Difference between revisions of "User:Rowechen"

Latest revision as of 20:57, 8 August 2024

Have fun. Don't die.

Problem 13

A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, $r_n$ , with its current predecessor and exchanging them if and only if the last term is smaller.

The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass. The numbers compared at each step are underlined.

$\underline{1 \quad 9} \quad 8 \quad 7$ $1 \quad {}\underline{9 \quad 8} \quad 7$ $1 \quad 8 \quad \underline{9 \quad 7}$ $1 \quad 8 \quad 7 \quad 9$

Suppose that $n = 40$ , and that the terms of the initial sequence $r_1, r_2, \dots, r_{40}$ are distinct from one another and are in random order. Let $p/q$ , in lowest terms, be the probability that the number that begins as $r_{20}$ will end up, after one bubble pass, in the $30^{\mbox{th}}$ place. Find $p + q$ .

Solution

Problem 14

Given a positive integer $n^{}_{}$ , it can be shown that every complex number of the form $r+si^{}_{}$ , where $r^{}_{}$ and $s^{}_{}$ are integers, can be uniquely expressed in the base $-n+i^{}_{}$ using the integers $1,2^{}_{},\ldots,n^2$ as digits. That is, the equation

$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$

is true for a unique choice of non-negative integer $m^{}_{}$ and digits $a_0,a_1^{},\ldots,a_m$ chosen from the set $\{0^{}_{},1,2,\ldots,n^2\}$ , with $a_m\ne 0^{}){}$ . We write

$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$

to denote the base $-n+i^{}_{}$ expansion of $r+si^{}_{}$ . There are only finitely many integers $k+0i^{}_{}$ that have four-digit expansions

$k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.$

Find the sum of all such $k^{}_{}$ .

Solution

Problem 13

Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum_{k=1}^{1995}\frac 1{f(k)}.$

Solution

Problem 14

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d_{}$ are positive integers and $d_{}$ is not divisible by the square of any prime number. Find $m+n+d.$

Solution

Problem 13

If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$

Solution

Problem 14

In triangle $ABC,$ it is given that angles $B$ and $C$ are congruent. Points $P$ and $Q$ lie on $\overline{AC}$ and $\overline{AB},$ respectively, so that $AP = PQ = QB = BC.$ Angle $ACB$ is $r$ times as large as angle $APQ,$ where $r$ is a positive real number. Find the greatest integer that does not exceed $1000r$ .

Solution

Problem 14

Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\ldots,f_m)$ , meaning that $k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m$ , where each $f_i$ is an integer, $0\le f_i\le i$ , and $0<f_m$ . Given that $(f_1,f_2,f_3,\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\cdots+1968!-1984!+2000!$ , find the value of $f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j$ .

Solution

Problem 13

In a certain circle, the chord of a $d$ -degree arc is 22 centimeters long, and the chord of a $2d$ -degree arc is 20 centimeters longer than the chord of a $3d$ -degree arc, where $d < 120.$ The length of the chord of a $3d$ -degree arc is $- m + \sqrt {n}$ centimeters, where $m$ and $n$ are positive integers. Find $m + n.$

Solution

Problem 13

Let $N$ be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when $N$ is divided by 1000.

Solution

Problem 15

A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left?

Solution

Problem 15

Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac pq,$ where $p$ and $q$ are relatively prime integers, find $p+q.$

Solution LOL

Problem 15

In triangle $ABC$ , $AC=13$ , $BC=14$ , and $AB=15$ . Points $M$ and $D$ lie on $AC$ with $AM=MC$ and $\angle ABD = \angle DBC$ . Points $N$ and $E$ lie on $AB$ with $AN=NB$ and $\angle ACE = \angle ECB$ . Let $P$ be the point, other than $A$ , of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE$ . Ray $AP$ meets $BC$ at $Q$ . The ratio $\frac{BQ}{CQ}$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m-n$ .

Solution

Problem 14

Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$ , $M_3$ , $M_5$ , and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$ , $\overline{A_3 A_4}$ , $\overline{A_5 A_6}$ , and $\overline{A_7 A_8}$ , respectively. For $i = 1, 3, 5, 7$ , ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$ , $R_3 \perp R_5$ , $R_5 \perp R_7$ , and $R_7 \perp R_1$ . Pairs of rays $R_1$ and $R_3$ , $R_3$ and $R_5$ , $R_5$ and $R_7$ , and $R_7$ and $R_1$ meet at $B_1$ , $B_3$ , $B_5$ , $B_7$ respectively. If $B_1 B_3 = A_1 A_2$ , then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$ , where $m$ and $n$ are positive integers. Find $m + n$ .

Solution

Problem 15

For some integer $m$ , the polynomial $x^3 - 2011x + m$ has the three integer roots $a$ , $b$ , and $c$ . Find $|a| + |b| + |c|$ .

Solution

Problem 15

Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions (a) $0\le A<B<C\le99$ , (b) there exist integers $a$ , $b$ , and $c$ , and prime $p$ where $0\le b<a<c<p$ , (c) $p$ divides $A-a$ , $B-b$ , and $C-c$ , and (d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$ .

Solution

Problem 14

For positive integers $n$ and $k$ , let $f(n, k)$ be the remainder when $n$ is divided by $k$ , and for $n > 1$ let $F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$ . Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$ .

Solution

Problem 14

In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when $N$ is divided by $1000$ .

Solution

Problem 14

For each integer $n \ge 2$ , let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and $0\le y \le x \left\lfloor \sqrt x \right\rfloor$ , where $\left\lfloor \sqrt x \right\rfloor$ is the greatest integer not exceeding $\sqrt x$ . Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.

Solution

Problem 13

Let $\triangle ABC$ have side lengths $AB=30$ , $BC=32$ , and $AC=34$ . Point $X$ lies in the interior of $\overline{BC}$ , and points $I_1$ and $I_2$ are the incenters of $\triangle ABX$ and $\triangle ACX$ , respectively. Find the minimum possible area of $\triangle AI_1I_2$ as $X$ varies along $\overline{BC}$ .

Solution

Problem 14

The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \frac{8}{5}a_n + \frac{6}{5}\sqrt{4^n - a_n^2}$ for $n \geq 0$ . Find the greatest integer less than or equal to $a_{10}$ .

Solution

Problem 14

The incircle $\omega$ of triangle $ABC$ is tangent to $\overline{BC}$ at $X$ . Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$ . Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$ , respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$ . Assume that $AP = 3$ , $PB = 4$ , $AC = 8$ , and $AQ = \dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 15

David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\text{ }B,\text{ }C$ , which can each be inscribed in a circle with radius $1$ . Let $\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$ , and define $\varphi_B$ and $\varphi_C$ similarly. Suppose that $\sin\varphi_A=\frac{2}{3}$ , $\sin\varphi_B=\frac{3}{5}$ , and $\sin\varphi_C=\frac{6}{7}$ . All three quadrilaterals have the same area $K$ , which can be written in the form $\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 15

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$ . Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$ , respectively, with line $AB$ closer to point $X$ than to $Y$ . Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$ . The three points $C$ , $Y$ , $D$ are collinear, $XC = 67$ , $XY = 47$ , and $XD = 37$ . Find $AB^2$ .

Solution

Problem 15

Let $\triangle ABC$ be an acute triangle with circumcircle $\omega,$ and let $H$ be the intersection of the altitudes of $\triangle ABC.$ Suppose the tangent to the circumcircle of $\triangle HBC$ at $H$ intersects $\omega$ at points $X$ and $Y$ with $HA=3,HX=2,$ and $HY=6.$ The area of $\triangle ABC$ can be written as $m\sqrt{n},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n.$

Solution

Problem 14

Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$

Solution

Problem 13

How many integers $N$ less than 1000 can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\ge 1$ ?

Solution

Problem 11

Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:

Exactly five of the entries are $1$ 's, and the remaining four entries are $0$ 's.
Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13},a_{22},a_{31}\}$ and $\{a_{11},a_{22},a_{33}\})$ , no more than one of the eight has all three entries equal.

Find the number of distinct T-grids.

Solution

Problem 12

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.

Solution

Problem 11

Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$ , $a_2$ , and $a_3$ be the medians of the numbers in rows $1$ , $2$ , and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$ . Let $Q$ be the number of arrangements for which $m = 5$ . Find the remainder when $Q$ is divided by $1000$ .

Solution

Problem 10

Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$ .

Solution

Problem 11

For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2))=g(f(4))=0.$

Solution

Problem 11

A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\sum^{28}_{k=1} a_k$ is divided by 1000.

Solution

Problem 13

Point $D$ lies on side $\overline{BC}$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC.$ The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F,$ respectively. Given that $AB=4,BC=5,$ and $CA=6,$ the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt{n}}p,$ where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p.$

Solution

Problem 15

In triangle $ABC$ , we have $BC = 13$ , $CA = 37$ , and $AB = 40$ . Points $D$ , $E$ , and $F$ are selected on $BC$ , $CA$ , and $AB$ respectively such that $AD$ , $BE$ , and $CF$ concur at the circumcenter of $ABC$ . The value of $\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF}$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Determine $m+n$ .

Problem 12

$ABC$ is a scalene triangle. The circle with diameter $AB$ intersects $BC$ at $D$ , and $E$ is the foot of the altitude from $C$ . $P$ is the intersection of $AD$ and $CE$ . Given that $AP = 136$ , $BP = 80$ , and $CP = 26$ , determine the circumradius of $ABC$ .

Problem 15

$ABCD$ is a convex quadrilateral in which $AB \parallel CD$ . Let $U$ denote the intersection of the extensions of $AD$ and $BC$ . $\Omega_1$ is the circle tangent to line segment $BC$ which also passes through $A$ and $D$ , and $\Omega_2$ is the circle tangent to $AD$ which passes through $B$ and $C$ . Call the points of tangency $M$ and $S$ . Let $O$ and $P$ be the points of intersection between $\Omega_1$ and $\Omega_2$ . Finally, $MS$ intersects $OP$ at $V$ . If $AB = 2$ , $BC = 2005$ , $CD = 4$ , and $DA = 2004$ , then the value of $UV^2$ is some integer $n$ . Determine the remainder obtained when $n$ is divided by $1000$ .

Problem 13

$P(x)$ is the polynomial of minimal degree that satisfies $P(k) = \frac{1}{k(k+1)}$

for $k = 1, 2, 3, . . . , 10$ . The value of $P(11)$ can be written as $-\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Determine $m + n$ .

Problem 12

$ABCD$ is a cyclic quadrilateral with $AB = 8$ , $BC = 4$ , $CD = 1$ , and $DA = 7$ . Let $O$ and $P$ denote the circumcenter and intersection of $AC$ and $BD$ respectively. The value of $OP^2$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime, positive integers. Determine the remainder obtained when $m + n$ is divided by $1000$ .

Problem 11

$10$ lines and $10$ circles divide the plane into at most $n$ disjoint regions. Compute $n$ .

@@ Line 1: / Line 1: @@
-Hey how did you get to this page? If you aren't me then I have to say hello. If you are me then I must be pretty conceited to waste my time looking at my own page.
+Have fun. Don't die.
-If you aren't me, seriously, how did you get to this page? This is pretty cool. Well, nice meeting you! I'm going to stop wasting my time typing this up and do some math. Gtg. Bye.
+== Problem 13 ==
+A given sequence <math>r_1, r_2, \dots, r_n</math> of distinct real numbers can be put in ascending order by means of one or more "bubble passes".  A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, <math>r_n</math>, with its current predecessor and exchanging them if and only if the last term is smaller.
-Here's the AIME compilation I will be doing:
+The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass.  The numbers compared at each step are underlined.
+<center><math>\underline{1 \quad 9} \quad 8 \quad 7</math></center>
+<center><math>1 \quad {}\underline{9 \quad 8} \quad 7</math></center>
+<center><math>1 \quad 8 \quad \underline{9 \quad 7}</math></center>
+<center><math>1 \quad 8 \quad 7 \quad 9</math></center>
+Suppose that <math>n = 40</math>, and that the terms of the initial sequence <math>r_1, r_2, \dots, r_{40}</math> are distinct from one another and are in random order.  Let <math>p/q</math>, in lowest terms, be the probability that the number that begins as <math>r_{20}</math> will end up, after one bubble pass, in the <math>30^{\mbox{th}}</math> place.  Find <math>p + q</math>.
-== Problem 2 ==
+[[1987 AIME Problems/Problem 13|Solution]]
-Find the number of [[ordered pair]]s <math>(x,y)</math> of positive integers that satisfy <math>x \le 2y \le 60</math> and <math>y \le 2x \le 60</math>.
+== Problem 14 ==
+Given a positive integer <math>n^{}_{}</math>, it can be shown that every complex number of the form <math>r+si^{}_{}</math>, where <math>r^{}_{}</math> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>1,2^{}_{},\ldots,n^2</math> as digits. That is, the equation
+<center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center>
+is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m</math> chosen from the set <math>\{0^{}_{},1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0^{}){}</math>. We write
+<center><math>r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}</math></center>
+to denote the base <math>-n+i^{}_{}</math> expansion of <math>r+si^{}_{}</math>. There are only finitely many integers <math>k+0i^{}_{}</math> that have four-digit expansions
+<center><math>k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.</math></center>
+Find the sum of all such <math>k^{}_{}</math>.
-[[1998 AIME Problems/Problem 2|Solution]]
+[[1989 AIME Problems/Problem 14|Solution]]
+== Problem 13 ==
+Let <math>f(n)</math> be the integer closest to <math>\sqrt[4]{n}.</math>  Find <math>\sum_{k=1}^{1995}\frac 1{f(k)}.</math>
-== Problem 5 ==
+[[1995 AIME Problems/Problem 13|Solution]]
-Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is <math> \frac{m + n \pi}{p}, </math> where <math> m, n, </math> and <math> p </math> are positive integers, and <math> n </math> and <math> p </math> are relatively prime, find <math> m + n + p. </math>
+== Problem 14 ==
+In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18.  The two chords divide the interior of the circle into four regions.  Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form <math>m\pi-n\sqrt{d},</math> where <math>m, n,</math> and <math>d_{}</math> are positive integers and <math>d_{}</math> is not divisible by the square of any prime number.  Find <math>m+n+d.</math>
-[[2003 AIME I Problems/Problem 5|Solution]]
+[[1995 AIME Problems/Problem 14|Solution]]
-== Problem 6 ==
+== Problem 13 ==
-The cards in a stack of <math> 2n </math> cards are numbered consecutively from 1 through <math> 2n </math> from top to bottom. The top <math> n </math> cards are removed, kept in order, and form pile <math> A. </math> The remaining cards form pile <math> B. </math> The cards are then restacked by taking cards alternately from the tops of pile <math> B </math> and <math> A, </math> respectively. In this process, card number <math> (n+1) </math> becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles <math> A </math> and <math> B </math> are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. Find the number of cards in the magical stack in which card number 131 retains its original position.
+If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], indexed so that <math>a_1 < a_2 < a_3 < \cdots < a_n,</math> its complex power sum is defined to be <math>a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,</math> where <math>i^2 = - 1.</math>  Let <math>S_n</math> be the sum of the complex power sums of all nonempty [[subset]]s of <math>\{1,2,\ldots,n\}.</math>  Given that <math>S_8 = - 176 - 64i</math> and <math> S_9 = p + qi,</math> where <math>p</math> and <math>q</math> are integers, find <math>|p| + |q|.</math>
-[[2005 AIME II Problems/Problem 6|Solution]]
+[[1998 AIME Problems/Problem 13|Solution]]
-== Problem 8 ==
+== Problem 14 ==
-How many different <math>4\times 4</math> arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?
+In triangle <math>ABC,</math> it is given that angles <math>B</math> and <math>C</math> are congruent. Points <math>P</math> and <math>Q</math> lie on <math>\overline{AC}</math> and <math>\overline{AB},</math> respectively, so that <math>AP = PQ = QB = BC.</math> Angle <math>ACB</math> is <math>r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find the greatest integer that does not exceed <math>1000r</math>.
-[[1997 AIME Problems/Problem 8|Solution]]
+[[2000 AIME I Problems/Problem 14|Solution]]
-== Problem 9 ==
+== Problem 14 ==
-Given a nonnegative real number <math>x</math>, let <math>\langle x\rangle</math> denote the fractional part of <math>x</math>; that is, <math>\langle x\rangle=x-\lfloor x\rfloor</math>, where <math>\lfloor x\rfloor</math> denotes the greatest integer less than or equal to <math>x</math>. Suppose that <math>a</math> is positive, <math>\langle a^{-1}\rangle=\langle a^2\rangle</math>, and <math>2<a^2<3</math>. Find the value of <math>a^{12}-144a^{-1}</math>.
+Every positive integer <math>k</math> has a unique factorial base expansion <math>(f_1,f_2,f_3,\ldots,f_m)</math>, meaning that <math>k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m</math>, where each <math>f_i</math> is an integer, <math>0\le f_i\le i</math>, and <math>0<f_m</math>. Given that <math>(f_1,f_2,f_3,\ldots,f_j)</math> is the factorial base expansion of <math>16!-32!+48!-64!+\cdots+1968!-1984!+2000!</math>, find the value of <math>f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j</math>.
-[[1997 AIME Problems/Problem 9|Solution]]
+[[2000 AIME II Problems/Problem 14|Solution]]
-== Problem 10 ==
+== Problem 13 ==
-Let <math>S</math> be the set of points whose coordinates <math>x,</math> <math>y,</math> and <math>z</math> are integers that satisfy <math>0\le x\le2,</math> <math>0\le y\le3,</math> and <math>0\le z\le4.</math>  Two distinct points are randomly chosen from <math>S.</math>  The probability that the midpoint of the segment they determine also belongs to <math>S</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
+In a certain circle, the chord of a <math>d</math>-degree arc is 22 centimeters long, and the chord of a <math>2d</math>-degree arc is 20 centimeters longer than the chord of a <math>3d</math>-degree arc, where <math>d < 120.</math>  The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n</math> are positive integers.  Find <math>m + n.</math>
+[[2001 AIME I Problems/Problem 13|Solution]]
+== Problem 13 ==
+Let <math> N </math> be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when <math> N </math> is divided by 1000.
-[[2001 AIME I Problems/Problem 10|Solution]]
+[[2003 AIME I Problems/Problem 13|Solution]]
+== Problem 15 ==
+A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left?
-== Problem 11 ==
+[[2004 AIME II Problems/Problem 15|Solution]]
-In a rectangular array of points, with 5 rows and <math>N</math> columns, the points are numbered consecutively from left to right beginning with the top row.  Thus the top row is numbered 1 through <math>N,</math> the second row is numbered <math>N + 1</math> through <math>2N,</math> and so forth.  Five points, <math>P_1, P_2, P_3, P_4,</math> and <math>P_5,</math> are selected so that each <math>P_i</math> is in row <math>i.</math>  Let <math>x_i</math> be the number associated with <math>P_i.</math>  Now renumber the array consecutively from top to bottom, beginning with the first column.  Let <math>y_i</math> be the number associated with <math>P_i</math> after the renumbering.  It is found that <math>x_1 = y_2,</math> <math>x_2 = y_1,</math> <math>x_3 = y_4,</math> <math>x_4 = y_5,</math> and <math>x_5 = y_3.</math>  Find the smallest possible value of <math>N.</math>
+== Problem 15 ==
+Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </math> respectively. Let <math> m </math> be the smallest positive value of <math> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally tangent to <math> w_2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math>
-[[2001 AIME I Problems/Problem 11|Solution]]
+[[2005 AIME II Problems/Problem 15|Solution]]
+LOL
-== Problem 12 ==
+== Problem 15 ==
-A sphere is inscribed in the tetrahedron whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</math>  The radius of the sphere is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers.  Find <math>m + n.</math>
+In triangle <math>ABC</math>, <math>AC=13</math>, <math>BC=14</math>, and <math>AB=15</math>. Points <math>M</math> and <math>D</math> lie on <math>AC</math> with <math>AM=MC</math> and <math>\angle ABD = \angle DBC</math>. Points <math>N</math> and <math>E</math> lie on <math>AB</math> with <math>AN=NB</math> and <math>\angle ACE = \angle ECB</math>. Let <math>P</math> be the point, other than <math>A</math>, of intersection of the circumcircles of <math>\triangle AMN</math> and <math>\triangle ADE</math>. Ray <math>AP</math> meets <math>BC</math> at <math>Q</math>. The ratio <math>\frac{BQ}{CQ}</math> can be written in the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m-n</math>.
-[[2001 AIME I Problems/Problem 12|Solution]]
+[[2010 AIME II Problems/Problem 15|Solution]]
-== Problem 9 ==
+== Problem 14 ==
-The system of equations
+Let <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8</math> be a regular octagon.  Let <math>M_1</math>, <math>M_3</math>, <math>M_5</math>, and <math>M_7</math> be the midpoints of sides <math>\overline{A_1 A_2}</math>, <math>\overline{A_3 A_4}</math>, <math>\overline{A_5 A_6}</math>, and <math>\overline{A_7 A_8}</math>, respectively.  For <math>i = 1, 3, 5, 7</math>, ray <math>R_i</math> is constructed from <math>M_i</math> towards the interior of the octagon such that <math>R_1 \perp R_3</math>, <math>R_3 \perp R_5</math>, <math>R_5 \perp R_7</math>, and <math>R_7 \perp R_1</math>.  Pairs of rays <math>R_1</math> and <math>R_3</math>, <math>R_3</math> and <math>R_5</math>, <math>R_5</math> and <math>R_7</math>, and <math>R_7</math> and <math>R_1</math> meet at <math>B_1</math>, <math>B_3</math>, <math>B_5</math>, <math>B_7</math> respectively.  If <math>B_1 B_3 = A_1 A_2</math>, then <math>\cos 2 \angle A_3 M_3 B_1</math> can be written in the form <math>m - \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers.  Find <math>m + n</math>.
-<cmath>\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\
-\log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\
-\log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\
-\end{eqnarray*}</cmath>
-has two solutions <math>(x_{1},y_{1},z_{1})</math> and <math>(x_{2},y_{2},z_{2})</math>. Find <math>y_{1} + y_{2}</math>.
+[[2011 AIME I Problems/Problem 14|Solution]]
+== Problem 15 ==
+For some integer <math>m</math>, the polynomial <math>x^3 - 2011x + m</math> has the three integer roots <math>a</math>, <math>b</math>, and <math>c</math>.  Find <math>|a| + |b| + |c|</math>.
-[[2000 AIME I Problems/Problem 9|Solution]]
+[[2011 AIME I Problems/Problem 15|Solution]]
+==Problem 15==
+Let <math>N</math> be the number of ordered triples <math>(A,B,C)</math> of integers satisfying the conditions (a) <math>0\le A<B<C\le99</math>, (b) there exist integers <math>a</math>, <math>b</math>, and <math>c</math>, and prime <math>p</math> where <math>0\le b<a<c<p</math>, (c) <math>p</math> divides <math>A-a</math>, <math>B-b</math>, and <math>C-c</math>, and (d) each ordered triple <math>(A,B,C)</math> and each ordered triple <math>(b,a,c)</math> form arithmetic sequences. Find <math>N</math>.
-== Problem 12 ==
+[[2013 AIME I Problems/Problem 15|Solution]]
-Given a function <math>f</math> for which
+==Problem 14==
-<center><math>f(x) = f(398 - x) = f(2158 - x) = f(3214 - x)</math></center>
+For positive integers <math>n</math> and <math>k</math>, let <math>f(n, k)</math> be the remainder when <math>n</math> is divided by <math>k</math>, and for <math>n > 1</math> let <math>F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)</math>. Find the remainder when <math>\sum\limits_{n=20}^{100} F(n)</math> is divided by <math>1000</math>.
-holds for all real <math>x,</math> what is the largest number of different values that can appear in the list <math>f(0),f(1),f(2),\ldots,f(999)</math>?
-[[2000 AIME I Problems/Problem 12|Solution]]
+[[2013 AIME II Problems/Problem 14|Solution]]
 == Problem 14 ==
-There are <math>2n</math> complex numbers that satisfy both <math>z^{28} - z^{8} - 1 = 0</math> and <math>|z| = 1</math>. These numbers have the form <math>z_{m} = \cos\theta_{m} + i\sin\theta_{m}</math>, where <math>0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360</math> and angles are measured in degrees. Find the value of <math>\theta_{2} + \theta_{4} + \ldots + \theta_{2n}</math>.
+In a group of nine people each person shakes hands with exactly two of the other people from the group. Let <math>N</math> be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when <math>N</math> is divided by <math>1000</math>.
-[[2001 AIME II Problems/Problem 14|Solution]]
+[[2012 AIME II Problems/Problem 14|Solution]]
-== Problem 13 ==
+==Problem 14==
-In triangle <math>ABC</math> the medians <math>\overline{AD}</math> and <math>\overline{CE}</math> have lengths 18 and 27, respectively, and <math>AB = 24</math>.  Extend <math>\overline{CE}</math> to intersect the circumcircle of <math>ABC</math> at <math>F</math>.  The area of triangle <math>AFB</math> is <math>m\sqrt {n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime.  Find <math>m + n</math>.
+For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane defined by the inequalities <math>1\le x \le n</math> and <math>0\le y \le x \left\lfloor \sqrt x \right\rfloor</math>, where <math>\left\lfloor \sqrt x \right\rfloor</math> is the greatest integer not exceeding <math>\sqrt x</math>. Find the number of values of <math>n</math> with <math>2\le n \le 1000</math> for which <math>A(n)</math> is an integer.
+[[2015 AIME I Problems/Problem 14|Solution]]
+==Problem 13==
+Let <math>\triangle ABC</math> have side lengths <math>AB=30</math>, <math>BC=32</math>, and <math>AC=34</math>. Point <math>X</math> lies in the interior of <math>\overline{BC}</math>, and points <math>I_1</math> and <math>I_2</math> are the incenters of <math>\triangle ABX</math> and <math>\triangle ACX</math>, respectively. Find the minimum possible area of <math>\triangle AI_1I_2</math> as <math>X</math> varies along <math>\overline{BC}</math>.
-[[2002 AIME I Problems/Problem 13|Solution]]
+[[2018 AIME I Problems/Problem 13 | Solution]]
 == Problem 14 ==
-The perimeter of triangle <math>APM</math> is <math>152</math>, and the angle <math>PAM</math> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <math>\overline{AM}</math> and <math>\overline{PM}</math>. Given that <math>OP=m/n</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>.
+The sequence <math>(a_n)</math> satisfies <math>a_0=0</math> and <math>a_{n + 1} = \frac{8}{5}a_n + \frac{6}{5}\sqrt{4^n - a_n^2}</math> for <math>n \geq 0</math>. Find the greatest integer less than or equal to <math>a_{10}</math>.
-[[2002 AIME II Problems/Problem 14|Solution]]
+[[2009 AIME II Problems/Problem 14|Solution]]
+==Problem 14==
+The incircle <math>\omega</math> of triangle <math>ABC</math> is tangent to <math>\overline{BC}</math> at <math>X</math>. Let <math>Y \neq X</math> be the other intersection of <math>\overline{AX}</math> with <math>\omega</math>. Points <math>P</math> and <math>Q</math> lie on <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, so that <math>\overline{PQ}</math> is tangent to <math>\omega</math> at <math>Y</math>. Assume that <math>AP = 3</math>, <math>PB = 4</math>, <math>AC = 8</math>, and <math>AQ = \dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+[[2018 AIME II Problems/Problem 14 | Solution]]
+==Problem 15==
+David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, <math>A,\text{ }B,\text{ }C</math>, which can each be inscribed in a circle with radius <math>1</math>. Let <math>\varphi_A</math> denote the measure of the acute angle made by the diagonals of quadrilateral <math>A</math>, and define <math>\varphi_B</math> and <math>\varphi_C</math> similarly. Suppose that <math>\sin\varphi_A=\frac{2}{3}</math>, <math>\sin\varphi_B=\frac{3}{5}</math>, and <math>\sin\varphi_C=\frac{6}{7}</math>. All three quadrilaterals have the same area <math>K</math>, which can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
+[[2018 AIME I Problems/Problem 15 | Solution]]
+==Problem 15==
+Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>X</math> and <math>Y</math>. Line <math>\ell</math> is tangent to <math>\omega_1</math> and <math>\omega_2</math> at <math>A</math> and <math>B</math>, respectively, with line <math>AB</math> closer to point <math>X</math> than to <math>Y</math>. Circle <math>\omega</math> passes through <math>A</math> and <math>B</math> intersecting <math>\omega_1</math> again at <math>D \neq A</math> and intersecting <math>\omega_2</math> again at <math>C \neq B</math>. The three points <math>C</math>, <math>Y</math>, <math>D</math> are collinear, <math>XC = 67</math>, <math>XY = 47</math>, and <math>XD = 37</math>. Find <math>AB^2</math>.
+[[2016 AIME I Problems/Problem 15 | Solution]]
+==Problem 15==
+Let <math>\triangle ABC</math> be an acute triangle with circumcircle <math>\omega,</math> and let <math>H</math> be the intersection of the altitudes of <math>\triangle ABC.</math> Suppose the tangent to the circumcircle of <math>\triangle HBC</math> at <math>H</math> intersects <math>\omega</math> at points <math>X</math> and <math>Y</math> with <math>HA=3,HX=2,</math> and <math>HY=6.</math> The area of <math>\triangle ABC</math> can be written as <math>m\sqrt{n},</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math>
+[[2020 AIME I Problems/Problem 15 | Solution]]
+==Problem 14==
+Let <math>P(x)</math> be a quadratic polynomial with complex coefficients whose <math>x^2</math> coefficient is <math>1.</math> Suppose the equation <math>P(P(x))=0</math> has four distinct solutions, <math>x=3,4,a,b.</math> Find the sum of all possible values of <math>(a+b)^2.</math>
+[[2020 AIME I Problems/Problem 14 | Solution]]
 == Problem 13 ==
-Let <math> N </math> be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when <math> N </math> is divided by 1000.
+How many integers <math> N </math> less than 1000 can be written as the sum of <math> j </math> consecutive positive odd integers from exactly 5 values of <math> j\ge 1 </math>?
+[[2006 AIME II Problems/Problem 13|Solution]]
+== Problem 11 ==
+Define a <i>T-grid</i> to be a <math>3\times3</math> matrix which satisfies the following two properties:
+<OL>
+<LI>Exactly five of the entries are <math>1</math>'s, and the remaining four entries are <math>0</math>'s.</LI>
+<LI>Among the eight rows, columns, and long diagonals (the long diagonals are <math>\{a_{13},a_{22},a_{31}\}</math> and <math>\{a_{11},a_{22},a_{33}\})</math>, no more than one of the eight has all three entries equal.</LI></OL>
+Find the number of distinct <i>T-grids</i>.
+[[2010 AIME II Problems/Problem 11|Solution]]
+== Problem 12 ==
+Six men and some number of women stand in a line in random order.  Let <math>p</math> be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man.  Find the least number of women in the line such that <math>p</math> does not exceed 1 percent.
+[[2011 AIME I Problems/Problem 12|Solution]]
+==Problem 11==
+Consider arrangements of the <math>9</math> numbers <math>1, 2, 3, \dots, 9</math> in a <math>3 \times 3</math> array. For each such arrangement, let <math>a_1</math>, <math>a_2</math>, and <math>a_3</math> be the medians of the numbers in rows <math>1</math>, <math>2</math>, and <math>3</math> respectively, and let <math>m</math> be the median of <math>\{a_1, a_2, a_3\}</math>. Let <math>Q</math> be the number of arrangements for which <math>m = 5</math>. Find the remainder when <math>Q</math> is divided by <math>1000</math>.
+[[2017 AIME I Problems/Problem 11 | Solution]]
+==Problem 10==
+Find the number of functions <math>f(x)</math> from <math>\{1, 2, 3, 4, 5\}</math> to <math>\{1, 2, 3, 4, 5\}</math> that satisfy <math>f(f(x)) = f(f(f(x)))</math> for all <math>x</math> in <math>\{1, 2, 3, 4, 5\}</math>.
+[[2018 AIME II Problems/Problem 10 | Solution]]
+==Problem 11==
+For integers <math>a,b,c</math> and <math>d,</math> let <math>f(x)=x^2+ax+b</math> and <math>g(x)=x^2+cx+d.</math> Find the number of ordered triples <math>(a,b,c)</math> of integers with absolute values not exceeding <math>10</math> for which there is an integer <math>d</math> such that <math>g(f(2))=g(f(4))=0.</math>
+[[2020 AIME I Problems/Problem 11 | Solution]]
+== Problem 11 ==
+A sequence is defined as follows <math> a_1=a_2=a_3=1, </math> and, for all positive integers <math> n, a_{n+3}=a_{n+2}+a_{n+1}+a_n. </math> Given that <math> a_{28}=6090307, a_{29}=11201821, </math> and <math> a_{30}=20603361, </math> find the remainder when <math>\sum^{28}_{k=1} a_k </math> is divided by 1000.
+[[2006 AIME II Problems/Problem 11|Solution]]
+==Problem 13==
+Point <math>D</math> lies on side <math>\overline{BC}</math> of <math>\triangle ABC</math> so that <math>\overline{AD}</math> bisects <math>\angle BAC.</math> The perpendicular bisector of <math>\overline{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given that <math>AB=4,BC=5,</math> and <math>CA=6,</math> the area of <math>\triangle AEF</math> can be written as <math>\tfrac{m\sqrt{n}}p,</math> where <math>m</math> and <math>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p.</math>
+[[2020 AIME I Problems/Problem 13 | Solution]]
+==Problem 15==
+In triangle <math>ABC</math>, we have <math>BC = 13</math>, <math>CA = 37</math>, and <math>AB = 40</math>. Points <math>D</math>, <math>E</math>, and <math>F</math> are selected
+on <math>BC</math>, <math>CA</math>, and <math>AB</math> respectively such that <math>AD</math>, <math>BE</math>, and <math>CF</math> concur at the circumcenter of <math>ABC</math>. The value of <math>\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF}</math> can be expressed as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m+n</math>.
+==Problem 12==
+<math>ABC</math> is a scalene triangle. The circle with diameter <math>AB</math> intersects <math>BC</math> at <math>D</math>, and <math>E</math> is the foot of the altitude from <math>C</math>. <math>P</math> is the intersection of <math>AD</math> and <math>CE</math>. Given that <math>AP = 136</math>, <math>BP = 80</math>, and <math>CP = 26</math>, determine the circumradius of <math>ABC</math>.
+==Problem 15==
+<math>ABCD</math> is a convex quadrilateral in which <math>AB \parallel CD</math>. Let <math>U</math> denote the intersection of the extensions of <math>AD</math> and <math>BC</math>. <math>\Omega_1</math> is the circle tangent to line segment <math>BC</math> which also passes through <math>A</math> and <math>D</math>, and <math>\Omega_2</math> is the circle tangent to <math>AD</math> which passes through <math>B</math> and <math>C</math>. Call the points of tangency <math>M</math> and <math>S</math>. Let <math>O</math> and <math>P</math> be the points of intersection between <math>\Omega_1</math> and <math>\Omega_2</math>.
+Finally, <math>MS</math> intersects <math>OP</math> at <math>V</math>. If <math>AB = 2</math>, <math>BC = 2005</math>, <math>CD = 4</math>, and <math>DA = 2004</math>, then the value of <math>UV^2</math> is some integer <math>n</math>. Determine the remainder obtained when <math>n</math> is divided by <math>1000</math>.
+==Problem 13==
+<math>P(x)</math> is the polynomial of minimal degree that satisfies
+<cmath>P(k) = \frac{1}{k(k+1)}</cmath>
+for <math>k = 1, 2, 3, . . . , 10</math>. The value of <math>P(11)</math> can be written as <math>-\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively
+prime positive integers. Determine <math>m + n</math>.
+==Problem 12==
-[[2003 AIME I Problems/Problem 13|Solution]]
+<math>ABCD</math> is a cyclic quadrilateral with <math>AB = 8</math>, <math>BC = 4</math>, <math>CD = 1</math>, and <math>DA = 7</math>. Let <math>O</math> and <math>P</math> denote the circumcenter and intersection of <math>AC</math> and <math>BD</math> respectively. The value of <math>OP^2</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime, positive integers. Determine the remainder obtained when <math>m + n</math> is divided by <math>1000</math>.
-== Problem 14 ==
-The decimal representation of <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers and <math> m < n, </math> contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of <math> n </math> for which this is possible.
-[[2003 AIME I Problems/Problem 14|Solution]]
+==Problem 11==
+<math>10</math> lines and <math>10</math> circles divide the plane into at most <math>n</math> disjoint regions. Compute <math>n</math>.

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Difference between revisions of "User:Rowechen"

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