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2008 AIME I Problems/Problem 11
...math>B</math>, and <math>AABAA</math>, while <math>BBAB</math> is not such a sequence. How many such sequences have length 14? ...nding two <math>B</math>s to a string of length <math>n-2</math> ending in a <math>B</math>. Thus, we have the [[recursion]]s

7 KB (1,173 words) - 22:39, 28 November 2023
2008 AIME I Problems/Problem 15
.... When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of t triple A=(0,0,0),B=(1,0,0),C=(1,1,0),D=(0,1,0);

6 KB (1,041 words) - 00:54, 1 February 2024
2008 AIME II Problems/Problem 5
...>, let <math>BC = 1000</math> and <math>AD = 2008</math>. Let <math>\angle A = 37^\circ</math>, <math>\angle D = 53^\circ</math>, and <math>M</math> and Extend <math>\overline{AB}</math> and <math>\overline{CD}</math> to meet at a point <math>E</math>. Then <math>\angle AED = 180 - 53 - 37 = 90^{\circ}</m

8 KB (1,338 words) - 23:15, 28 November 2023
2008 AIME II Problems/Problem 11
pair A=(0,96),B=(-28,0),C=(28,0),X=C-(64/3,0),Y=B+(4*r/3,0),P=X+(0,16),Q=Y+(0,r),M D(A--B--C--cycle); D(Y--Q--P--X); D(Q--M); D(P--C,dashed);

6 KB (1,065 words) - 20:12, 9 August 2022
2003 IMO Problems/Problem 2
Determine all pairs of positive integers <math>(a,b)</math> such that <cmath> \frac{a^2}{2ab^2-b^3+1} </cmath>

2 KB (397 words) - 00:23, 16 December 2017
2001 Iran NMO (Round 3) Problems/Problem 5
...math>, respectively. Prove that <math>S</math>, <math>T</math>, and <math>A'</math> are collinear. pair A=(1,2.7), B=(0,0), C=(3,0);

3 KB (437 words) - 15:47, 27 April 2008
2008 USAMO Problems/Problem 2
...ath>F</math>, inside of triangle <math>ABC</math>. Prove that points <math>A</math>, <math>N</math>, <math>F</math>, and <math>P</math> all lie on one c ...\angle OFA = 90^{\circ}</math>, as this would immediately prove that <math>A,P,O,F,N</math> are concyclic.

20 KB (3,565 words) - 11:54, 1 May 2024
2003 AMC 12A Problems/Problem 17
...h>M</math> intersects a circle with radius <math>4</math> and center <math>A</math> at points <math>P</math> and <math>D</math>. What is the distance fr pair A,B,C,D,M,P;

6 KB (1,026 words) - 22:35, 29 March 2023
Spiral similarity
...milarity is a plane transformation composed of a rotation of the plane and a dilation of the plane having the common center. The order in which the com Any two directly similar figures are related either by a translation or by a spiral similarity (directly similar figures are similar and have the same o

28 KB (4,863 words) - 00:29, 16 December 2023
1992 USAMO Problems/Problem 4
...>, <math>B'</math>, <math>C'</math>, and <math>P</math>. Prove that <math>AA'=BB'=CC'</math>. ...y symmetry, <math>BB'=CC'</math> and <math>CC'=AA'</math>, and hence <math>AA'=BB'=CC'</math> as desired.

5 KB (807 words) - 18:37, 25 June 2021
2002 AMC 10A Problems/Problem 20
Points <math>A,B,C,D,E</math> and <math>F</math> lie, in that order, on <math>\overline{AF <math>\text{(A)}\ 5/4 \qquad \text{(B)}\ 4/3 \qquad \text{(C)}\ 3/2 \qquad \text{(D)}\ 5/3

3 KB (439 words) - 22:15, 9 June 2023
1962 IMO Problems/Problem 3
...> begin their motion at the same instant from the starting positions <math>A</math> and <math>B'</math>, respectively. Determine and draw the locus of t ...math>t</math> being time, and the average of two such lines must also have a linear parametric form.

3 KB (581 words) - 02:00, 6 May 2023
1976 USAMO Problems/Problem 2
...>AX</math> and <math>BY</math>. You may assume that <math>AB</math> is not a diameter. pair A=r*expi(5*pi/6), B=r*expi(pi/6), X=r*expi(pi/3), X1=r*expi(-pi/12), Y=r*expi

5 KB (848 words) - 23:41, 6 July 2020
2000 AMC 10 Problems/Problem 16
label("$A$",(0,3),NW); <math>\textbf{(A)}\ \frac{4\sqrt{5}}{3} \qquad\textbf{(B)}\ \frac{5\sqrt{5}}{3} \qquad\textb

7 KB (966 words) - 23:22, 31 July 2023
2002 AMC 12A Problems/Problem 18
\text{(A) }15 ...ing the same pattern, has its center, <math>D_2</math>, at (-15,0) and has a radius of <math>\sqrt{81}</math> = 9. So we can construct this diagram:

3 KB (485 words) - 03:13, 1 September 2023
2009 AIME II Problems/Problem 3
pair A=(0,10), B=(0,0), C=(14,0), D=(14,10), Q=(0,5); draw (A--B--C--D--cycle);

4 KB (572 words) - 16:47, 2 August 2023
2009 AIME II Problems/Problem 10
...D</math> and <math>B</math> are equal. The number of kilometers from <math>A</math> to <math>D</math> is given by <math>\frac {p\sqrt{q}}{r}</math>, whe ...),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
1972 AHSME Problems/Problem 18
Let <math>ABCD</math> be a trapezoid with the measure of base <math>AB</math> twice that of base <math <math>\textbf{(A) }3\textstyle\frac{2}{3}\qquad \textbf{(B) }3\frac{3}{4}\qquad \textbf{(C)

2 KB (267 words) - 04:54, 23 June 2022
2009 IMO Problems/Problem 2
Let <math>ABC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <m dot("A", (40, 100), N);

2 KB (280 words) - 01:16, 19 November 2023
2002 AMC 12A Problems/Problem 23
<math>\text{(A)}\ 14 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 14\sqr pair A, B, C, D, M;

6 KB (899 words) - 01:41, 5 July 2023

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