The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?

Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.

Six musicians gathered at a chamber music festival . At each scheduled concert some of these musicians played while the others listened as members of the audience . What is the least number of such concerts which would need to be scheduled in order to enable each musician to listen , as a member of the audience, to all the other musicians? (Canadian origin)

$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?

[b]a)[/b] Show that the expression $ x^3-5x^2+8x-4 $ is nonegative, for every $ x\in [1,\infty ) . $ [b]b)[/b] Determine $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $

A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?

On the sides $BC,CA$ and $AB$ of a triangle $ABC$ of area $S$ are taken points $A' ,B' ,C'$ respectively such that $AC' /AB = BA' /BC = CB' /CA = p$, where $0 < p < 1$ is variable. (a) Find the area of triangle $A' B' C'$ in terms of $ p$. (b) Find the value of $p$ which minimizes this area. (c) Find the locus of the intersection point $P$ of the lines through $A' $ and $C'$ parallel to $AB$ and $AC$ respectively.

Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$ for any positive integer $ n $ not less than 2 . Find the minimum of $m.$

Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.

If $x$ and $y$ are nonnegative real numbers such that $x+y=1$, determine minimal and maximal value of $$A=x\sqrt{1+y}+y\sqrt{1+x}$$

A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N$ this will be possible, regardless of the number of tasks on Olympiad and the student successes?

Two straight lines intersecting at an angle of $46^o$ are the axes of symmetry of the figure $F$ on the plane. What is the smallest number of axes of symmetry this figure can have?

a) Each of the numbers $x_1,x_2,...,x_n$ can be $1, 0$, or $-1$. What is the minimal possible value of the sum of all products of couples of those numbers. b) Each absolute value of the numbers $x_1,x_2,...,x_n$ doesn't exceed $1$. What is the minimal possible value of the sum of all products of couples of those numbers.

The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice can win the game no matter how Bob plays.

For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$. The [i]sum product value[/i] of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together. For example, if we take $n = 2$ and the sequence $0,1, 1, -1$, then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$. Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$. The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$, which equals $0$. Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have. [i]Attention: you are required to prove that a smaller sum product value is impossible.[/i]

Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$. In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?