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• 2010 AMC 12A Problems/Problem 24
  ...\left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)</math>. The intersection of the domain of <math>f(x)</math> with ...align*}\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x) &= 0\\
  9 KB (1,434 words) - 17:54, 17 August 2022
• 2010 AMC 10A Problems
  The area of a circle whose circumference is <math>24\pi</math> is <math>k\pi</math>. What is the value of <math>k</math>? ...}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qquad \textbf{(D)}\ \frac{4}{\pi} \qquad \textbf{(E)}\ 4</math>
  13 KB (1,902 words) - 11:20, 5 March 2023
• 2010 AMC 10B Problems
  Markala attended two meetings during her <math>9</math>-hour work day. The first meeting took <m \textbf{(A)}\ \dfrac{\pi}{3}-1
  12 KB (1,817 words) - 22:44, 22 December 2020
• Proofs without words
  ...4)+shiftR--(n-1,-n+1)+r*expi(pi/4)+shiftR^^r*expi(5*pi/4)+shiftR--r*expi(5*pi/4)+(n-1,-n+1)+shiftR,linetype("4 4")); fill((n-1.5,-1.5) -- (n-1.5,-1.5)+r*expi(5.2*pi/6) -- (n-1.5,-1.5)+r*expi(3.3*pi/6) -- cycle); /* manual arrowhead? avoid resizing */
  55 KB (7,986 words) - 17:04, 20 December 2018
• 2010 AIME I Problems
  ...ll the positive divisors of <math>2010^2</math>. She then randomly selects two distinct divisors from this list. Let <math>p</math> be the probability tha Jackie and Phil have two fair coins and a third coin that comes up heads with probability <math>\fra
  8 KB (1,243 words) - 21:58, 10 August 2020
• 2010 AIME I Problems/Problem 11
  ...x = 15</math>, the [[volume]] of the resulting [[solid]] is <math>\frac {m\pi}{n\sqrt {p}}</math>, where <math>m</math>, <math>n</math>, and <math>p</mat ...about the line <math>y = x/3+5</math>, the resulting solid is the union of two right [[cone]]s that share the same base and axis.
  4 KB (636 words) - 16:46, 25 November 2023
• 2010 AIME I Problems/Problem 13
  ...ively. Line <math>\ell</math> divides region <math>\mathcal{R}</math> into two regions with areas in the ratio <math>1: 2</math>. Suppose that <math>AU = <math>Y = \frac {1}{3}\pi(3)^2 = 3\pi</math>
  10 KB (1,418 words) - 23:05, 20 October 2021
• Mock AIME 2 2010 Answer Key
  ...hooses one of the other four to shoot. The probability that there are some two people shooting each other can be expressed in the form <math>\frac{a}{b}</ ...r of arrangements, <math>4^5</math>. For any pair to shoot each other, the two shoot each other in exactly one way, while the other three people shoot any
  36 KB (6,214 words) - 20:22, 13 July 2023
• LaTeX:LaTeX on AoPS
  LaTeX uses a special "math mode" to display mathematics. There are two types of this "math mode": Besides displaying in-line vs. displaying centered and on a new line, the two modes render differently in other ways. Note that <code>$\sum_{k=1}^n k^2$<
  8 KB (1,356 words) - 22:35, 26 June 2020
• 2010 USAMO Problems/Problem 3
  multiplying any two and dividing by the third, we get: &= \frac{\Gamma^4(\frac{1}{4})}{8\pi^2}
  11 KB (1,889 words) - 13:45, 4 July 2013
• 2010 USAMO Problems/Problem 4
  We know that angle <math>BIC = 135^{\circ}</math>, as the other two angles in triangle <math>BIC</math> add to <math>45^{\circ}</math>. Assume markangle(Label("$\scriptstyle{\frac{\pi}{4} - \frac{\theta}{2}}$"), radius=40, I, C, B);
  7 KB (1,230 words) - 19:47, 31 January 2024
• 2007 AMC 8 Problems/Problem 25
  ...rt will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regio ...> or <math>9\pi</math>. Note that the area of the whole circle is <math>36\pi</math>.
  3 KB (510 words) - 10:08, 18 June 2024
• 2010 AMC 12B Problems/Problem 18
  ...and either circle's center (symmetry, you chose!). The intersection of the two circles should form a geometrical lens shape. By sectors, <cmath>f(\theta)=\frac{\theta}{2\pi}\cdot2\pi=\theta</cmath>
  6 KB (1,105 words) - 13:39, 9 January 2024
• 2004 AMC 8 Problems
  Find the number of two-digit positive integers whose digits total <math>7</math>. ...of the five numbers in a list is <math>54</math>. The average of the first two
  13 KB (1,860 words) - 18:51, 13 June 2024
• 2006 AMC 8 Problems
  ...{4}</math> revolutions. In what direction does the spinner point after the two moves? The letter T is formed by placing two <math> 2 \times 4 </math> inch rectangles next to each other, as shown. Wha
  16 KB (2,215 words) - 19:18, 10 April 2024
• 2010 AMC 10B Problems/Problem 20
  Two circles lie outside regular hexagon <math>ABCDEF</math>. The first is tange ...area of the first circle is <math>(\frac{\sqrt{3}}{6})^2 \cdot \pi =\frac{\pi}{12}</math>.
  3 KB (483 words) - 18:41, 4 May 2024
• 2004 JBMO Problems/Problem 4
  ...have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the tr Prove that there are two more black triangles that white ones.
  2 KB (451 words) - 01:03, 23 December 2018
• 2010 AMC 10B Problems/Problem 19
  A circle with center <math>O</math> has area <math>156\pi</math>. Triangle <math>ABC</math> is equilateral, <math>\overline{BC}</math The formula for the area of a circle is <math>\pi r^2</math> so the radius of this circle is <math>\sqrt{156}.</math>
  3 KB (443 words) - 18:34, 4 May 2024
• 2010 AMC 10B Problems/Problem 16
  ...c{\pi}{18} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{2\pi}{9}</math> ...f by the two radii - area of the triangle with sides on the square and the two radii).
  3 KB (544 words) - 20:54, 24 March 2024
• 2011 AMC 12A Problems
  ...nts per text message sent, plus <math>10</math> cents for each minute used over <math>30</math> hours. In January Michelle sent <math>100</math> text messa ...ber of successful free throws was one more than their number of successful two-point shots. The team's total score was <math>61</math> points. How many fr
  13 KB (1,994 words) - 13:52, 3 July 2021

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