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Millennium Problems
...nstitute (which also had organized the meeting), had selected the problems over the previous several months. They were led by Arthur Jaffe, the first direc ...thoroughly examine the solution. This committee would consist of at least two world-renowned mathematicians and at least one member of the SAB.

13 KB (1,969 words) - 17:57, 22 February 2024
2008 iTest Problems
...{120}</math>. How many dollars did the family save by buying a family pass over buying single day passes for every member of the family? The difference between two prime numbers is <math>11</math>. Find their sum.

71 KB (11,749 words) - 01:31, 2 November 2023
2005 PMWC Problems/Problem T7
...? Calculate an approximate answer by using <math>\pi=3.14</math> or <math>\pi=22/7</math>. ...</math>. Adding, we get <math>\frac{8}{3}\pi+\frac{\pi}{6}+\frac{\pi}{6}=3\pi</math>. Then approximate.

854 bytes (145 words) - 23:28, 21 October 2022
2005 AMC 8 Problems
...t Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack brings up &#3 The Little Twelve Basketball League has two divisions, with six teams in each division. Each team plays each of the oth

13 KB (1,821 words) - 22:18, 5 December 2023
Van Aubel's Theorem
..._{AB}, P_{BC}, P_{CD}, P_{DA}</math>. Van Aubel's Theorem states that the two line segments connecting opposite centers are perpendicular and equal lengt ...by the complex number <math>x</math>. Note that <math>\angle PAB = \frac{\pi}{4}</math> and that <math>PA = \frac{\sqrt{2}}{2}AB</math>, and similarly f

2 KB (410 words) - 14:01, 4 March 2023
2002 AMC 10A Problems
...\pi \qquad \text{(C)}\ 2\pi \qquad \text{(D)}\ 3\pi \qquad \text{(E)}\ 3.5\pi</math> Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 p

11 KB (1,733 words) - 11:04, 12 October 2021
2002 AMC 10B Problems
...d \mathrm{(C) \ } 6\pi\qquad \mathrm{(D) \ } 9\pi\qquad \mathrm{(E) \ } 12\pi </math> ...ne}\qquad \mathrm{(C) \ } \text{two}\qquad \mathrm{(D) \ } \text{more than two, but finitely many}\qquad \mathrm{(E) \ } \text{infinitely many} </math>

10 KB (1,540 words) - 22:53, 19 December 2023
2002 AMC 10A Problems/Problem 19
...ase that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the ...i \qquad \text{(C)}\ 5\pi/2 \qquad \text{(D)}\ 8\pi/3 \qquad \text{(E)}\ 3\pi</math>

1 KB (204 words) - 20:45, 28 May 2023
2004 AMC 10B Problems
How many two-digit positive integers have at least one <math>7</math> as a digit? ...75+100\pi \qquad \mathrm{(D) \ } 100+100\pi \qquad \mathrm{(E) \ } 100+125\pi </math>

14 KB (2,137 words) - 15:29, 9 June 2024
1974 USAMO Problems
Two boundary points of a ball of radius 1 are joined by a curve contained in th A father, mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are:

3 KB (427 words) - 18:55, 3 July 2013
1962 IMO Problems/Problem 6
...ircle. Prove that the distance <math>d</math> between the centers of these two circles is ...cdot IL = PI\cdot QI</math> by Power of a Point. Therefore, <math>2r\rho = PI \cdot QI = (PO + OI)(QO - OI) = (r + d)(r - d)</math>, and we have <math>2r

2 KB (308 words) - 06:29, 16 December 2023
1963 IMO Problems/Problem 3
...ath>-axis and the entire shape is above the <math>x</math>-axis. There are two cases: the number of sides is even, and the number of sides is odd: ...h>y\textrm{-coordinate} = \sum_{k = 1}^{\frac{n}{2}}a_k \cdot \sin \frac{2\pi(k-1)}{n}.\textbf{ (1)}</cmath>

4 KB (836 words) - 17:58, 7 December 2022
1963 IMO Problems
Prove that <math>\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}</math>. ...tter. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did s

2 KB (342 words) - 21:16, 20 August 2020
2021 GMC 10B
The radius of a circle that has an area of <math>\frac{\pi}{\sqrt{2}}</math> is <math>r</math>. Find <math>r^{2}</math> A three digit natural number is <math>Alternative</math> if it has two even digits and one odd digit as its number digits. Find the number of alte

11 KB (1,695 words) - 14:33, 7 March 2022
1986 AJHSME Problems/Problem 23
The large circle has diameter <math>\text{AC}</math>. The two small circles have their centers on <math>\text{AC}</math> and just touch a The small circle has radius <math>1</math>, thus its area is <math>\pi</math>.

2 KB (277 words) - 21:32, 3 July 2013
2004 AMC 10B Problems/Problem 25
...in the picture, that is outside the smaller circle and inside each of the two larger circles? \mathrm{(A) \ } \frac{5}{3} \pi - 3\sqrt 2

4 KB (715 words) - 18:24, 26 June 2024
2004 AMC 12B Problems/Problem 10
An annulus is the region between two concentric circles. The concentric circles in the figure have radii <math>b ...hrm{(C) \ } \pi c^2 \qquad \mathrm{(D) \ } \pi d^2 \qquad \mathrm{(E) \ } \pi e^2 </math>

2 KB (340 words) - 14:35, 23 April 2023
2009 AMC 12A Problems
...>(6m,0)</math>, and the line <math>y = mx</math> divides the triangle into two triangles of equal area. What is the sum of all possible values of <math>m< ...^3 + \cdots</math> and <math>a + ar_2 + ar_2^2 + ar_2^3 + \cdots</math> be two different infinite geometric series of positive numbers with the same first

13 KB (2,105 words) - 13:13, 12 August 2020
2009 AMC 10A Problems
\mathrm{(B)}\ \frac{\pi}{6} \mathrm{(C)}\ \frac{2}{\pi}

14 KB (2,126 words) - 17:46, 13 June 2024
2009 AMC 12A Problems/Problem 25
The first two terms of a sequence are <math>a_1 = 1</math> and <math>a_2 = \frac {1}{\sqr As the number of possible consecutive two terms is finite, we know that the sequence <math>b_n</math> is periodic. Wr

7 KB (990 words) - 07:23, 24 October 2022

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