Find, with proof, all functions $f$ mapping integers to integers with the property that for all integers $m,n$, $$f(m)+f(n)= \max\left(f(m+n),f(m-n)\right).$$

Calculate the least integer greater than $ 5^{(\minus{}6)(\minus{}5)(\minus{}4)...(2)(3)(4)}$.

Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$ and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.

In $\triangle ABC$, $\left|\overline{AB}\right|=\left|\overline{AC}\right|$, $D$ is the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from $B$ to $AC$, then $\textbf{(A)}~\left|\overline{BC}\right|^3>\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(B)}~\left|\overline{BC}\right|^3<\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(C)}~\left|\overline{BC}\right|^3=\left|\overline{BD}\right|^3+\left|\overline{BE}\right|^3$ $\textbf{(D)}~\text{None of the above}$

The smallest sum one could get by adding three different numbers from the set $ \{7,25,\minus{}1,12,\minus{}3 \}$ is \[ \textbf{(A)}\ \minus{}3 \qquad \textbf{(B)}\ \minus{}1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 21 \]

Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

An angle is considered as a point and two rays coming out of it. Find the largest value on $n$ such that it is possible to place $n$ $60$ degree angles on the plane in a way that any pair of these angles have exactly $4$ intersection points.

Here are $n^2$ numbers: $a_{11},a_{12},a_{13},\cdots,a_{1n}\\ a_{21},a_{22},a_{23},\cdots,a_{2n}\\ \cdots\\ a_{n1},a_{n2},a_{n3},\cdots,a_{nn}$ Numbers in each line are arithmetic sequence, numbers in each column are geometric series. If $a_{24}=1,a_{42}=\frac{1}{8},a_{43}=\frac{3}{16}$, find $a_{11}+a_{22}+\cdots+a_{nn}$.

Will the number $1/1996$ decrease or increase and by how many times if in the decimal notation of this number the first non-zero digit after the decimal point is crossed out?

Find the greatest natural number $n$ such that $n\leq 2008$ and \[(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]\] is a perfect square.

Let \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3. \] For any reals $r,s,t$, we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|. \] Find the minimum value of $g(r,s,t)$.

Compute the number of functions $f : \{1, 2,... , 2550\} \to \{61, 80, 84\}$ such that $\sum_{k=1}^{2550} f(k)$ is divisible by $3$.

$ABC$ is a triangle that $2\angle B < \angle A <90^{\circ}$, and $P$ is a point on $AB$ satisfying $\angle A=2\angle APC$. If $BC=a$, $AC=b$, $BP=1$, express $AP$ as a function of $a, b$.

Let $a_1,a_2,\ldots,a_n$ are different integer numbers in the range $[100,200]$ for which: $a_1+a_2+\ldots+a_n\ge11100$. Prove that it can be found at least number from the given in the representation of decimal system on which there are at least two equal digits. [i]L. Davidov[/i]

Let $ \alpha$ be a rational number with $ 0 < \alpha < 1$ and $ \cos (3 \pi \alpha) \plus{} 2\cos(2 \pi \alpha) \equal{} 0$. Prove that $ \alpha \equal{} \frac {2}{3}$.

What are the last two digits of $2017^{2017}$?

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{b^{2}}= b^{a}.\]

Alex places a rook on any square of an empty $8\times8$ chessboard. Then he places additional rooks one rook at a time, each attacking an odd number of rooks which are already on the board. A rook attacks to the left, to the right, above and below, and only the first rook in each direction. What is the maximum number of rooks Alex can place on the chessboard?

Let $ABCD$ be a cyclic quadrilateral with perpendicular diagonals and circumcenter $O$. Let $g$ be the line obtained by reflection of the diagonal $AC$ along the angle bisector of $\angle BAD$. Prove that the point $O$ lies on the line $g$. [i]Proposed by Theresia Eisenkölbl[/i]

There are ten apples and $p$ pears in a basket. Anna eats two apples, and she finds that there are now more pears than apples. She then eats four pears. After eating the pears, she notices that there are more apples than pears. What is the sum of all possible values of $p$? $\textbf{(A) } 19\qquad\textbf{(B) } 28\qquad\textbf{(C) } 30\qquad\textbf{(D) } 42\qquad\textbf{(E) } 45$

Find the largest positive $p$ ($p>1$) such, that $\forall a,b,c\in[\frac1p,p]$ the following inequality takes place \[9(ab+bc+ca)(a^2+b^2+c^2)\geq(a+b+c)^4\]

Compute the smallest positive integer $n$ that is a multiple of $29$ with the property that for every positive integer that is relatively prime to $n$, $k^{n}\equiv 1\pmod{n}.$

Let $ABC$ be a triangle and $M$ be the middle point of $BC$. Let $\Omega$ be the circumference such as $A,B,C \in \Omega$. Let $P$ be the intersection of $\Omega$ and $AM$. $AF$ is a hight of the triangle, with $F\in BC$, and $H$ the orthocenter.Additionally the intersections of $MH$ and $PF$ with $\Omega$ are $K$ and $T$ respectibly. Demonstrate that the circumscribed circumference of the traingle $KTF$ is tangent with $BC$.

Consider the following increasing sequence $1,3,5,7,9,…$ of all positive integers consisting only of odd digits. Find the $2017$ -th term of the above sequence.

Let $n\ge 1$ be an integer. In town $X$ there are $n$ girls and $n$ boys, and each girl knows each boy. In town $Y$ there are $n$ girls, $g_1,g_2,\ldots ,g_n$, and $2n-1$ boys, $b_1,b_2,\ldots ,b_{2n-1}$. For $i=1,2,\ldots ,n$, girl $g_i$ knows boys $b_1,b_2,\ldots ,b_{2i-1}$ and no other boys. Let $r$ be an integer with $1\le r\le n$. In each of the towns a party will be held where $r$ girls from that town and $r$ boys from the same town are supposed to dance with each other in $r$ dancing pairs. However, every girl only wants to dance with a boy she knows. Denote by $X(r)$ the number of ways in which we can choose $r$ dancing pairs from town $X$, and by $Y(r)$ the number of ways in which we can choose $r$ dancing pairs from town $Y$. Prove that $X(r)=Y(r)$ for $r=1,2,\ldots ,n$.