A fair six sided die is rolled to give a number $n$. A fair two sided coin is then flipped $n^2$ times. Find the expected number of heads flipped. Express your answer as a common fraction.

Let $a_1,a_2,\cdots,a_6,a_7$ be seven positive integers. Let $S$ be the set of all numbers of the form $a_i^2+a_j^2$ where $1\leq i<j\leq 7$. Prove that there exist two elements of $S$ which have the same remainder on dividing by $36$.

Let $n$ be a positive integer and let $G$ be the set of points $(x, y)$ in the plane such that $x$ and $y$ are integers with $1 \leq x, y \leq n$. A subset of $G$ is called [i]parallelogram-free[/i] if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of $G$ can have?

A pentagon $ABCDE$ is such that $ABCD$ is cyclic, $BE\parallel CD$, and $DB=DE$. Let us fix the points $B,C,D,E$ and vary $A$ on the circumcircle of $BCD$. Let $P=AC\cap BE$, and $Q=BC\cap DE$. Prove that the second intersection of circles $(ABE)$ and $(PQE)$ lie on a fixed circle.

Let $\triangle BC$ be a triangle with side lengths $AB = 9, BC = 10, CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC, E = BO \cap CA, F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.

Ana and Natalia alternately play on a $ n \times n$ board (Ana rolls first and $n> 1$). At the beginning, Ana's token is placed in the upper left corner and Natalia's in the lower right corner. A turn consists of moving the corresponding piece in any of the four directions (it is not allowed to move diagonally), without leaving the board. The winner is whoever manages to place their token on the opponent's token. Determine if either of them can secure victory after a finite number of turns.

Show that if $a,b$ are positive real numbers such that $a^{2000}+b^{2000}=a^{1998}+b^{1998}$ then $a^2+b^2 \le 2$.

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

Determine the smallest real number $C$ such that the inequality $$\frac x{\sqrt{yz}}\cdot\frac1{x+1}+\frac y{\sqrt{zx}}\cdot\frac1{y+1}+\frac z{\sqrt{xy}}\cdot\frac1{x+1}\le C$$holds for all positive real numbers $x,y$ and $z$ with $\frac1{x+1}+\frac1{y+1}+\frac1{z+1}=1$.

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

Fix positive integers $m$ and $n$. Suppose that $a_1, a_2, \dots, a_m$ are reals, and that pairwise distinct vectors $v_1, \dots, v_m\in \mathbb{R}^n$ satisfy $$\sum_{j\neq i} a_j \frac{v_j-v_i}{||v_j-v_i||^3}=0$$ for $i=1,2,\dots,m$. Prove that $$\sum_{1\le i<j\le m} \frac{a_ia_j}{||v_j-v_i||}=0.$$

We define the number $s$ as \[s=\sum^{\infty}_{i=1} \dfrac{1}{10^i-1}=\dfrac{1}{9}+\dfrac{1}{99}+\dfrac{1}{999}+\dfrac{1}{9999}+\ldots=0.12232424...\] We can determine the $n$th digit right of the decimal point of $s$ without summing the entire infinite series because after summing the first $n$ terms of the series, the rest of the series sums to less than $\dfrac{2}{10^{n+1}}$. Determine the smallest prime number $p$ for which the $p$th digit right of the decimal point of $s$ is greater than 2. Justify your answer.

A hundred of silver coins are laid down in a line. A wizard can convert silver coin into golden one in $3$ seconds. Each golden coin, which is near the coin being converted, reduces this time by $1$ second. What minimal time is required for the wizard to convert all coins to gold?

Is it possible to write natural numbers at all points of the plane with integer coordinates so that three points with integer coordinates lie on the same line if and only if the numbers written in them had a common divisor greater than one?

Prove that for each $x,y,z \in R$ it holds that $$\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0$$

Let $f(x) = x^2 + 6x + c$ for all real number s$x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots?

Find positive integers $n$ and $a_1, a_2, \dots, a_n$ such that $$a_1+a_2+\dots a_n=1979$$ and the product $a_1a_2\dots a_n$ as large as possible.

Find all possible values of $x-\lfloor x\rfloor$ if $\sin \alpha = 3/5$ and $x=5^{2003}\sin {(2004\alpha)}$.

Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$

Suppose that $f: \mathbb{R}\rightarrow\mathbb{R}$ fulfils $\left|\sum^n_{k=1}3^k\left(f(x+ky)-f(x-ky)\right)\right|\le1$ for all $n\in\mathbb{N},x,y\in\mathbb{R}$. Prove that $f$ is a constant function.

Prove that for any natural numbers $0 \leq i_1 < i_2 < \cdots < i_k$ and $0 \leq j_1 < j_2 < \cdots < j_k$, the matrix $A = (a_{rs})_{1\leq r,s\leq k}$, $a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!}$ ($1\leq r,s\leq k$) is nonsingular.

How many a) bishops b) horses can be positioned on a chessboard at most, so that no one threatens another?

Let us put pieces on some squares of $2n \times 2n$ chessboard in such a way that on every horizontal and vertical line there is an odd number of pieces. Prove that the whole number of pieces on the black squares is even.

Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.