The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.

On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types: (a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$. (b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$. Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves. [i]D. Fon-der-Flaas[/i]

Let $P(x)$ be a polynomial with integer coefficients that has at least one rational root. Let $n$ be a positive integer. Alan and Allan are playing a game. First, Alan writes down $n$ integers at $n$ different locations on a board. Then Allan may make moves of the following kind: choose a position that has integer $a$ written, then choose a different position that has integer $b$ written, then at the first position erase $a$ and in its place write $a+P(b)$. After any nonnegative number of moves, Allan may choose to end the game. Once Allan ends the game, his score is the number of times the mode (most common element) of the integers on the board appears. Find, in terms of $P(x)$ and $n$, the maximum score Allan can guarantee. [i]Henrick Rabinovitz[/i]

Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers. Prove that \[ \left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2. \] Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.

A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

A 3 × 3 grid of blocks is labeled from 1 through 9. Cindy paints each block orange or lime with equal probability and gives the grid to her friend Sophia. Sophia then plays with the grid of blocks. She can take the top row of blocks and move it to the bottom, as shown. 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 1 2 3 Grid A Grid A0 She can also take the leftmost column of blocks and move it to the right end, as shown. 1 2 3 4 5 6 7 8 9 2 3 1 5 6 4 8 9 7 Grid B Grid B0 Sophia calls the grid of blocks citrus if it is impossible for her to use a sequence of the moves described above to obtain another grid with the same coloring but a different numbering scheme. For example, Grid B is citrus, but Grid A is not citrus because moving the top row of blocks to the bottom results in a grid with a different numbering but the same coloring as Grid A. What is the probability that Sophia receives a citrus grid of blocks?

[i]George the grasshopper[/i] lives of the real line, starting at $0$ . He is given the following sequence of numbers: $2, 3, 4, 8, 9, ... ,$ which are all the numbers of the form $2^k$ or $3^l$, $k, l \in \mathbb{N}$, arranged in increasing order. Starting from $2$, for each number $x$ in the sequence in order, he (currently at $a$) must choose to jump to either $a+x$ or $a-x$. Show that [i]George the grasshopper[/i] can jump in a way that he reaches every integer on the real line.

Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$. Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have \[ n\mid a_i \minus{} b_i \minus{} c_i \] [i]Proposed by Ricky Liu & Zuming Feng, USA[/i]

The circles $ C_{1}$ and $ C_{2}$ touch externally at $ M$ and the radius of $ C_{2}$ is larger than that of $ C_{1}$. $ A$ is any point on $ C_{2}$ which does not lie on the line joining the centers of the circles. $ B$ and $ C$ are points on $ C_{1}$ such that $ AB$ and $ AC$ are tangent to $ C_{1}$. The lines $ BM$, $ CM$ intersect $ C_{2}$ again at $ E$, $ F$ respectively. $ D$ is the intersection of the tangent at $ A$ and the line $ EF$. Show that the locus of $ D$ as $ A$ varies is a straight line.

A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$.

Let $n$ cards are placed in a circle. Each card has a white side and a black side. On each move, you pick one card with black side up, flip it over, and also flip over the two neighboring cards. Suppose initially, there are only one black-side-up card. (a)If $n=2015$ , can you make all cards white-side-up through a finite number of moves? (b)If $n=2016$ , can you make all cards white-side-up through a finite number of moves?

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]