$2016$ numbers written on the board: $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, ..., \frac{2016}{2016}$. In one move, it is allowed to choose any two numbers $a$ and $b$ written on the board, delete them, and write the number $3ab - 2a - 2b + 2$ instead. Determine what number will remain written on the board after $2015$ moves.

Given natural numbers $m, n$ and $X$ such that $X \ge m$ and $X \ge n$. Prove that one can find two integers $u$ and $v$ such that $|u| + |v| > 0$, $|u| \le \sqrt{X}$, $|v| \le \sqrt{X}$ and $$0 \le mu + nv \le 2 \sqrt{X}.$$

Given a polynomial $P$ of degree $2016$ with real coefficients and a quadratic polynomial $Q$ with real coefficients. Is it possible that the roots of the polynomial $P (Q(x))$ are exactly all these numbers: $$-2015, -2014, . . . , -2, -1, 1, 2, . . . , 2016, 2017?$$

Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds: $$f(2^x+2y) =2^yf(f(x))f(y).$$

Given real positive numbers $a, b, c$ and $d$, for which the equalities $a^2 + ab + b^2 = 3c^2$ and $a^3 + a^2b + ab^2 + b^3 = 4d^3$ are fulfilled. Prove that $$a + b + d \le 3c.$$

Given a natural number $n$, for which we can find a prime number less than $\sqrt{n}$ that is not a divisor of $n$. The sequence $a_1, a_2,... ,a_n$ is the numbers $1, 2,... ,n$ arranged in some order. For this sequence, we will find the longest ascending subsequense $a_{i_1} < a_{i_2} < ... < a_{i_k}$, ($i_1 <...< i_k$) and the longest decreasing substring $a_{j_1} > ... > a_{j_l}$, ($j_1 < ... < j_l$) . Prove that at least one of these two subsequnsces $a_{i_1} , . . . , a_{i_k}$ and $a_{j_1} > ... > a_{j_l}$ contains a number that is not a divisor of $n$.

In the parliament of Nekurnekadzeme, all activities take place in commissions, which consist of exactly three members. The constitution stipulates that any three commissions must have at least five members. We will call a family of commissions a clique if every two of them have exactly two members in common, but if any other commission is added to this family, this condition is no longer fulfilled. Prove that two different cliques cannot have more than one commission in common.

$3n - 2$ participants took part in the chess festival, some of them played one game of chess with each other. Prove that at least one of the following statements holds: (A) One can find $n$ chess players $A_1 , A_2 , . . . , A_n$ suchthat Ai has played a game with $A_{i+1}$ for all $i = 1, ...,n -1$. (B) Seven chess players can be found in $B_1 , . . . , B_7$, who have not played with each other, except perhaps three pairs $(B_1, B_2)$, $(B_3, B_4)$ and $(B_5, B_6)$, each of whom may or may not have played a game of chess.

The numbers from$ 1$ to $2016$ are divided into three (disjoint) subsets $A, B$ and $C$, each one contains exactly $672$ numbers. Prove that you can find three numbers, each from a different subset, such that the sum of two of them is equal to the third. original wordingSkaitļi no 1 līdz 2016 ir sadalīti trīs (nešķeļošās) apakškopās A, B un C, katranotām satur tieši 672 skaitļus. Pierādīt, ka var atrast trīs tādus skaitļus, katru no citas apakškopas, ka divu no tiem summa ir vienāda ar trešo.

On an infinite sheet of tiles, an infinite number of $1 \times 2$ tile rectangles are placed, their edges follow the lines of the tiles, and they do not touch each other, not even the corners. Is it true that the remaining checkered sheet can be completely covered with $1 \times 2$ checkered rectangles? original wordingUz bezgalīgas rūtiņu lapas ir novietoti bezgaglīgi daudzi 1 x 2 rūtiņu taisnstūri, to malas iet pa rūtiņu līnijām, un tie nesaskaras cits ar citu pat ne ar stūriem. Vai tiesa, ka atlikušo rūtiņu lapu var pilnībā noklāt ar 1 x 2 rūtiņu tainstūriem?

Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)

For what positive numbers $m$ and $n$ do there exist points $A_1, ..., Am$ and $B_1 ..., B_n$ in the plane such that, for any point $P$, the equation $$|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2 $$holds true?

Suppose that $A, B, C$, and $X$ are any four distinct points in the plane with $$\max \,(BX,CX) \le AX \le BC.$$Prove that $\angle BAC \le 150^o$.

Let $ABC$ be a scalene triangle. Let $D$ and $E$ be the points where the incircle touches sides $BC$ and $CA$, respectively. Let $K$ be the common point of line $BC$ and the bisector of the angle $\angle BAC$. Let $AD$ intersect $EK$ in $P$. Prove that $PC$ is perpendicular to $AK$.

Let $ABC$ be a triangle. Let its altitudes $AD$, $BE$ and $CF$ concur at $H$. Let $K, L$ and $M$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Prove that, if $\angle BAC = 60^o$, then the midpoints of the segments $AH$, $DK$, $EL$, $FM$ are concyclic.

What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$for naturals $n$? Click to reveal hidden textoriginal wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n?

Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

Solve the system of equations in integers: $$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$

Prove that for equation $$x^{2015} + y^{2015} = z^{2016}$$there are infinitely many solutions where $x,y$ and $z$ are different natural numbers.

For what pairs of natural numbers $(a, b)$ is the expression $$(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)$$the power of a prime number?