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2015 IMO Problems
We say that a finite set <math>\mathcal{S}</math> in the plane is balanced if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is

4 KB (692 words) - 22:33, 15 February 2021
Fermat's Little Theorem
...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.

16 KB (2,658 words) - 16:02, 8 May 2024
Number theory/Advanced
...vertically upward as the point at infinity. We denote it by <math>\mathcal{O}</math>. ...points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commuta

5 KB (849 words) - 16:14, 18 May 2021
Law of Sines
The '''Law of Sines''' is a useful identity in a [[triangle]], which, along with the [[law of cosines]] and the [[law of tan ...<math>B</math>, <math>c</math> opposite to <math>C</math>, and where <math>R</math> is the circumradius:

4 KB (658 words) - 16:19, 28 April 2024
Trigonometry
...s and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way towards ...definition of trigonometry is preferred in order to extend trigonometry to a complex domain.

8 KB (1,217 words) - 20:15, 7 September 2023
Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
...power]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says ...</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power.

5 KB (912 words) - 20:06, 14 March 2023
Power of a Point Theorem
...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2 =

5 KB (827 words) - 17:30, 21 February 2024
Algebraic geometry
...> be a [[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine alge ...s are algebraic varieties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying

2 KB (361 words) - 01:59, 24 January 2020
Mock AMC
...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==

51 KB (6,175 words) - 20:58, 6 December 2023
2006 AIME I Problems/Problem 13
...st integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. .../math> but not by <math>2^n, \ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are divisible by <math>2^1</math>

10 KB (1,702 words) - 00:45, 16 November 2023
LaTeX:Symbols
This article provides a short list of commonly used LaTeX symbols. .../math> on the web, (technically an AJAX library simulating it.)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list o

16 KB (2,324 words) - 16:50, 19 February 2024
2005 AMC 12A Problems
{{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})

13 KB (1,971 words) - 13:03, 19 February 2020
2000 AMC 12 Problems
...2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\

13 KB (1,948 words) - 12:26, 1 April 2022
2003 AMC 12B Problems
\text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qqu ...ks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?

13 KB (1,987 words) - 18:53, 10 December 2022
2005 AIME II Problems
{{AIME Problems|year=2005|n=II}} ...imes the number of possible sets of 3 cards that can be drawn. Find <math> n. </math>

7 KB (1,119 words) - 21:12, 28 February 2020
2005 AIME II Problems/Problem 12
...math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...abel("$x$",E/2+G/2,(0,1));label("$y$",G/2+F/2,(0,1)); label("$450$",(O+G)/2,(-1,1));

13 KB (2,080 words) - 21:20, 11 December 2022
2005 AIME I Problems/Problem 11
...mum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> We note that aligning the base of the semicircle with a side of the square is certainly non-optimal. If the semicircle is tangent

4 KB (707 words) - 11:11, 16 September 2021
2005 AIME II Problems/Problem 15
...h> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </ma Let <math>w_3</math> have center <math>(x,y)</math> and radius <math>r</math>. Now, if two circles with radii <math>r_1</math> and <math>r_2</mat

12 KB (2,000 words) - 13:17, 28 December 2020
2004 AIME I Problems/Problem 14
...are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> ...-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h);

4 KB (729 words) - 01:00, 27 November 2022
2004 AIME II Problems/Problem 12
...> k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> ...of the center circle be <math>r</math> and its center be denoted as <math>O</math>.

3 KB (431 words) - 23:21, 4 July 2013
1983 AIME Problems
...th> and <math>z</math> all exceed <math>1</math> and let <math>w</math> be a positive number such that <math>\log_xw=24</math>, <math>\log_y w = 40</mat ...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B<

7 KB (1,104 words) - 03:13, 27 May 2024
1992 AIME Problems
A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s

8 KB (1,117 words) - 05:32, 11 November 2023
1996 AIME Problems
...mn, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>. ...at <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?

6 KB (931 words) - 17:49, 21 December 2018
2000 AIME II Problems
{{AIME Problems|year=2000|n=II}} ...and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.

6 KB (947 words) - 21:11, 19 February 2019
2002 AIME II Problems
{{AIME Problems|year=2002|n=II}} ...a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube?

7 KB (1,177 words) - 15:42, 11 August 2023
1983 AIME Problems/Problem 4
...hat of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math>B< real r=10;

11 KB (1,741 words) - 22:40, 23 November 2023
1983 AIME Problems/Problem 15
...rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? pair A=(-0.91,-0.41);

20 KB (3,497 words) - 15:37, 27 May 2024
1989 AIME Problems/Problem 10
...h>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find pair A = (0,0), B = (3, 0), C = (1, 4);

8 KB (1,401 words) - 21:41, 20 January 2024
1990 AIME Problems/Problem 14
...ments <math>\overline{CP}</math> and <math>\overline{DP}</math>, we obtain a [[triangular pyramid]], all four of whose faces are [[isosceles triangle]]s pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);

7 KB (1,086 words) - 08:16, 29 July 2023
1991 AIME Problems/Problem 12
...t terms, denote the [[perimeter]] of <math>ABCD^{}_{}</math>. Find <math>m+n^{}_{}</math>. ...)",S,W); label("$15$",B/2+P/2,N);label("$20$",B/2+Q/2,E);label("$O$",O,SW); </asy></center>

8 KB (1,270 words) - 23:36, 27 August 2023
1991 AIME Problems/Problem 11
...ath>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. for (int a=1; a<24; a+=2)

4 KB (740 words) - 17:46, 24 May 2024
1993 AIME Problems/Problem 13
...enny and Kenny can see each other again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ...math>P</math> and <math>Q</math> be the points of tangency of circle <math>O</math> to <math>AC</math> and <math>BD</math> respectively.

8 KB (1,231 words) - 20:06, 26 November 2023
1996 AIME Problems/Problem 15
...ce as large as angle <math>DBA</math>, and angle <math>ACB</math> is <math>r</math> times as large as angle <math>AOB</math>. Find <math>\lfloor 1000r \ ...=(0,0), A=expi(pi/4), C=IP(A--A + 2*expi(17*pi/12), B--(3,0)), D=A+C, O=IP(A--C,B--D);

5 KB (710 words) - 21:04, 14 September 2020
1996 AIME Problems/Problem 4
...th>x</math> centimeters directly above an upper [[vertex]], the cube casts a shadow on the horizontal surface. The area of the shadow, which does not in real r = 2.8;

2 KB (257 words) - 17:50, 4 January 2016
1999 AIME Problems/Problem 15
...The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle along the sides of ...F\)",foot(C,A,B),NW);label("$P$",P,NW);label("$Q$",Q,NE);label("$R$",R,SE);</asy><asy>import three; defaultpen(linewidth(0.6));

7 KB (1,169 words) - 15:28, 13 May 2024
2000 AIME I Problems/Problem 13
...and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. Let the intersection of the highways be at the origin <math>O</math>, and let the highways be the x and y axes. We consider the case wher

3 KB (571 words) - 00:38, 13 March 2014
2002 AIME II Problems/Problem 15
...me, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>. ...<math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that

7 KB (1,182 words) - 09:56, 7 February 2022
2002 AIME II Problems/Problem 14
...nd <math>n</math> are [[relatively prime]] positive integers, find <math>m+n</math>. ...MPA</math> are similar. Also note that <math>AM = BM</math> by [[power of a point]]. Using the fact that the ratio of corresponding sides in similar tr

4 KB (658 words) - 19:15, 19 December 2021
2002 AIME II Problems/Problem 12
...<math>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left(a+b+c\right)</math>. ...ne <math>y=\frac{2x}{3}</math>. We can find the number of such paths using a Pascal's Triangle type method below, computing the number of paths to each

7 KB (1,127 words) - 13:34, 19 June 2022
2000 AIME II Problems/Problem 12
...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n+k</math>. Let <math>D</math> be the foot of the [[perpendicular]] from <math>O</math> to the plane of <math>ABC</math>. By the [[Pythagorean Theorem]] on

3 KB (532 words) - 13:14, 22 August 2020
2000 AIME II Problems/Problem 10
A [[circle]] is [[inscribe]]d in [[quadrilateral]] <math>ABCD</math>, [[tange ...drawing the lines from <math>O</math> tangent to the sides and from <math>O</math> to the vertices of the quadrilateral, four pairs of congruent [[righ

2 KB (399 words) - 17:37, 2 January 2024
University of South Carolina High School Math Contest/1993 Exam/Problems
If the width of a particular rectangle is doubled and the length is increased by 3, then the <cmath> \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2 \qquad \mathrm{(C) \ } 3 \qquad \mathrm{(D)

14 KB (2,102 words) - 22:03, 26 October 2018
Mock AIME 1 2006-2007 Problems/Problem 14
...}\angle AMP</math> for fixed <math>m</math> and <math>n</math> where <math>n>m</math>. If <math>f(m,49)</math> is an integer, find the sum of all possib ...ee below an attempted solution to understand why this problem doesn't have a solution:

3 KB (541 words) - 17:32, 22 November 2023
Mock AIME 1 2006-2007/Problems
...th> and <math>n</math> are relatively prime positive intgers, find <math>m+n</math>. (Note that <math>[ABC]</math> denotes the area of <math>\triangle A ...>\mathcal{S}</math>, we have that <math>\star (n)=12</math> and <math>0\le n< 10^{7}</math>. If <math>m</math> is the number of elements in <math>\mathc

8 KB (1,355 words) - 14:54, 21 August 2020
2005 Canadian MO Problems/Problem 3
Let <math>S</math> be a set of <math>n\ge 3</math> points in the interior of a circle. ...math> on the circle such that <math>a</math> is (strictly) closer to <math>A</math> than any other point in <math>S</math>, <math>b</math> is closer to

2 KB (460 words) - 13:35, 9 June 2011
2007 iTest Problems
A twin prime pair is a set of two primes <math>(p, q)</math> such that <math>q</math> is <math>2</ <math>\mathrm{(A)}\, 4</math>

30 KB (4,794 words) - 23:00, 8 May 2024
Euler line
...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...ath>, and <math>\triangle CH_AH_B</math> [[concurrence | concur]] at <math>N</math>, the nine-point circle of <math>\triangle ABC</math>.

59 KB (10,203 words) - 04:47, 30 August 2023
2003 AMC 10A Problems/Problem 25
...ided by <math>100</math>. For how many values of <math>n</math> is <math>q+r</math> divisible by <math>11</math>? <math> \mathrm{(A) \ } 8180\qquad \mathrm{(B) \ } 8181\qquad \mathrm{(C) \ } 8182\qquad \math

6 KB (943 words) - 20:57, 29 May 2023
Sector
pair O=origin, A=dir(angle2), B=dir(angle1); path sector=O--B--arc(O,1,angle1,angle2)--A--cycle;

1 KB (178 words) - 21:12, 24 April 2008
Mock AIME 1 Pre 2005 Problems
...<math>n</math> are relatively prime positive integers. Determine <math>m + n</math>. <math>A, B, C, D,</math> and <math>E</math> are collinear in that order such that <

6 KB (1,100 words) - 22:35, 9 January 2016
Mock AIME 3 Pre 2005 Problems
...igits are in increasing order. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. (Repeated digits are allowed.) A function <math>f(x)</math> is defined for all real numbers <math>x</math>.

7 KB (1,135 words) - 23:53, 24 March 2019
Mock AIME 3 Pre 2005 Problems/Problem 5
...math>-letter Zuminglish words. Determine the remainder obtained when <math>N</math> is divided by <math>1000</math>. ...te the number of <math>n</math>-letter words ending in a vowel followed by a constant (<tt>VC</tt> - the only other combination, two vowels, is impossib

5 KB (795 words) - 16:03, 17 October 2021
2007 AIME II Problems/Problem 15
...nd <math>n</math> are [[relatively prime]] positive integers. Find <math>m+n.</math> pair A,B,C,X,Y,Z,P,Q,R;

11 KB (2,099 words) - 17:51, 4 January 2024
2007 USAMO Problems/Problem 2
...sists of all [[point]]s <math>(m,n)</math>, where <math>m</math> and <math>n</math> are [[integer]]s. Is it possible to cover all grid points by an inf ...ngent]] circles with radius greater than or equal to 5, one can always fit a circle with radius greater than <math>\frac{1}{\sqrt{2}}</math> between tho

5 KB (754 words) - 03:41, 7 August 2014
Circumradius
...very triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]]. ==Formula for a Triangle==

4 KB (729 words) - 16:52, 19 February 2024
2005 AMC 12A Problems/Problem 15
...\overline{DC} \perp \overline{AB}</math> and <math>\overline{DE}</math> is a second diameter. What is the [[ratio]] of the area of <math>\triangle DCE</ pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0);

14 KB (1,970 words) - 17:02, 18 August 2023
2006 AIME II Problems/Problem 12
...visible]] by the [[perfect square | square]] of any prime, find <math> p+q+r. </math> 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;

6 KB (1,033 words) - 02:36, 19 March 2022
1971 Canadian MO Problems
...of a circle such that <math>DE=3</math> and <math>EB=5 .</math> Let <math>O</math> be the center of the circle. Join <math>OE</math> and extend <math>O <math>ABCD</math> is a quadrilateral with <math>AD=BC</math>. If <math>\angle ADC</math> is greate

3 KB (519 words) - 08:58, 13 September 2012
2007 AMC 12B Problems
<math> \mathrm{(A) \ } 678\qquad \mathrm{(B) \ } 768\qquad \mathrm{(C) \ } 786\qquad \mathrm{ A college student drove his compact car 120 miles home for the weekend and av

12 KB (1,814 words) - 12:58, 19 February 2020
Asymptote: Useful functions
The bundled <code>markers</code> package provides a useful function for marking angles. The exact same function exists in the < Label ''L''="", int ''n''=1, real ''radius''=0, real ''space''=0, 

4 KB (646 words) - 21:18, 26 March 2024
2000 AMC 12 Problems/Problem 24
...ters at <math>B</math> and <math>A</math>, respectively, then there exists a circle tangent to both <math>\overarc {AC}</math> and <math>\overarc{BC}</m label("A", (0,0), W);

2 KB (263 words) - 19:59, 18 April 2024
1995 AHSME Problems
...7, 83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will <math> \mathrm{(A) \ \text{remain the same} } \qquad \mathrm{(B) \ \text{increase by 1} } \qq

17 KB (2,387 words) - 22:44, 26 May 2021
1951 AHSME Problems
The percent that <math>M</math> is greater than <math>N</math> is: ...\textbf{(D) \ } \frac {M - N}{M} \qquad \textbf{(E) \ } \frac {100(M + N)}{N} </math>

23 KB (3,641 words) - 22:23, 3 November 2023
1966 AHSME Problems
<math>\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qqua When the base of a triangle is increased 10% and the altitude to this base is decreased 10%, t

19 KB (3,159 words) - 22:10, 11 March 2024
2004 AMC 12B Problems/Problem 19
...d cone has horizontal bases with radii <math>18</math> and <math>2</math>. A sphere is tangent to the top, bottom, and lateral surface of the truncated <math>\mathrm{(A)}\ 6

3 KB (520 words) - 19:12, 20 November 2023
2007 AMC 10B Problems/Problem 11
...s of an isosceles triangle that has two sides of length <math>3</math> and a base of length <math>2</math>. What is the area of this circle? <math>\textbf{(A) } 2\pi \qquad\textbf{(B) } \frac{5}{2}\pi \qquad\textbf{(C) } \frac{81}{32

5 KB (851 words) - 22:02, 26 July 2021
2008 AMC 10A Problems
{{AMC10 Problems|year=2008|ab=A}} A bakery owner turns on his doughnut machine at <math>\text{8:30}\ {\small\te

14 KB (2,138 words) - 15:08, 18 February 2023
2008 AMC 12A Problems/Problem 13
...rcle]] centered at <math>O</math>, and <math>\angle AOB = 60^\circ</math>. A second circle is internally [[tangent]] to the first and tangent to both <m <math>\mathrm{(A)}\ \frac{1}{16}\qquad\mathrm{(B)}\ \frac{1}{9}\qquad\mathrm{(C)}\ \frac{1}{

4 KB (630 words) - 20:32, 4 June 2021
2008 AMC 12A Problems/Problem 20
<math>\mathrm{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right)\qquad\mathrm{(B)}\ \frac{3}{56}\lef pair C=(0,0),A=(0,3),B=(4,0),D=(4-2.28571,1.71429);

6 KB (951 words) - 16:31, 2 August 2019
2008 AMC 12B Problems
A basketball player made <math>5</math> baskets during a game. Each basket was worth either <math>2</math> or <math>3</math> points. <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \

14 KB (2,199 words) - 13:43, 28 August 2020
2008 AIME I Problems/Problem 14
...a</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. Suppose <math>\overline{AB} = 18</math>, an ...= O - (9,0), A= O + (9,0), C=A+(18,0), T = 9 * expi(-1.2309594), P = foot(A,C,T);

8 KB (1,333 words) - 00:18, 1 February 2024
Mock AIME 5 Pre 2005 Problems
...math> and <math>n</math> are relatively prime integers, computer <math>m + n</math>. &\text {ii. all ten digits, a through j are all distinct.}\\

6 KB (909 words) - 07:27, 12 October 2022
Mock AIME 1 2007-2008 Problems/Problem 14
...and <math>c</math> is not divisible by the square of any prime, find <math>a+b+c+d</math>. real r = (2+26^0.5)/198;

2 KB (375 words) - 15:23, 25 November 2009
Mock AIME 1 2007-2008 Problems
...h> that can be written in the form <cmath>{n\choose k} : {n\choose k+1} : {n \choose k+2}</cmath> Find the sum of all possible values of <math>n+k</math>.

6 KB (992 words) - 14:15, 13 February 2018
2008 AIME II Problems/Problem 9
...ckwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</math> units in the positive <math>x</math>-directi ...the x-axis. If <math>(x', y')</math> is the position of the particle after a move from <math>P</math>, then we have two equations for <math>x'</math> an

5 KB (725 words) - 22:37, 28 January 2024
2008 AIME II Problems/Problem 11
...essed in the form <math>m - n\sqrt {k}</math>, where <math>m</math>, <math>n</math>, and <math>k</math> are positive integers and <math>k</math> is the real r=44-6*35^.5;

6 KB (1,065 words) - 20:12, 9 August 2022
Carnot's Theorem
...>O</math> to the sides (i.e., signed lengths of the pedal lines from <math>O</math>) is: <math>OO_A+OO_B+OO_C=R+r</math>

4 KB (723 words) - 01:45, 18 February 2021
2008 iTest Problems
...s as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fue When the Kubiks went on vacation to San Diego last year, they spent a day at the San Diego Zoo.

71 KB (11,749 words) - 01:31, 2 November 2023
Van Aubel's Theorem
...: <math>ABA'B'</math>, <math>BCB'C'</math>, <math>CDC'D'</math>, <math>DAD'A'</math>; yielding centers <math>P_{AB}, P_{BC}, P_{CD}, P_{DA}</math>. Van pair A, B, C, D, O, P, Q, R, SS;

2 KB (410 words) - 14:01, 4 March 2023
2001 USAMO Problems/Problem 4
Let <math>P</math> be a point in the plane of triangle <math>ABC</math> such that the segments <mat Set <math>A(0,0)</math>, <math>B(1,0)</math>, <math>C(x,y)</math>, <math>P(p,q)</math>.

8 KB (1,470 words) - 22:24, 18 June 2022
1976 USAMO Problems/Problem 2
...>AX</math> and <math>BY</math>. You may assume that <math>AB</math> is not a diameter. real r=10;

5 KB (848 words) - 23:41, 6 July 2020
2000 AMC 10 Problems/Problem 13
...s, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (h <math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5!\cdot 4!\cdot 3!\cdot 2!\c

2 KB (308 words) - 21:32, 27 July 2022
2004 AMC 10B Problems/Problem 22
A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those cir <math> \mathrm{(A) \ } \frac{3\sqrt{5}}{2} \qquad \mathrm{(B) \ } \frac{7}{2} \qquad \mathrm{

5 KB (700 words) - 13:46, 6 April 2024
2009 AMC 10B Problems
...nt muffin or a <math>75</math>-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? <math>\text{(A) } 1\qquad\text{(B) } 2\qquad\text{(C) } 3\qquad\text{(D) } 4\qquad\text{(E

15 KB (2,262 words) - 00:53, 18 June 2021
2009 AMC 10B Problems/Problem 22
...pieces as shown in this top view, where <math>M</math> is the midpoint of a top edge. The piece whose top is triangle <math>B</math> contains <math>c</ label("$A$",(-0.4,0.7));

12 KB (1,868 words) - 03:36, 30 September 2023
2009 AIME I Problems/Problem 12
...and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. ...th> and <math>DB=\dfrac{35^{2}}{37}</math> by similar triangles. Let <math>O</math> be the center of <math>\omega</math>; notice that <cmath>\tan(\angle

12 KB (1,970 words) - 22:53, 22 January 2024
2009 AIME II Problems/Problem 10
...e by the square of any prime. Find <math>p</math> + <math>q</math> + <math>r</math>. ...),NW,f); D(rightanglemark(C,B,A,20)); D(anglemark(D,A,E,35));D(anglemark(C,A,D,30));

9 KB (1,415 words) - 13:56, 18 December 2022
2009 AIME II Problems/Problem 13
...oduct of the lengths of these twelve chords. Find the remainder when <math>n</math> is divided by <math>1000</math>. ...th>A</math> and <math>B</math>. Assume <math>C_1</math> is closer to <math>A</math> instead of <math>B</math>. <math>\angle AOC_1</math> = <math>\frac

8 KB (1,279 words) - 20:27, 17 May 2024
Radical axis
The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairl ...rtains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. B

12 KB (2,125 words) - 08:38, 23 May 2024
2009 AIME II Problems/Problem 15
...and <math>t</math> is not divisible by the square of any prime. Find <math>r+s+t</math>. ...c{3}{5}</math>, point <math>B</math> lies between <math>M</math> and <math>A</math> on the semicircular arc. We will first compute the length of <math>\

11 KB (1,862 words) - 21:23, 23 May 2024
1968 AHSME Problems/Problem 35
MP("O",(0,0),S); MP("a",(5,0),S);

3 KB (597 words) - 01:52, 16 August 2023
2009 IMO Problems/Problem 2
Let <math>ABC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <math>Q</math> are interior points of dot("O", (50, 38), N);

2 KB (280 words) - 01:16, 19 November 2023
Schroeder-Bernstein Theorem
...orem''' (sometimes called the '''Cantor-Schroeder-Bernstein Theorem''') is a result from [[set theory]], named for Ernst Schröder and Felix Bernste ...ath> and <math>g : B \to A</math>, then there is a [[bijection]] <math>h : A \to B</math>.

4 KB (805 words) - 13:09, 20 February 2024
1992 AJHSME Problems
<math>\text{(A)}\ -1 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 9 \qquad <math>\text{(A)}\ \dfrac{10}{8} \qquad \text{(B)}\ 1\dfrac{1}{4} \qquad \text{(C)}\ 1\dfra

17 KB (2,346 words) - 13:36, 19 February 2020
Characteristic polynomial
The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eige ...nomial in <math>t</math>. Here, <math>I</math> refers to the <math>n\times n</math> [[identity matrix]].

19 KB (3,412 words) - 14:57, 21 September 2022
2010 AIME I Problems/Problem 13
...<math>n</math> is not divisible by the square of any prime. Find <math>m + n</math>. label("$A$",(-16.43287,-9.3374),NE/2);

10 KB (1,418 words) - 23:05, 20 October 2021
Mock AIME 2 2010 Answer Key
{{AIME box|year = Mock|n = 2 2010|before=[[Mock AIME 1 2010]]|after=[[Mock AIME 1 2011]]}} ...an be easily proved through strong induction. Starting from 2010, which is a multiple of 15, we must first purge 1 lemming. We can then purge 4 lemmings

36 KB (6,214 words) - 20:22, 13 July 2023
2010 AIME II Problems/Problem 14
...<math>r</math> is not divisible by the square of any prime. Find <math>p+q+r</math>. Let <math>O</math> be the [[circumcenter]] of <math>ABC</math> and let the intersection

10 KB (1,507 words) - 00:31, 19 November 2023
2010 AIME II Problems/Problem 9
...<math>n</math> are [[relatively prime]] positive integers. Find <math>m + n</math>. pair A,B,C,D,E,F;

5 KB (857 words) - 22:22, 27 August 2023
2010 USAMO Problems
...d <math>RS</math> is half the size of <math>\angle XOZ</math>, where <math>O</math> is the midpoint of segment <math>AB</math>. ...to make more than <math>\binom{n}{3}</math> such switches before reaching a position in which no further switches are possible.

3 KB (525 words) - 13:44, 4 July 2013
2010 USAJMO Problems
...Find with proof the smallest <math>n</math> such that <math>P(n)</math> is a multiple of <math>2010</math>. ...be an integer. Find, with proof, all sequences <math>x_1, x_2, \ldots, x_{n-1}</math> of positive integers with the following three properties:

3 KB (538 words) - 13:55, 16 June 2020
2010 USAMO Problems/Problem 1
Let <math>AXYZB</math> be a convex pentagon inscribed in a semicircle of diameter <math>AB</math>. Denote by <math>P, Q, R, S</math> the feet of the perpendiculars from <math>Y</math> onto

13 KB (2,271 words) - 14:08, 23 May 2024
1978 USAMO Problems
Given that <math>a,b,c,d,e</math> are real numbers such that <math>a+b+c+d+e=8</math>,

2 KB (372 words) - 19:06, 3 July 2013
1996 AJHSME Problems
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad <math>\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kare

13 KB (1,880 words) - 13:35, 19 February 2020
1999 AMC 8 Problems/Problem 15
...from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. <math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 40 \qq

3 KB (516 words) - 14:50, 21 December 2022
2010 AMC 10B Problems/Problem 19
...th> is a chord on the circle, <math>OA = 4\sqrt{3}</math>, and point <math>O</math> is outside <math>\triangle ABC</math>. What is the side length of <m <math>\textbf{(A)}\ 2\sqrt{3} \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 4\sqrt{3} \qquad \

3 KB (443 words) - 18:34, 4 May 2024
2011 AMC 12A Problems
{{AMC12 Problems|year=2011|ab=A}} A cell phone plan costs <math>\$20</math> dollars each month, plus <math>5</m

13 KB (1,994 words) - 13:52, 3 July 2021
2011 AMC 12A Problems/Problem 13
...rline{AB}</math> at <math>M</math> and <math>\overline{AC}</math> at <math>N.</math> What is the perimeter of <math>\triangle AMN?</math> \textbf{(A)}\ 27 \qquad

4 KB (683 words) - 03:12, 23 January 2023
2011 AMC 12A Problems/Problem 25
...and <math>AC \geq AB</math>. Let <math>H</math>, <math>I</math>, and <math>O</math> be the orthocenter, incenter, and circumcenter of <math>\triangle AB \textbf{(A)}\ 60^{\circ} \qquad

6 KB (1,046 words) - 13:05, 28 June 2022
1958 AHSME Problems
<math> \textbf{(A)}\ 5\qquad <math> \textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qq

25 KB (3,872 words) - 14:21, 20 February 2020
1993 AHSME Problems
...<math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals <math>\text{(A)} \ -4 \qquad \text{(B)} \ -2 \qquad \text{(C)} \ 0 \qquad \text{(D)} \ 2 \

20 KB (2,814 words) - 08:15, 27 June 2021
2011 AMC 12B Problems/Problem 20
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \q Let <math>A = (5,12)</math>, <math>B = (0,0)</math>, <math>C = (14, 0)</math>, <math>X=

8 KB (1,200 words) - 19:31, 7 August 2023
2011 AIME I Problems/Problem 4
...overline{AC} </math> at point <math>K</math>. Let <math>M</math> and <math>N</math> be the feet of the perpendiculars from <math>C</math> to <math> \ove pair A,B,C,D,K,L,M,N,P,Q;

6 KB (1,068 words) - 18:52, 2 August 2023
2011 AIME II Problems/Problem 10
...[[relatively prime]] positive integers. Find the remainder when <math>m + n</math> is divided by 1000. The line through the midpoint of a chord of a circle and the center of that circle is perpendicular to that chord, so <ma

11 KB (1,720 words) - 03:12, 18 December 2023
1999 AMC 8 Problems
<math>\text{(A)} \div \qquad \text{(B)}\ \times \qquad \text{(C)} + \qquad \text{(D)}\ - \ ...e degree measure of the smaller angle formed by the hands of a clock at 10 o clock?

17 KB (2,394 words) - 19:51, 8 May 2023
2011 AMC 10B Problems
<math> \textbf{(A)}\ -1\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad <math> \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 85 \qquad\textbf{(D)}\ 90

13 KB (2,090 words) - 18:05, 7 January 2021
2011 AMC 10B Problems/Problem 17
real r=3; pair A=(-3cos(80),-3sin(80));

5 KB (699 words) - 04:53, 21 January 2023
2007 AMC 10B Problems
<math>\textbf{(A) } 678 \qquad\textbf{(B) } 768 \qquad\textbf{(C) } 786 \qquad\textbf{(D) } Define the operation <math>\star</math> by <math>a \star b = (a+b)b.</math> What is <math>(3 \star 5) - (5 \star 3)?</math>

15 KB (2,297 words) - 12:57, 19 February 2020
2002 AMC 10B Problems/Problem 24
...any seconds does it take a rider to travel from the bottom of the wheel to a point <math>10</math> vertical feet above the bottom? <math> \mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 7.5\qquad \mathrm{(D)

3 KB (559 words) - 02:44, 8 February 2024
Olympiad books
Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to trai *''Inequalities: A Mathematical Olympiad Approach'' - '''Radmila Bulajich Manfrino, Jose Anton

17 KB (2,261 words) - 00:30, 22 April 2024
Mock Amc 12
...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob ...r more information on Mock AMCs, as well as tips on how to write them, and a complete list of all Mock AMC 10s and Mock AMC 12s.

18 KB (2,206 words) - 19:41, 24 December 2020
1999 AMC 8 Problems/Problem 4
for (int a = 0; a < 6; ++a) dot((4*a,3*b));

1 KB (196 words) - 21:02, 14 April 2017
1967 AHSME Problems
...umber <math>5b9</math>. If <math>5b9</math> is divisible by 9, then <math>a+b</math> equals <math> \text{(A)}\ 2\qquad\text{(B)}\ 4\qquad\text{(C)}\ 6\qquad\text{(D)}\ 8\qquad\text{(E

20 KB (3,108 words) - 14:14, 20 February 2020
2003 AMC 8 Problems
Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of <math>\mathrm{(A)}\ 12 \qquad\mathrm{(B)}\ 16 \qquad\mathrm{(C)}\ 20 \qquad\mathrm{(D)}\ 22

16 KB (2,236 words) - 12:02, 19 February 2024
1999 AMC 8 Problems/Problem 8
...en, and W = white). They are hinged together as shown, then folded to form a cube. The face opposite the white face is label("R",(.5,2.3),N);

1 KB (176 words) - 00:34, 5 July 2013
2009 AMC 8 Problems
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. The <math> \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 11\qquad\

18 KB (2,551 words) - 18:46, 27 February 2024
1992 AJHSME Problems/Problem 11
The bar graph shows the results of a survey on color preferences. What percent preferred blue? for (int a = 1; a <= 6; ++a)

2 KB (226 words) - 00:09, 5 July 2013
1997 AHSME Problems
If <math>\texttt{a}</math> and <math>\texttt{b}</math> are digits for which <math> \begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array} </m

17 KB (2,590 words) - 13:38, 19 February 2020
1997 AHSME Problems/Problem 19
A circle with center <math>O</math> is tangent to the coordinate axes and to the hypotenuse of the <math pair A = origin;

3 KB (422 words) - 19:00, 9 August 2015
1979 USAMO Problems/Problem 4
...ath> and <math>OB</math>. Find <math>Q</math> on <math>OA</math> and <math>R</math> on <math>OB</math> collinear with <math>P</math> so that <math>\frac ...ath>(O_1),(O_2)</math> again at the inverses <math>Q',R'</math> of <math>Q,R.</math> Hence

3 KB (568 words) - 12:24, 11 March 2018
1996 AHSME Problems
<math> \text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E ...After <math>10</math> days of doing his chores daily, Walter has received a total of <math>36</math> dollars. On how many days did Walter do them excep

15 KB (2,343 words) - 13:39, 19 February 2020
2003 AMC 8 Problems/Problem 18
...one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following pair a=(102,256), b=(68,131), c=(162,101), d=(134,150);

2 KB (250 words) - 22:17, 5 January 2024
2009 PUMaC Problems
'''Algebra A''' ...for which all the numbers <cmath>\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots</cmath> are negative.

25 KB (4,154 words) - 16:27, 2 September 2011
1985 AHSME Problems
<math> \mathrm{(A)\ } 15 \qquad \mathrm{(B) \ }16 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{( In an arcade game, the "monster" is the shaded sector of a circle of radius <math>1</math> cm, as shown in the figure. The missing pie

17 KB (2,488 words) - 03:26, 20 March 2024
2010 AMC 8 Problems
<math> \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qq If <math>a @ b = \frac{a\times b}{a+b}</math> for <math>a,b</math> positive integers, then what is <math>5 @ 10</math>?

18 KB (2,768 words) - 21:05, 9 January 2024
2011 AMC 8 Problems
...math> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? <math>\textbf{(A) }\ \textdollar 1.50 \qquad \textbf{(B) }\ \textdollar 2.00 \qquad \textbf{

16 KB (2,371 words) - 17:34, 9 January 2024
Barycentric coordinates
...triangle <math> \Delta{A_1}{A_2}{A_3} </math>. These masses then determine a point <math> P </math>, which is the geometric centroid of the three masses The Central NC Math Group published a lecture concerning this topic at https://www.youtube.com/watch?v=KQim7-wrwL

25 KB (5,106 words) - 11:00, 24 May 2024
2012 AMC 12A Problems/Problem 22
Distinct planes <math>p_1,p_2,....,p_k</math> intersect the interior of a cube <math>Q</math>. Let <math>S</math> be the union of the faces of <math> <math> \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qqua

5 KB (940 words) - 17:13, 4 April 2020
2012 AIME I Problems
{{AIME Problems|year=2012|n=I}} ...s with three not necessarily distinct digits, <math>abc</math>, with <math>a \neq 0</math> and <math>c \neq 0</math> such that both <math>abc</math> and

10 KB (1,617 words) - 14:49, 2 June 2023
2007 iTest Problems/Problem 15
Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and ...one of its sides coincides with a side of the square. Then "circumscribe" a circle around the

3 KB (539 words) - 21:01, 29 July 2018
Mock AIME I 2012 Problems
...<math>q</math> and <math>r</math> are relatively prime. What is <math>p+q+r</math>? A permutation of the numbers <math>\{2</math>, <math>3</math>, <math>4</math>

7 KB (1,309 words) - 11:13, 8 April 2012
2012 USAJMO Problems/Problem 1
...h>, <math>S</math> are concyclic (in other words, these four points lie on a circle). pair A, B, C, O, P, Q, R, S;

4 KB (613 words) - 20:50, 19 December 2023
2015 IMO Problems/Problem 1
We say that a finite set <math>\mathcal{S}</math> in the plane is balanced if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is

4 KB (773 words) - 08:14, 19 July 2016
2002 Romanian NMO Problems
Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours h Prove that any real number <math>0<x<1</math> can be written as a difference of two positive and less than <math>1</math> irrational numbers.

10 KB (1,695 words) - 10:03, 10 May 2012
1974 AHSME Problems
<math> \mathrm{(A)\ } 4x+3y=xy \qquad \mathrm{(B) \ }y=\frac{4x}{6-y} \qquad \mathrm{(C) \ } <math> \mathrm{(A)\ } -\frac{h}{3} \qquad \mathrm{(B) \ }\frac{h}{3} \qquad \mathrm{(C) \ }

15 KB (2,151 words) - 14:04, 19 February 2020
1974 AHSME Problems/Problem 23
...t of tangency. If <math> TP=4 </math> and <math> T'Q=9 </math> then <math> r </math> is pair O = (0,0); pair T = dir(90);

2 KB (337 words) - 12:44, 5 July 2013
1954 AHSME Problems
<math>\textbf{(A)}\ y^2-5\sqrt{y^2-25} \qquad \textbf{(B)}\ -y^2 \qquad \textbf{(C)}\ y^2 \\ <math>\textbf{(A)}\ 4 \text{ and }1 \qquad \textbf{(B)}\ \text{only }1 \qquad \textbf{(C)}\

23 KB (3,535 words) - 16:29, 24 April 2020
1955 AHSME Problems
<math> \textbf{(A)}\ 3.75\times 10^{-7}\qquad\textbf{(B)}\ 3\frac{3}{4}\times 10^{-7}\qquad\t The smaller angle between the hands of a clock at <math>12:25</math> p.m. is:

22 KB (3,509 words) - 21:29, 31 December 2023
1998 USAMO Problems/Problem 6
...A_{i}A_{i+1}A_{i+2}A_{i+3}</math> have an inscribed circle. (Here <math>A_{n+j} = A_{j}</math>.) ...ath> and <math>A_{i+2}A_{i+3}A_{i+4}A_{i+5}</math> in an equiangular <math>n</math>-gon are tangential, and <math>A_iA_{i+3}</math> is the longest side

5 KB (871 words) - 18:59, 10 May 2023
1999 USAMO Problems
Some checkers placed on an <math>n\times n</math> checkerboard satisfy the following conditions: (a) every square that does not contain a checker shares a side with one that does;

3 KB (465 words) - 13:35, 4 July 2013
2008 AMC 8 Problems
<math>\textbf{(A)}\ 12 \qquad <math>\textbf{(A)}\ 8671 \qquad

14 KB (2,035 words) - 15:23, 26 January 2024
2012 AMC 8 Problems
...many pounds of meat does she need to make <math> 24 </math> hamburgers for a neighborhood picnic? <math> \textbf{(A)}\hspace{.05in}6\qquad\textbf{(B)}\hspace{.05in}6\frac{2}3\qquad\textbf{(C)

13 KB (1,835 words) - 08:51, 8 March 2024
2012 AMC 8 Problems/Problem 5
pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0);

2 KB (208 words) - 20:16, 3 January 2024
1982 USAMO Problems/Problem 5
...h <math>A</math>, and so that two spheres can be constructed through <math>A</math>, <math>B</math>, and <math>C</math> which are both tangent to <math> ...an be flattened to the plane <math>P'</math> through <math>A</math>, <math>O</math>, <math>O_1</math> and <math>O_2</math>.

3 KB (524 words) - 16:08, 11 July 2018
2013 AIME I Problems
{{AIME Problems|year=2013|n=I}} ...les twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

9 KB (1,580 words) - 13:07, 24 February 2024
1991 AHSME Problems/Problem 10
Point <math>P</math> is <math>9</math> units from the center of a circle of radius <math>15</math>. How many different chords of the circle c (A) 11 (B) 12 (C) 13 (D) 14 (E) 29

2 KB (387 words) - 14:27, 23 June 2021
2013 AIME II Problems
{{AIME Problems|year=2013|n=II}} ...\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>.

8 KB (1,402 words) - 12:17, 13 March 2020
2013 AIME II Problems/Problem 8
A hexagon that is inscribed in a circle has side lengths <math>22</math>, <math>22</math>, <math>20</math>, real a;

8 KB (1,357 words) - 09:23, 11 March 2024
2013 AIME II Problems/Problem 10
...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c+d</math>. pair A, B, K, L;

5 KB (823 words) - 17:57, 29 December 2023
Mock AIME 6 2006-2007 Problems
...of all positive integers of the form <math>2^r\cdot3^s</math>, where <math>r</math> and <math>s</math> are nonnegative integers that do not exceed <math Draw in the diagonals of a regular octagon. What is the sum of all distinct angle measures, in degree

7 KB (1,173 words) - 21:04, 7 December 2018
2014 AIME II Problems/Problem 11
...me positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>. ...ft(0, \frac{\sqrt{3}}{2}\right).</math> <math>M =</math> midpoint<math>(D, R) = \left(\frac{1}{4}, \frac{\sqrt{3}}{4}\right)</math> and the slope of <ma

6 KB (1,059 words) - 18:24, 20 January 2024
2014 AIME II Problems/Problem 10
...</math>, where <math>n</math> is an integer. Find the remainder when <math>n</math> is divided by <math>1000</math>. ...{i \theta_w}</math> and likewise for <math>z</math>. Consider circle <math>O</math> with the origin as the center and radius 2014 on the complex plane.

6 KB (1,045 words) - 13:08, 21 January 2024
ASIA TEAM Problems
...1</math> for the first time in <math>r</math> jumps. In particular, <math>o(1)=0</math> because Kelvin is already at lily pad <math>1</math>. ...first fly, he places it on <math>a_1</math>. After that, when he catches a fly, he places it on <math>a_i</math> for which <math>i</math> is the least

10 KB (1,710 words) - 23:23, 10 January 2020
Mock AIME I 2015 Problems
...ath>. Finally, Palmer announces the product of the numbers in boxes D and A. If <math>k</math> is the number that Palmer says, what is <math>20k</math ...th> and <math>b</math> are positive relatively prime integers. Find <math>a+b</math>.

9 KB (1,463 words) - 14:48, 12 February 2017
2013 AMC 8 Problems
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \ ...lf-pound packages for just \$3 per package." What is the regular price for a full pound of fish, in dollars? (Assume that there are no deals for bulk)

15 KB (2,162 words) - 20:05, 8 May 2023
1968 AHSME Problems
Let <math>P</math> units be the increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diameter <math>\text{(A) } \frac{1}{\pi}\quad\text{(B) } \pi\quad\text{(C) } \frac{\pi^2}{2}\quad\t

16 KB (2,571 words) - 14:13, 20 February 2020
2014 USAMO Problems/Problem 6
..., j\in\{0, 1, \ldots n\}</math>, then<cmath>\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.</cmath> ...</math> and in each cell place a prime <math>p</math> dividing <math>\gcd (a+i, b+j)</math>.

2 KB (361 words) - 11:55, 25 June 2020
1960 AHSME Problems
If <math>2</math> is a solution (root) of <math>x^3+hx+10=0</math>, then <math>h</math> equals: <math>\textbf{(A)}10\qquad

21 KB (3,242 words) - 21:27, 30 December 2020
2014 AMC 10B Problems/Problem 23
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is <math>\text{(A) } \dfrac32 \quad \text{(B) } \dfrac{1+\sqrt5}2 \quad \text{(C) } \sqrt3 \q

6 KB (1,060 words) - 19:15, 11 August 2023
2014 AMC 12B Problems/Problem 24
...nd <math>n</math> are relatively prime positive integers. What is <math>m+n</math> ? <math>\textbf{(A) }129\qquad

7 KB (1,037 words) - 08:09, 1 January 2023
2014 AMC 12B Problems/Problem 19
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is real r=(3+sqrt(5))/2;

4 KB (703 words) - 16:24, 9 September 2022
2014 AIME I Problems/Problem 13
...nts <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, pair A = (0,sqrt(850));

9 KB (1,404 words) - 21:07, 13 October 2023
2014 AIME I Problems/Problem 15
...is a positive integer not divisible by the square of any prime. Find <math>a+b+c</math>. ...=90^\circ</math>. But <math>DF=FE</math>, so <math>\triangle DEF</math> is a 45-45-90 triangle. Letting <math>DG=3x</math>, we have that <math>EG=4x</ma

10 KB (1,643 words) - 22:30, 28 January 2024
1973 AHSME Problems
A chord which is the perpendicular bisector of a radius of length 12 in a circle, has length <math> \textbf{(A)}\ 3\sqrt3\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 6\sqrt3\qquad\textbf{(D

18 KB (2,788 words) - 13:55, 20 February 2020
1990 AHSME Problems/Problem 28
...0,90,130</math> and <math>110</math> is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to th <math>\text{(A) } 12\quad

2 KB (335 words) - 11:51, 5 October 2019
1969 AHSME Problems
<math>\frac{a}{b},a \ne b,b \ne 0</math>, the value of the fraction is changed to <math>\frac{c <math>\text{(A) } \frac{1}{c-d}\quad

16 KB (2,662 words) - 14:12, 20 February 2020
1970 AHSME Problems
<math>\textbf{(A) }\sqrt{2}+\sqrt{3}\qquad A square and a circle have equal perimeters. The ratio of the area of the circle to the ar

15 KB (2,366 words) - 07:52, 26 December 2023
1986 AHSME Problems
<math>\textbf{(A)}\ 2y \qquad <math>\textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad

17 KB (2,512 words) - 18:30, 12 October 2023
1982 AHSME Problems
<math>\textbf{(A)} \ 2 \qquad If a number eight times as large as <math>x</math> is increased by two, then one

17 KB (2,500 words) - 19:05, 11 September 2023
1979 AHSME Problems
label("$A$",(-2,1),W); <math>\textbf{(A) }8\qquad

17 KB (2,664 words) - 01:34, 19 March 2022
1977 AHSME Problems
<math>\text{(A)}\ x \qquad <math>\textbf{(A)}\ \text{ equiangular}\qquad

15 KB (2,412 words) - 05:09, 27 November 2020
1976 AHSME Problems
<math>\textbf{(A) }-2\qquad For how many real numbers <math>x</math> is <math>\sqrt{-(x+1)^2}</math> a real number?

17 KB (2,835 words) - 14:36, 8 September 2021
1975 AHSME Problems
<math>\textbf{(A)}\ 3/4 \qquad <math>\textbf{(A)}\ \text{all }m\qquad

16 KB (2,512 words) - 04:48, 27 November 2021
2007 iTest Problems/Problem 19
...rechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!" ...son a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.

3 KB (397 words) - 01:18, 29 October 2023
2007 iTest Problems/Problem 18
...ic with a double root at <math>a</math> and another root at b, where <math>a</math> and <math>b</math> are real numbers. If <math>p=-6</math> and <math>q=9</math>, what is <math>r</math>?

2 KB (393 words) - 17:01, 10 June 2018
2007 iTest Problems/Problem 21
James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's. <math>\text{(A) }0\qquad

2 KB (247 words) - 03:34, 14 June 2018
2007 iTest Problems/Problem 22
<math>\text{(A) } 0 \quad <math>\text{(N) } 13 \quad

929 bytes (137 words) - 22:05, 10 January 2019
2007 iTest Problems/Problem 25
<math>\text{(A) } 0\qquad \text{(N) } 13\qquad

2 KB (270 words) - 14:35, 29 July 2018
2013 Canadian MO Problems
is a constant polynomial. ...+\cdots + a_n</math> all have di fferent remainders when divided by <math>n + 1</math>?

2 KB (381 words) - 13:45, 8 October 2014
2007 iTest Problems/Problem 23
<math>\text{(A) } 0\quad \text{(N) } 3+2i\quad

1 KB (202 words) - 16:48, 24 November 2018
2007 iTest Problems/Problem 24
...such that <math>2008N</math> is a perfect square and <math>2007N</math> is a perfect cube. Find the remainder when <math>N</math> is divided by <math>25</math>.

2 KB (340 words) - 19:49, 30 June 2018
2007 IMO Problems/Problem 4
...tor of <math>\angle{BCA}</math> intersects the circumcircle again at <math>R</math>, the perpendicular bisector of <math>BC</math> at <math>P</math>, an real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;

8 KB (1,480 words) - 14:52, 5 August 2022
2014 IMO Problems/Problem 3
...}</math>. Point <math>H</math> is the foot of the perpendicular from <math>A</math> to <math>BD</math>. Points <math>S</math> and <math>T</math> lie on pair A = (0,1), C=(0,-1);

6 KB (1,071 words) - 03:58, 8 September 2018
1964 AHSME Problems
<math>\textbf{(A)}\ \log_{10}50 \qquad <math>\textbf{(A)}\ \text{a parabola} \qquad

19 KB (2,907 words) - 14:16, 20 February 2020
1963 AHSME Problems
<math>\textbf{(A)}\ (0,0)\qquad let <math>n=x-y^{x-y}</math>. Find <math>n</math> when <math>x=2</math> and <math>y=-2</math>.

20 KB (3,122 words) - 14:17, 20 February 2020
1957 AHSME Problems
...lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is: <math>\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qqua

26 KB (3,950 words) - 21:09, 31 August 2020
1986 AHSME Problems/Problem 28
...P, AQ</math> and <math>AR</math> are the perpendiculars dropped from <math>A</math> onto <math>CD, CB</math> extended and <math>DE</math> extended, respectively. Let <math>O</math> be the center of the pentagon. If <math>OP = 1</math>, then <math>AO

4 KB (702 words) - 17:13, 17 April 2020
2015 AIME II Problems
{{AIME Problems|year=2015|n=II}} .../math> percent greater than another integer. Find the remainder when <math>N</math> is divided by <math>1000</math>.

8 KB (1,326 words) - 19:15, 13 January 2024
2015 AIME II Problems/Problem 11
...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. pair A,B,C,O, Q, P, M, N;

9 KB (1,249 words) - 21:59, 26 November 2023
2016 AMC 10A Problems/Problem 24
A quadrilateral is inscribed in a circle of radius <math>200\sqrt{2}</math>. Three of the sides of this quadr <math>\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\te

24 KB (3,861 words) - 14:17, 26 April 2024
2015 USAMO Problems/Problem 2
...n segment <math>\overline{PQ}</math>, show that <math>M</math> moves along a circle. <cmath>A=(1,0) P=(1-a,b), Q=(1-a,-b)</cmath>,

5 KB (902 words) - 09:58, 20 August 2021
Aops font
The AoPS font is a font that can be used to make certain special symbols in the forum. You enc | a

6 KB (924 words) - 21:31, 22 February 2024
2004 IMO Problems/Problem 1
...circumcircles of the triangles <math>BMR</math> and <math>CNR</math> have a common point lying on the side <math>BC</math>. pair A,B,C,O,K,M,N,R,D;

5 KB (941 words) - 00:51, 19 November 2023
2017 AIME I Problems
{{AIME Problems|year=2017|n=I}} ...points are designated on <math>\triangle ABC</math>: the 3 vertices <math>A</math>, <math>B</math>, and <math>C</math>; <math>3</math> other points on

7 KB (1,163 words) - 16:43, 2 June 2022
2019 AMC 12A Problems
{{AMC12 Problems|year=2019|ab=A}} ...za with radius <math>3</math> inches. What is the integer closest to <math>N</math>?

14 KB (2,180 words) - 22:25, 25 April 2024
2016 AMC 10B Problems
...is the value of <math>\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}</math> when <math>a= \tfrac{1}{2}</math>? <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{5}{2}\qquad\textbf{(D)}\

12 KB (1,868 words) - 17:50, 30 December 2023
2016 AIME I Problems/Problem 6
...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. real c=8.1,a=5*(c+sqrt(c^2-64))/6,b=5*(c-sqrt(c^2-64))/6;

14 KB (2,397 words) - 20:04, 27 August 2023
2016 AIME II Problems/Problem 14
...planes). There is a point <math>O</math> whose distance from each of <math>A,B,C,P,</math> and <math>Q</math> is <math>d</math>. Find <math>d</math>. ...ngle ABC</math>, and <math>L</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. We must have <math>\tan(\angle KLP+ \angle QLK)=

15 KB (2,560 words) - 01:44, 1 July 2023
2016 AIME II Problems/Problem 9
We have <math>a_k=r^{k-1}</math> and <math>b_k=(k-1)d</math>. First, <math>b_{k-1}<c_{k-1}=100< ...r values of <math>r</math> such that <math>j(r) -i(r)>1</math>. Let's make a table:

6 KB (983 words) - 01:18, 2 February 2023
2006 iTest Problems
<math>\mathrm{(A)}\, 8</math> <math>\mathrm{(A)}\, 15\quad\mathrm{(B)}\, \frac{40}{3}</math>

31 KB (4,811 words) - 00:02, 4 November 2023
1954 AHSME Problems/Problem 37
...ath>, <math>PQ</math> extended to <math>D</math> and <math>\angle n</math> a right angle, then: path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false)

3 KB (560 words) - 10:11, 3 May 2020
1982 AHSME Problems/Problem 18
In the adjoining figure of a rectangular solid, <math>\angle DHG=45^\circ</math> and <math>\angle FHB=60 real r=sqrt(3);

2 KB (294 words) - 08:42, 15 April 2016
2016 USAMO Problems
...ots, X_{100}</math> be a sequence of mutually distinct nonempty subsets of a set <math>S</math>. Any two sets <math>X_i</math> and <math>X_{i+1}</math> ...ABC</math> be an acute triangle, and let <math>I_B, I_C,</math> and <math>O</math> denote its <math>B</math>-excenter, <math>C</math>-excenter, and cir

4 KB (608 words) - 13:49, 22 November 2023
2016 USAJMO Problems
...on the arc <math>\stackrel{\frown}{BC}</math> that does not contain <math>A</math>, and let <math>I_B</math> and <math>I_C</math> denote the incenters ...the circumcircle of triangle <math>\triangle PI_BI_C</math> passes through a fixed point.

3 KB (414 words) - 16:43, 5 August 2023
2016 USAMO Problems/Problem 3
...ABC</math> be an acute triangle, and let <math>I_B, I_C,</math> and <math>O</math> denote its <math>B</math>-excenter, <math>C</math>-excenter, and cir 1. Let <math>I_A</math> be the <math>A</math>-excenter, then <math>I_A,O,</math> and <math>P</math> are colinear. This can be proved by the Trigonom

6 KB (998 words) - 21:36, 17 October 2022
2016 USAMO Problems/Problem 5
...<math>M\in\overline{AB},</math> <math>Q\in\overline{AC},</math> and <math>N, P\in\overline{BC}.</math> Let <math>S</math> be the intersection of lines ...t <math>\overline{OI}</math> is parallel to <math>\ell,</math> where <math>O</math> is the circumcenter of triangle <math>ABC,</math> and <math>I</math>

8 KB (1,449 words) - 00:09, 12 October 2023
1976 AHSME Problems/Problem 18
O=(0,0), c1=circle(O, r1);

2 KB (361 words) - 08:05, 9 April 2023
Mock AIME 4 Pre 2005/Problems
...n>1</math> is it possible to express <math>2005</math> as the sum of <math>n</math> distinct positive integers? ...> is the geometric mean of the previous <math>n - 1</math> terms for <math>n > 3</math> and <math>b_{2005} = 6.</math> If <math>a_i = b_i</math> for <ma

7 KB (1,094 words) - 15:39, 24 March 2019
Mock AIME 3 2010 Problems
-1. Find the sum of all distinct integers n such that n|243. ...can distribute 1 piece of candy among 243 hungry schoolchildren sitting in a row

14 KB (2,904 words) - 18:24, 16 May 2017
2017 AMC 12A Problems/Problem 24
Quadrilateral <math>ABCD</math> is inscribed in circle <math>O</math> and has side lengths <math>AB=3, BC=2, CD=6</math>, and <math>DA=8</ ...math>\overline{AC}</math>. Let <math>G</math> be the point on circle <math>O</math> other than <math>C</math> that lies on line <math>CX</math>. What is

8 KB (1,322 words) - 03:24, 7 October 2023
2017 AMC 12A Problems/Problem 16
...ly tangent to each other and internally tangent to the largest semicircle. A circle centered at <math>P</math> is drawn externally tangent to the two sm <math> \textbf{(A)}\ \frac{3}{4}

13 KB (1,982 words) - 17:12, 20 December 2022
2017 AIME I Problems/Problem 4
...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. ...<math>O</math>. Note that <math>O</math> is equidistant from each of <math>A</math>, <math>B</math>, and <math>C</math>. Then,

7 KB (1,177 words) - 15:55, 5 January 2024
2017 AIME I Problems/Problem 8
...and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. ...th diameter <math>\overline{OP}</math> and points <math>Q</math> and <math>R</math> on the semicircle. Since the radius of the semicircle is <math>100</

9 KB (1,539 words) - 15:47, 17 February 2024
2017 AIME I Problems/Problem 15
...d <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p.</math> pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0);

22 KB (3,622 words) - 17:11, 6 January 2024
2017 AIME II Problems
{{AIME Problems|year=2017|n=II}} ...he triangle is closer to vertex <math>B</math> than to either vertex <math>A</math> or vertex <math>C</math> can be written as <math>\frac{p}{q}</math>,

7 KB (1,200 words) - 15:02, 8 September 2020
2017 AIME II Problems/Problem 12
...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. ...t(\frac{r-r^2}{r^2+1}\right)^2}=\frac{1-r}{\sqrt{r^2+1}}.</math> Let <math>r=\frac{11}{60}</math>, and the distance from the origin is <math>\frac{49}{6

6 KB (908 words) - 02:35, 23 January 2024
2017 AIME II Problems/Problem 3
...he triangle is closer to vertex <math>B</math> than to either vertex <math>A</math> or vertex <math>C</math> can be written as <math>\frac{p}{q}</math>, pair A,B,C,D,X,Z,P;

9 KB (1,411 words) - 22:18, 29 January 2024
2017 AIME II Problems/Problem 15
...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. ...<math>D</math> with respect to line <math>MN</math>. If <math>X</math> is a point in space and <math>X'</math> is the point symmetric to <math>X</math>

6 KB (971 words) - 02:08, 22 January 2024
Circular Inversion
...e transformed point on the ray extending from <math>O</math> through <math>A</math>. pair O = (0, 0);

16 KB (2,516 words) - 23:48, 15 January 2024
2017 UNCO Math Contest II Problems
A prime is an integer strictly greater than one that is evenly divisible by n pair A=24*dir(40),B=15*dir(220);

7 KB (1,192 words) - 15:14, 20 August 2020
2017 UNCO Math Contest II Problems/Problem 2
pair A=dir(60),B=dir(120),C=dir(180),D=dir(240),E=dir(300),F=dir(360),O=(0,0); pair G=(2/sqrt(3))*A,H=(2/sqrt(3))*B,I=(2/sqrt(3))*C,J=(2/sqrt(3))*D,K=(2/sqrt(3))*E,L=(2/sqrt(3

1 KB (195 words) - 18:22, 16 January 2023
2018 AMC 10B Problems/Problem 24
Let <math>ABCDEF</math> be a regular hexagon with side length <math>1</math>. Denote by <math>X</math>, <math>\textbf{(A)}\ \frac {3}{8}\sqrt{3} \qquad \textbf{(B)}\ \frac {7}{16}\sqrt{3} \qquad \

17 KB (2,562 words) - 03:38, 6 September 2022
2018 AMC 12B Problems/Problem 4
A circle has a chord of length <math>10</math>, and the distance from the center of the ci \textbf{(A) }25\pi \qquad

2 KB (244 words) - 22:06, 27 May 2023
1960 AHSME Problems/Problem 32
...E</math> is a straight line, <math>AP = AD</math>, and <math>AB</math> has a length twice the radius. Then: pair O=origin,E=dir(e),C=dir(c),B=dir(180+c),D=dir(180+e), rot=rotate(90,B)*O,A=extension(E,D,B,rot);

3 KB (508 words) - 20:33, 17 August 2022
2018 AIME I Problems/Problem 10
...that begin and end at point <math>A</math>. Find the remainder when <math>n</math> is divided by <math>1000.</math> real r = 0.05;

11 KB (1,934 words) - 12:18, 29 March 2024
1977 AHSME Problems/Problem 19
...the diagonals of convex quadrilateral <math>ABCD</math>, and let <math>P,Q,R</math>, and <math>S</math> be the centers of the circles <math>\textbf{(A) }PQRS\text{ is a parallelogram}\\

2 KB (305 words) - 08:38, 28 June 2018
2003 Indonesia MO Problems
...ath>a^9 - a</math> is divisible by <math>6</math> for every integers <math>a</math>. ...>, respectively. <math>PR</math> and <math>QS</math> intersects at <math>O</math>. Prove that <math>PO = OR</math> and <math>QO = OS</math>.

3 KB (497 words) - 02:20, 19 May 2024
1973 AHSME Problems/Problem 15
A sector with acute central angle <math> \theta</math> is cut from a circle of radius 6. The radius of the circle circumscribed about the sector <math> \textbf{(A)}\ 3\cos\theta \qquad

2 KB (306 words) - 14:00, 20 February 2020
2003 Indonesia MO Problems/Problem 2
...>, respectively. <math>PR</math> and <math>QS</math> intersects at <math>O</math>. Prove that <math>PO = OR</math> and <math>QO = OS</math>. pair A=(0,0), B=(60,0), C=(40,80), D=(10,50), P=(30,0), Q=(50,40), R=(25,65), SX=(5,25);

3 KB (583 words) - 17:52, 22 January 2024
2012 Indonesia MO Problems
...nd <math>b</math>, the number <cmath>n=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b</cmath> is an even non-negative integer. ...s ,a_n</math> be positive real numbers such that <math>a_{2}a_{3}\cdots a_{n}=1</math>. Prove that

5 KB (774 words) - 00:15, 13 August 2018
1973 AHSME Problems/Problem 35
...llel to the unit radius <math>OR</math> of the circle with center at <math>O</math>. Chords <math>MP</math>, <math>PQ</math>, and <math>NR</math> are ea label("O", (0,0),W);

7 KB (1,181 words) - 20:41, 17 April 2022
2006 iTest Problems/Problem 18
<math>\text{(A) Ptolemy; Archimedes}\qquad \text{(N) Cauchy; De Moivre}\qquad

2 KB (252 words) - 12:04, 27 November 2018
2019 AMC 10A Problems
{{AMC10 Problems|year=2019|ab=A}} <math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qqu

13 KB (2,024 words) - 16:07, 22 April 2024
2019 AIME II Problems/Problem 15
...d <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. pair A,B,C,P,Q,X,Y,O;

7 KB (1,115 words) - 03:11, 7 January 2024
2019 AIME II Problems/Problem 11
...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> pair A = (2,sqrt(45));

12 KB (1,985 words) - 19:52, 28 January 2024
2019 AIME II Problems/Problem 13
...circle is equal to <math>\tfrac{1}{8}-\tfrac{\sqrt2}{n}.</math> Find <math>n.</math> Let $r$ denote the radius of the circle, $O$ be the center of the circle. Then:

7 KB (1,051 words) - 20:45, 27 January 2024
1988 IMO Problems
...ath> R</math> and <math> r</math> (<math> R > r</math>) with centre <math> O.</math> Fix <math> P</math> on the small circle and consider the variable c ...s 0 and 1 in such a manner that each of the sets has exactly <math> \frac {n}{2}</math> zeros?

3 KB (499 words) - 13:29, 2 August 2021
2019 AMC 10A Problems/Problem 21
...of its sides is tangent to the sphere. What is the distance between <math>O</math> and the plane determined by the triangle? \textbf{(A) }2\sqrt{3}\qquad

7 KB (1,000 words) - 15:03, 23 October 2021
2019 AMC 10A Problems/Problem 16
...figure below shows <math>13</math> circles of radius <math>1</math> within a larger circle. All the intersections occur at points of tangency. What is t <math>\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi\left(3\

6 KB (871 words) - 08:20, 20 November 2023
2004 IMO Problems
...circumcircles of the triangles <math>BMR</math> and <math>CNR</math> have a common point lying on the side <math>BC</math>. ...omials <math>f</math> with real coefficients such that for all reals <math>a,b,c</math> such that <math>ab + bc + ca = 0</math> we have the following re

3 KB (488 words) - 04:36, 21 February 2021

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