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1966 AHSME Problems/Problem 4
...the radius of the smaller circle is <math>x</math>, so it's area is <math>\pi x^2</math>. ...be <math>2x^2 \pi</math>. Putting one over the other and dividing, you get two as the answer: or <math>\boxed{(B)}</math>.

1 KB (191 words) - 22:09, 14 January 2018
Liouville's Theorem (complex analysis)
<cmath> \lvert f'(z_0) \rvert = \biggl\lvert \frac{1}{2\pi i} point <math>z \in \mathbb{C}</math>. Now for any two complex numbers <math>A</math>

2 KB (412 words) - 20:30, 16 January 2024
2000 AMC 8 Problems
...e numbers lie in the interval between <math>\frac{5}{3}</math> and <math>2\pi?</math> Consider these two geoboard quadrilaterals. Which of the following statements is true?

15 KB (2,165 words) - 18:29, 5 June 2024
2009 AIME II Problems/Problem 13
...ath> instead of <math>B</math>. <math>\angle AOC_1</math> = <math>\frac {\pi}{7}</math>. Using the [[Law of Cosines]], <math>\overline {AC_1}^2</math> = <math>8 - 8 \cos \frac {\pi}{7}</math>,

8 KB (1,279 words) - 20:27, 17 May 2024
1965 IMO Problems
Determine all values <math>x</math> in the interval <math>0\leq x\leq 2\pi </math> which satisfy the inequality .../math> is equal to <math>k</math>. Compute the ratio of the volumes of the two solids obtained.

3 KB (497 words) - 12:39, 29 January 2021
Fundamental group
...example, the fundamental group of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a c We say that two binary operations <math>\circ, \cdot</math> on a

8 KB (1,518 words) - 20:11, 23 January 2017
2003 AMC 12B Problems/Problem 25
...ave to place the third point is a <math>60</math> degrees arc(if the first two are <math>60</math> degrees apart), with a <math>\frac{1}{6}</math> probabi ...ave to place the third point is a <math>120</math> degree arc(if the first two are the same point), with a <math>\frac{1}{3}</math> probability.

4 KB (635 words) - 11:46, 1 September 2022
2010 AMC 12A Problems
...}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qquad \textbf{(D)}\ \frac{4}{\pi} \qquad \textbf{(E)}\ 4</math> In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and

12 KB (1,817 words) - 15:00, 12 August 2020
2010 AMC 12A Problems/Problem 13
The image below shows the two curves for <math>k=4</math>. The blue curve is <math>x^2+y^2=k^2</math>, wh Clearly the only such integer is <math>k=1</math>, hence the two curves are only disjoint for <math>k=1</math> and <math>k=-1</math>.

9 KB (1,622 words) - 20:53, 11 September 2023
2010 AMC 12A Problems/Problem 17
...math>\triangle ABC</math>, we get that <math>AC^2 = r^2+1^2-2r\cos{\frac{2\pi}{3}} = r^2+r+1</math>. Therefore, the area of <math>\triangle ACE</math> is Note: To verify that the quadratic <math>r^2-6r+1</math> has two positive roots, we can either solve for the roots directly or note that dis

4 KB (690 words) - 10:13, 14 October 2022
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

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