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• Mock AIME 3 Pre 2005 Problems/Problem 1
  ...r centers is <math>84</math>, then the area of the third circle is <math>k\pi</math> for some integer <math>k</math>. Determine <math>k</math>. ...h>, so <math>r=14</math>. Thus the area of the circle is <math>\boxed{196}\pi</math>.
  795 bytes (129 words) - 10:22, 4 April 2012
• Arc
  ...the circle. More generally, an arc is a portion of a smooth curve joining two points. ..., in particular, the [[circumference]] of a circle is given by <math>C = 2\pi</math>.
  1 KB (170 words) - 00:24, 16 December 2023
• 2007 AIME II Problems/Problem 11
  ...t starts. The distance <math>x</math> can be expressed in the form <math>a\pi+b\sqrt{c},</math> where <math>a,</math> <math>b,</math> and <math>c</math> If it weren’t for the small tube, the larger tube would travel <math>144\pi</math>. Consider the distance from which the larger tube first contacts the
  2 KB (407 words) - 16:31, 29 February 2020
• 2007 AIME II Problems
  ...ers can produce <math>300</math> widgets and <math>200</math> whoosits. In two hours, <math>60</math> workers can produce <math>240</math> widgets and <ma ...t starts. The distance <math>x</math> can be expressed in the form <math>a\pi+b\sqrt{c},</math> where <math>a,</math> <math>b,</math> and <math>c</math>
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2007 AMC 12A Problems
  The larger of two consecutive odd integers is three times the smaller. What is their sum? <math>\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18</math>
  11 KB (1,750 words) - 13:35, 15 April 2022
• 2007 USAMO Problems/Problem 6
  ...i/2)--intersectionpoint(incircle(A,B,C),I--I+O-A)-abs(A-O)*expi(angle(A-O)-pi/2), q_a--q_a+O-A); ...i/2)--intersectionpoint(incircle(A,B,C),I--I+A-O)-abs(A-O)*expi(angle(A-O)-pi/2), O--A);
  7 KB (1,274 words) - 15:11, 31 August 2017
• Cylinder
  ...] from a line [[perpendicular]] to both planes. Thus, a '''cylinder''' has two circular flat surfaces--think of a can of soup. Typically, to solve for sur * Surface [[area]]: <math>2 \pi r^2 + 2 \pi r h</math>
  827 bytes (135 words) - 18:13, 27 July 2023
• 2007 Cyprus MO/Lyceum/Problem 21
  ...ent pulleys, that are connected with a strap. If the distances between any two pulley center points are <math>AB=3\,\mathrm{m}</math>, <math>AC=4\,\mathrm ...\qquad \mathrm{(B) \ } (12 + \pi)\,\mathrm{m}\qquad \mathrm{(C) \ } (12+ 4\pi)\,\mathrm{m}\qquad \mathrm{(D) \ } \left(12+\frac\pi5\right)\,\mathrm{m}\qq
  1 KB (175 words) - 22:21, 23 July 2020
• Pascal's Theorem
  ...l prove the theorem for a cyclic hexagon, using directed angles mod <math>\pi </math>. '''Lemma.''' Let <math>\omega_1, \omega_2 </math> be two circles which intersect at <math>M, N </math>, let <math>AB </math> be a ch
  4 KB (712 words) - 21:57, 25 April 2020
• 2007 AMC 12A Problems/Problem 24
  ...the [[equation]] <math>\sin{x}=\sin{(nx)}</math> on the interval <math>[0,\pi]</math>. What is <math>\sum_{n=2}^{2007} F(n)</math>? Notice that the solutions are basically reflections across <math>x = \frac{\pi}{2}</math>.
  4 KB (588 words) - 14:40, 23 August 2023
• 2006 AIME II Problems/Problem 12
  pair A=(0,0),B=(-3^.5,-3),C=(3^.5,-3),D=13*expi(-2*pi/3),E1=11*expi(-pi/3),F=E1+D; Because the ratio of the area of two similar figures is the square of the ratio of the corresponding sides, <mat
  6 KB (1,033 words) - 02:36, 19 March 2022
• 2005 PMWC Problems
  touches its two neighbors and the larger circle. What is the radius of the ...in the list <math>1, 2, 3, 4, 5, \dots, 10000</math> which contain exactly two consecutive <math>9</math>'s such as <math>993, 1992</math> and <math>9929<
  9 KB (1,449 words) - 20:49, 2 October 2020
• 1998 PMWC Problems
  ...>\angle BOA</math> is a right angle. <math>A</math> and <math>B</math> are two points on the circumference of circle of radius <math>OA</math>. Find the a .../3),6sin(4*pi/3))--(0,0))--(-3,-3)--intersectionpoint((8cos(5*pi/6),8sin(5*pi/6))--(0,0),(-3,3)--(-3,-3))--cycle,gray);
  11 KB (1,738 words) - 19:25, 10 March 2015
• 2000 PMWC Problems
  <math>p</math> is the product of two <math>4</math>-digit numbers formed by the digits <math>1,2,3,4,5,6,7,8</ma <math>a</math> and <math>b</math> are two numbers that have prime factors <math>3</math> and <math>5</math> only. <ma
  11 KB (1,713 words) - 22:47, 13 July 2023
• 2004 AMC 10A Problems/Problem 21
  ...e radian measure of the acute angle formed by the two lines? (Note: <math>\pi</math> radians is <math>180</math> degrees.) ...\pi}{6} \qquad \mathrm{(D) \ } \frac{\pi}{5} \qquad \mathrm{(E) \ } \frac{\pi}{4} </math>
  3 KB (476 words) - 03:50, 23 January 2023
• 2007 AMC 12B Problems
  ...t response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave the la ...lengths <math>AB = 5</math>, <math>BC = 6</math>, and <math>AC = 7</math>. Two bugs start simultaneously from <math>A</math> and crawl along the sides of
  12 KB (1,814 words) - 12:58, 19 February 2020
• 1960 IMO Problems/Problem 6
  The area of <math>T</math> may be computed in two different ways: <cmath>V_1 = \frac{1}{3}\pi R^2 h</cmath>
  7 KB (1,214 words) - 18:49, 29 January 2018
• Root (operation)
  ...t[n]{x}=\sqrt[n]{|x|}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right)</math> , where <math>k=0,1,2,...,n-1</math> and <math>x\in\mat ...h>\sqrt[4]{16}=\sqrt[4]{16}\left(\cos\frac{0+2\pi\cdot0}{4}+i\sin\frac{0+2\pi\cdot0}{4}\right)\implies\boxed{2}</math>
  3 KB (532 words) - 16:52, 20 May 2020
• 2007 AMC 10A Problems
  The larger of two consecutive odd integers is three times the smaller. What is their sum? ...005, <math>78</math> in 2006, and is <math>85</math> in 2007. Between what two consecutive years was there the largest percentage increase?
  13 KB (2,058 words) - 17:54, 29 March 2024
• 1995 AHSME Problems
  ...oint, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point ...n attendance. A league official who knows the actual numbers attending the two games notes that:
  17 KB (2,387 words) - 22:44, 26 May 2021
• 1959 AHSME Problems
  ...is drawn parallel to the base <math>AB</math>, dividing the triangle into two equal areas. ...mean (average) of a set of <math>50</math> numbers is <math>38</math>. If two numbers, namely, <math>45</math> and <math>55</math>, are discarded, the me
  22 KB (3,345 words) - 20:12, 15 February 2023
• 2002 AMC 12B Problems/Problem 25
  ...of the area of the circle, which is <math>\frac{1}{2} (\pi \cdot 4^2) = 8\pi \approx \boxed{\textbf{(E) }25}</math>. ...e area of the circle satisfying <math>f(x)+f(y) \le 0</math>, or <math>16 \pi</math>.
  2 KB (365 words) - 14:48, 7 March 2022
• 2007 AMC 10B Problems/Problem 11
  A circle passes through the three vertices of an isosceles triangle that has two sides of length <math>3</math> and a base of length <math>2</math>. What is ...\frac{81}{32}\pi \qquad\textbf{(D) } 3\pi \qquad\textbf{(E) } \frac{7}{2}\pi</math>
  5 KB (851 words) - 22:02, 26 July 2021
• 1994 USAMO Problems
  Let <math>\, k_1 < k_2 < k_3 < \cdots \,</math> be positive integers, no two consecutive, and let <math>\, s_m = k_1 + k_2 + \cdots + k_m \,</math> for ...f the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifica
  2 KB (391 words) - 07:58, 19 July 2016
• 2008 AMC 10A Problems
  Heather compares the price of a new computer at two different stores. Store <math>A</math> offers <math>15\%</math> off the sti ...}\ \left(2\sqrt{3}+3\right)^2\pi\\\mathrm{(E)}\ 9\left(\sqrt{3}+1\right)^2\pi</math>
  14 KB (2,138 words) - 15:08, 18 February 2023
• 1976 IMO Problems/Problem 2
  ...tions, where 1 is positive and 1 is negative. The absolute values of these two solutions are also both less than 2. This proves the base case. ...l}{2^n+1}\cdot\pi</math>. As we can choose the range <math>0\leq\theta\leq\pi</math> to ensure no duplications, we get that, upon rearranging, <math>0\le
  3 KB (596 words) - 16:19, 28 July 2015
• 2008 AMC 12A Problems/Problem 25
  We can make two observations from this: ...<math>z_n = 2e^{i\pi/6}z_{n-1}</math>, so <math>z_{n-1} = \frac{1}{2}e^{-i\pi/6}z_n</math>. This is the reverse transformation. We have
  5 KB (745 words) - 10:58, 9 December 2022
• 2008 AMC 10B Problems
  ...diagonal are to be added. What will be the positive difference between the two diagonal sums? ...- 2ax + b = 0</math> has two real solutions. What is the average of these two solutions?
  12 KB (1,838 words) - 16:52, 7 October 2022
• 2008 AMC 12B Problems
  ...diagonal are to be added. What will be the positive difference between the two diagonal sums? ...is steps. The pedometer records up to <math>99999</math> steps, then flips over to <math>00000</math> on the next step. Pete plans to determine his mileage
  14 KB (2,199 words) - 13:43, 28 August 2020
• 2008 AIME I Problems
  ...ngle entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entr ...\arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.</cmath>
  9 KB (1,536 words) - 00:46, 26 August 2023
• 2008 AIME II Problems
  ...Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the Let <math>a = \pi/2008</math>. Find the smallest positive integer <math>n</math> such that
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AIME I Problems/Problem 8
  ...\arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.</cmath> Applying this to the first two terms, we get <math>\arctan{\dfrac{1}{3}} + \arctan{\dfrac{1}{4}} = \arctan
  3 KB (490 words) - 22:36, 28 November 2023
• 2008 AIME I Problems/Problem 10
  ...whose angle at the longer base <math>\overline{AD}</math> is <math>\dfrac{\pi}{3}</math>. The [[diagonal]]s have length <math>10\sqrt {21}</math>, and po pair A=(0,0), D=(4,0), B= A+2 expi(1/3*pi), C= D+2expi(2/3*pi), E=(-4/3,0), F=(3,0);
  4 KB (629 words) - 22:38, 28 November 2023
• 2008 AIME II Problems/Problem 5
  pair A=(0,0), D=(2*r, 0), N=(r,0), E=N+r*expi(74*pi/180); pair A=(0,0), D=(2*r, 0), N=(r,0), E=N+r*expi(74*pi/180);
  8 KB (1,338 words) - 23:15, 28 November 2023
• 2008 AIME II Problems/Problem 9
  ...efine a ''move'' for the particle as a counterclockwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</ma ...he position of the particle after a move from <math>P</math>, then we have two equations for <math>x'</math> and <math>y'</math>:
  5 KB (725 words) - 22:37, 28 January 2024
• 1974 USAMO Problems/Problem 5
  Consider the two triangles <math>\triangle ABC</math> and <math>\triangle PQR</math> shown i pair P=(7,0), Q=(14,0), R=P+7expi(pi/3), M=(10,1.2);
  7 KB (1,221 words) - 18:57, 3 July 2013
• 2005 USAMO Problems/Problem 3
  ...be an acute-angled triangle, and let <math>P</math> and <math>Q</math> be two points on side <math>BC</math>. Construct point <math>C_1 </math> in such a ...th>Q' =Q</math>. All our angles will be directed, and measured mod <math>\pi</math>.
  6 KB (1,117 words) - 01:17, 11 October 2021
• Cone
  <center><!-- NOTE: two different versions --><asy> size(120); ...ath> and height <math>h</math> has [[volume]] <math>V = \frac{1}{3} \cdot \pi r^2 \cdot h</math>. This is a special case of the general formula for the
  7 KB (1,128 words) - 20:12, 27 September 2022
• 2008 USAMO Problems/Problem 4
  This diagonal [[partition]]s <math>\mathcal{P}</math> into two regions <math>\mathcal{Q},\, \mathcal{R}</math>, and is the side of an isos int n = 17; real r = 1; real rad = pi/2;
  8 KB (1,318 words) - 12:37, 20 April 2022
• 2003 USAMO Problems/Problem 6
  ...by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after w label("$c$",expi(pi/3),(0,0),red);
  5 KB (739 words) - 13:39, 4 July 2013
• 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 &#36;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>
  13 KB (1,988 words) - 20:19, 15 May 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 (703 words) - 22:37, 2 November 2022
• 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 (2,130 words) - 11:32, 7 November 2021
• 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
• 2009 AMC 12A Problems/Problem 21
  <center><cmath>p(2009 + 9002\pi i) = p(2009) = p(9002) = 0</cmath></center> From the three zeroes, we have <math>p(x) = (x - (2009 + 9002\pi i))(x - 2009)(x - 9002)</math>.
  2 KB (322 words) - 10:25, 29 July 2020
• 2009 AMC 12A Problems/Problem 20
  pair D=MP("D",(0,0)),C=MP("C",(12,0)),A=MP("A",C+14*expi(145*pi/180),N),B=MP("B",A+(9,0),N),E=IP(A--C,B--D);MP("9",(A+B)/2,N);MP("12",(C+D) ...\triangle AEB \sim \triangle EDC</math> (which should be trivial given the two equal triangles) we have that
  3 KB (543 words) - 21:09, 23 October 2023
• 2009 AMC 12A Problems/Problem 19
  ...Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were <math>A</math> and <math>B</math>, respectively. Each polygon ...isosceles triangle formed by the center of the polygon <math>S</math> and two consecutive vertices <math>X</math> and <math>Y</math>. We are given that <
  4 KB (630 words) - 21:27, 30 December 2023
• 2009 AMC 10A Problems/Problem 21
  ...e area of the large circle is <math>L = \pi R^2 = \pi r^2 (1+\sqrt 2)^2 = \pi r^2 (3+2\sqrt 2)</math>. The area of four small circles is <math>S = 4\pi r^2</math>. Hence their ratio is:
  3 KB (474 words) - 12:50, 29 September 2023
• 2009 AMC 10A Problems/Problem 19
  ...h> as it rolls once around the circumference of circle <math>A</math>. The two circles have the same points of tangency at the beginning and end of circle ...cumference of circle <math>B</math> with radius <math>r</math> is <math>2r\pi</math>. Since circle <math>B</math> makes a complete revolution and ''ends
  2 KB (276 words) - 09:57, 8 June 2021
• 2001 AMC 12 Problems/Problem 8
  ...^\circ</math> sector of a circle of radius <math>10</math> by aligning the two straight sides? The length of the red line is <math>\dfrac{252}{360}\cdot 2\pi \cdot 10 = 14\pi</math>. This is the circumference of a circle with radius <math>7</math>.
  2 KB (279 words) - 00:32, 30 December 2023
• 2009 AMC 12B Problems
  A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remain Kiana has two older twin brothers. The product of their ages is <math>128</math>. What is
  13 KB (2,030 words) - 03:04, 5 September 2021
• 2009 AMC 12B Problems/Problem 23
  ...34 + \frac34i</math> into polar form as <math>\frac{3\sqrt{2}}{4}e^{\frac{\pi}{4}i}</math>. Restated using geometric probabilities, we are trying to find ...ath>2</math> by <math>2</math> square centered at the origin. Graphing our two equations gives us the four lines <cmath>x-y=\frac{4}{3},</cmath> <cmath>x-
  2 KB (422 words) - 13:25, 20 January 2020
• 2009 AMC 12B Problems/Problem 24
  For how many values of <math>x</math> in <math>[0,\pi]</math> is <math>\sin^{ - 1}(\sin 6x) = \cos^{ - 1}(\cos x)</math>? ...in^{-1}(x) \leq \pi/2</math> and <math>\forall x: 0\leq \cos^{-1}(x) \leq \pi</math>.
  7 KB (1,287 words) - 07:09, 22 December 2022
• 2009 AIME I Problems/Problem 12
  ...>O</math> be center of the circle and <math>P</math>,<math>Q</math> be the two points of tangent such that <math>P</math> is on <math>BI</math> and <math> Since the ratios between corresponding lengths of two similar diagrams are equal, we can let <math>AD = 144, CD = 420</math> and
  12 KB (1,970 words) - 22:53, 22 January 2024
• 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) - 03:32, 13 April 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
• 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 (516 words) - 13:40, 23 January 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) - 19:58, 8 May 2023
• 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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