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2015 IMO Problems
We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> 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) - 12:53, 6 July 2022
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);

19 KB (3,221 words) - 01:05, 7 February 2023
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) - 19:33, 28 December 2022
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,<br/>

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

10 KB (1,797 words) - 02:05, 24 October 2023
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 (1,849 words) - 19:43, 2 January 2023
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

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