Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $. $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}$$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.

A billiards table is in the figure of regular hexagon $ABCDEF$. $P$ is the midpoint of $AB$. We shut the ball at $P$, then it touches $Q$ on side $BC$, then it touches side $CD,DE,EF,FA$. Finally, the ball touches side $AB$ again. Let $\theta=\angle BPQ$, find the value range of $\theta$.

Solve the equation in postive integers $$x^2+y^2+1998=1997x-1999y.$$

In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product.

Let $P(x)$ be a polynomial with integer coefficient such that $P(1) = 10$ and $P(-1) = 22$. (a) Give an example of $P(x)$ such that $P(x) = 0$ has an integer root. (b) Suppose that $P(0) = 4$, prove that $P(x) = 0$ does not have an integer root.

Every day from day 2, neighboring cubes (cubes with common faces) to red cubes also turn red and are numbered with the day number.

Let $\triangle MAB$ be a triangle with circumcenter $O$. $P$ and $Q$ lie on line $AB$ (both interior or exterior) such that $\angle PMA = \angle BMQ$. Let $D$ be a point on the perpendicular line through $M$ to $AB$. $E$ is the second intersection of the two circles $(DAB)$ and $(DPQ)$. The line $MO$ intersects $AB$ at $J$. Show that the circumcenter of $\triangle EMJ$ lies on line $AB$. [i](Proposed by Tan Rui Xuen)[/i]

Let $a, b, c$ be complex numbers such that $a+b+c = 1$, $a^2+b^2+c^2 = 2$ and $a^3+b^3+c^3 = 3$. Find the value of $a^4 + b^4 + c^4$.

Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds: $$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$

Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?

Real numbers $b>a>0$ are given. Find the number $r$ in $[a,b]$ which minimizes the value of $\max\left\{\left|\frac{r-x}x\right||a\le x\le b\right\}$.

Write a positive integer in each box so that: All six numbers are different. The sum of the six numbers is $100$. If each number is multiplied by its neighbor (in a clockwise direction) and the six results of those six multiplications are added, the smallest possible value is obtained. Explain why a lower value cannot be obtained. [img]https://cdn.artofproblemsolving.com/attachments/7/1/6fdadd6618f91aa03cdd6720cc2d6ee296f82b.gif[/img]

Aumann, Bill, and Charlie each roll a fair $6$-sided die with sides labeled $1$ through $6$ and look at their individual rolls. Each flips a fair coin and, depending on the outcome, looks at the roll of either the player to his right or the player to his left, without anyone else knowing which die he observed. Then, at the same time, each of the three players states the expected value of the sum of the rolls based on the information he has. After hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Then, for the third time, after hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Prove that Aumann, Bill, and Charlie say the same number the third time.

Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$. (Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ ) Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$, their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$. Can we find two non-zero super-integers with zero product? (a zero super-integer has all its digits zero)

It is possible to place the $2014$ points in the plane so that we can construct $1006^2$ parralelograms with vertices among these points, so that the parralelograms have area 1?

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $ m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is $ \textbf{(A)}\ \frac {1}{2m \plus{} 1} \qquad \textbf{(B)}\ m \qquad \textbf{(C)}\ 1 \minus{} m \qquad \textbf{(D)}\ \frac {1}{4m} \qquad \textbf{(E)}\ \frac {1}{8m^2}$

A function $f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}$ is called [i]good[/i] if the function $g(n)=|f(n)-n|$ is injective. Furthermore, a good function $f$ is called [i]excellent[/i] if there exists another good function $f'$ such that $f(n)-f'(n)$ is nonzero for exactly one value of $n$. Let $N$ be the number of good functions that are not excellent. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Nathan Ramesh

Let $f(n) = \sum^n_{i=1}\frac{gcd(i,n)}{n}$. Find the sum of all positive integers $ n$ for which $f(n) = 6$.

There are ten points marked on a circumference, numbered from $1$ to $10$ and join all points with segments. I color the segments, with red someones and others with blue. Without changing the colors of the segments, renumber all the points from the $1$ to the $10$. Will be possible to color the segments and to renumber the points so that those numbers that were jointed with red are jointed now with blue and the numbers that were jointed with blue they are jointed now with red?

Let $ H$ be the intersection of altitudes in triangle $ ABC$. If $ \angle B\equal{}\angle C\equal{}\alpha$ and $ O$ is the center of circle passing through $ A$, $ H$ and $ C$, then find $ \angle HOC$ in terms of $ \alpha$. $\textbf{(A)}\ 90{}^\circ \minus{}\alpha \qquad\textbf{(B)}\ 90{}^\circ \plus{}\frac{\alpha }{2} \qquad\textbf{(C)}\ 180{}^\circ \minus{}\alpha \\ \qquad\textbf{(D)}\ 180{}^\circ \minus{}\frac{\alpha }{2} \qquad\textbf{(E)}\ 180{}^\circ \minus{}2\alpha$

The set of all $10$-digit numbers may be represented as a union of two subsets: the subset $M$ consisting of all $10$-digit numbers, each of which may be represented as a product of two $5$-digit numbers, and the subset $N$ , containing the remaining $10$-digit numbers . Which of the sets $M$ and $N$ contains more elements? (S. Fomin , Leningrad)

Let $n$ be a non-negative integer. Find all non-negative integers $a,b,c,d$ such that \[a^2+b^2+c^2+d^2=7\cdot 4^n\]

Prove that there exists a set of infinitely many positive integers such that the elements of no finite subset of this set add up to a perfect square.

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

Prove that if $ a $ and $ b $ are real numbers and the polynomial $ ax^3 - ax^2 + 9bx - b $ has three positive roots, then they are equal.