)+p'(a_0)a_1p+(a_1p)^2b</cmath></center> ...>, (2) <math>\hat{x}\equiv x\pmod{p^{n-k}}</math> and (3) <math>k=v(p'(x))=v(p'(\hat{x}))</math>.
  13 KB (2,298 words) - 23:34, 28 May 2023
• 2023 USAJMO Problems/Problem 2
  x=\frac{2b^2-2c^2}{a^2-3b^2-c^2} P=\left(\frac{b^2-c^2}{a^2-2b^2-2c^2},\frac{a^2-3b^2-c^2}{2a^2-4b^2-4c^2},\frac{a^2-3b^2-c^2}{2a^2-4b^2-4
  7 KB (1,280 words) - 01:22, 6 February 2024
• 2023 USAMO Problems/Problem 1
  x=\frac{2b^2-2c^2}{a^2-3b^2-c^2} P=\left(\frac{b^2-c^2}{a^2-2b^2-2c^2},\frac{a^2-3b^2-c^2}{2a^2-4b^2-4c^2},\frac{a^2-3b^2-c^2}{2a^2-4b^2-4
  4 KB (684 words) - 01:55, 6 February 2024
• 2023 USAMO Problems/Problem 6
  https://youtube.com/watch?v=LAuyU2OuVzE ...and <math>w=\frac{2b^3cx_0z_0}{c^2y_0^2-b^2z_0^2}</math>, so <cmath>\frac{v}{w}=\frac{b^2z_0}{c^2y_0},</cmath> which is the required condition for <mat
  14 KB (2,600 words) - 23:37, 10 March 2024
• 2024 AIME I Problems/Problem 7
  https://www.youtube.com/watch?v=ejmrAJ9TpvM&ab_channel=MegaMathChannel &=(75a-117b)+(116a+75b)i+48\left(\dfrac{2a+3b+(3a-2b)i}{a^2+b^2}\right) \\
  12 KB (1,957 words) - 09:04, 18 May 2024
• Bisector
  ...frac{a+c}{a+2b+c} \implies \frac {B'D}{BD} = \frac {BB' - BD}{BD} = \frac{2b}{a+c} = 2\frac {B'I}{BI}.</cmath> The points <math>U \in AA''</math> and <math>V \in CC''</math> be such points that <math>L \in UV, UV \perp BB''.</math>
  14 KB (2,490 words) - 06:51, 6 June 2024
• DVI exam
  DVI is an exam in mathematics at the Moscow State University named after M.V. Lomonosov. The first four problems have a standard level. <cmath>\left\{\begin{array}{l} u = \sqrt {2} (x + y),\\v = 3y.\end{array}\right.</cmath>
  33 KB (5,611 words) - 02:05, 19 June 2024
• Cubic formula
  ...{a}\left(x-\frac{b}{3a}\right)+\frac{d}{a}=x^3+\frac{3ac-b^2}{3a^2}x+\frac{2b^3+27a^2d-9abc}{27a^3}</math>. .../math>, <math>3uv(u+v)</math> matches up with <math>-px</math>, and <math>(v^3+w^3)</math> matches up with <math>-q</math>. Things are about to get exci
  3 KB (644 words) - 15:57, 24 June 2024

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