Suppose $\omega$ is a circle centered at $O$ with radius $8$. Let $AC$ and $BD$ be perpendicular chords of $\omega$. Let $P$ be a point inside quadrilateral $ABCD$ such that the circumcircles of triangles $ABP$ and $CDP$ are tangent, and the circumcircles of triangles $ADP$ and $BCP$ are tangent. If $AC = 2\sqrt{61}$ and $BD = 6\sqrt7$,then $OP$ can be expressed as $\sqrt{a}-\sqrt{b}$ for positive integers $a$ and $b$. Compute $100a + b$.

Find the area of a figure consisting of points whose coordinates satisfy the inequality $$(y^3 - arcsin x)(x^3 + arcsin y) \ge 0.$$

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

Find the number of all integers $n$ with $4\le n\le 1023$ which contain no three consecutive equal digits in their binary representations.

There is a finite number of polygons in a plane and each two of them have a point in common. Prove that there exists a line which crosses every polygon.

Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$

Let $ f(x) \equal{} \frac{x^2\plus{}1}{2x}$ for $ x \neq 0.$ Define $ f^{(0)}(x) \equal{} x$ and $ f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x))$ for all positive integers $ n$ and $ x \neq 0.$ Prove that for all non-negative integers $ n$ and $ x \neq \{\minus{}1,0,1\}$ \[ \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.\]

Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$

$ ABCD$ is a square. Let $ E$ be a point on the segment $ BC$ and $ F$ be a point on the segment $ ED$. If $ DF \equal{} BF$ and $ EF \equal{} BE$, then $ \angle DFA$ is $\textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 60^\circ \qquad\textbf{(C)}\ 75^\circ \qquad\textbf{(D)}\ 80^\circ \qquad\textbf{(E)}\ 85^\circ$

Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.

A poll shows that $ 70\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work? $ \textbf{(A)}\ 0.063 \qquad \textbf{(B)}\ 0.189 \qquad \textbf{(C)}\ 0.233 \qquad \textbf{(D)}\ 0.333 \qquad \textbf{(E)}\ 0.441$

In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores: (a) Michael K had an average test score of $90$, Michael M had an average test score of $91$, and Michael R had an average test score of $92$. (b) Michael K took more tests than Michael M, who in turn took more tests than Michael R. (c) Michael M got a higher total test score than Michael R, who in turn got a higher total test score than Michael K. (The total test score is the sum of the test scores over all tests) What is the least number of tests that Michael K, Michael M, and Michael R could have taken combined? [i]Proposed by James Lin[/i]

Find all pairs of natural numbers $(a, b)$ that fulfill $a^b=(a+b)^a$.

In triangle $\triangle ABC$, $D$ lies between $A$ and $C$ and $AC=3AD$, $E$ lies between $B$ and $C$ and $BC=4EC$. $B,G,F,D$ in that order, are on a straight line and $BD=5GF=5FD$. Suppose the area of $\triangle ABC$ is $900$, find the area of the triangle $\triangle EFG$.

Let $D$ be a plane region bounded by a circle of radius $r.$ Let $(x,y)$ be a point of $D$ and consider a circle of radius $\delta$ and center at $(x,y).$ Denote by $l(x,y)$ the length of that arc of the circle which is outside $D.$ Find $$\lim_{\delta \to 0} \frac{1}{\delta^{2}} \int_{D} l(x,y)\; dx\; dy.$$

Find all triplets (a, b, c) of positive integers so that $a^2b$, $b^2c$ and $c^2a$ devide $a^3+b^3+c^3$

Let $ a,b,c$ be positive real numbers. Prove the inequality: $ \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}$

Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.

In the non-isosceles triangle $ABC$ an altitude from $A$ meets side $BC$ in $D$ . Let $M$ be the midpoint of $BC$ and let $N$ be the reflection of $M$ in $D$ . The circumcirle of triangle $AMN$ intersects the side $AB$ in $P\ne A$ and the side $AC$ in $Q\ne A$ . Prove that $AN,BQ$ and $CP$ are concurrent.

Prove: there are polynomials $S_1, S_2, \ldots$ in the variables $x_1, x_2, \ldots,y_1, y_2,\ldots$ with integer coefficients satisfying, for every integer $n \ge 1$, $$\sum_{d \mid n} d \cdot S_d ^{n/d}=\sum_{d \mid n} d \cdot (x_d ^{n/d}+y_d ^{n/d}) \quad (*)$$ Here, the sums run through the positive divisors $d$ of $n$. For example, the first two polynomials are $S_1 = x_1 + y_1$ and $S_2 = x_2 + y_2 - x_1y_1$, which verify identity $(*)$ for $n = 2$: $S_1^2 + 2S_2 = (x_1^2 + y_1^2) + 2 \cdot(x_2 + y_2)$.

For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.

Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is [asy] real xscl = 1.2; int[] x = {0,1,2,4,5},y={0,1,3,4,5}; for(int a:x){ for(int b:y) { dot((a*xscl,b)); } } for(int a:x) { pair prev = (a,y[0]); for(int i = 1;i<y.length;++i) { pair p = (a,y[i]); pen pen = linewidth(.7); if(y[i]-prev.y!=1){ pen+=dotted; } draw((xscl*prev.x,prev.y)--(xscl*p.x,p.y),pen); prev = p; } }for(int a:y) { pair prev = (x[0],a); for(int i = 1;i<x.length;++i) { pair p = (x[i],a); pen pen = linewidth(.7); if(x[i]-prev.x!=1){ pen+=dotted; } draw((xscl*prev.x,prev.y)--(p.x*xscl,p.y),pen); prev = p; } } path lblx = (0,-.7)--(5*xscl,-.7); draw(lblx); label("$10$",lblx); path lbly = (5*xscl+.7,0)--(5*xscl+.7,5); draw(lbly); label("$20$",lbly);[/asy] $\text{(A)} \ 30 \qquad \text{(B)} \ 200 \qquad \text{(C)} \ 410 \qquad \text{(D)} \ 420 \qquad \text{(E)} \ 430$

Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that: \[ a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3\] and $a(1)=a(2)=1$. Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$.

Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of the solid by a rotation of the figure enclosed by the curve $C$, the line $y=\frac{a}{a+k}$ and the $y$-axis around the $y$-axis. Find the ratio $\frac{V_2}{V_1}.$