Define the [i]mexth[/i] of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that [list] [*]for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and [*]for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). [/list] [i]Proposed by Espen Slettnes[/i]

Henry invited $2N$ guests to his birthday party. He has $N$ white hats and $N$ black hats. He wants to place hats on his guests and split his guests into one or several dancing circles so that in each circle there would be at least two people and the colors of hats of any two neighbours would be different. Prove that Henry can do this in exactly $(2N)!$ different ways. (All the hats with the same color are identical, all the guests are obviously distinct, $(2N)! = 1 \cdot 2 \cdot . . . \cdot (2N)$.) Gleb Pogudin

Let $k$ be a positive integer such that $1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}$. Find the remainder when $k$ is divided by $7$.

In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis. (1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section. (2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$. (3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis. [i]1987 Sophia University entrance exam/Science and Technology[/i]

Box A contains $3$ black and $4$ blue marbles. Box B has $7$ black and $1$ blue, whereas Box C has $2$ black, $3$ blue and $1$ green marble. I close my eyes and pick two marbles from $2$ different boxes. If it turns out that I get $1$ black and $1$ blue marble, what is the probability that the black marble is from box A and the blue one is from C?

Prair has a box with some combination of red and green balls. If she randomly draws two balls out of the box (without replacement), the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors! How many possible values between $200$ and $1000$ are there for the total number of balls in the box? [i]Proposed by Andrew Wu[/i]

Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.

.Let $CH$ be an altitude of right-angled triangle $ABC$ ($\angle C = 90^o$), $HA_1$, $HB_1$ be the bisectors of angles $CHB$, $AHC$ respectively, and $E, F$ be the midpoints of $HB_1$ and $HA_1$ respectively. Prove that the lines $AE$ and $BF$ meet on the bisector of angle $ACB$.

Are there natural number(s) $n$, such that $3^n+1$ has a divisor in the form $24k+20$

Let $ X_1,X_2,...,X_n$ be (not necessary independent) discrete random variables. Prove that there exist at least $ n^2/2$ pairs $ (i,j)$ such that \[ H(X_i\plus{}X_j) \geq \frac 13 \min_{1 \leq k \leq n} \{ H(X_k) \},\] where $ H(X)$ denotes the Shannon entropy of $ X$. [i]GY. Katona[/i]

A group of $101$ Dalmathians participate in an election, where they each vote independently on either candidate $A$ or $B$ with equal probability. If $X$ Dalmathians voted for the winning candidate, the expected value of $X^2$ can be expressed as $\tfrac{a}{b}$ for positive integers $a,b$ with $\gcd(a,b) = 1.$ Find the unique positive integer $k \le 103$ such that $103 | a-bk.$

Let $ a_1 <a_2 <... <a_n $ consecutive positive integers (with $ n> 2 $). A grasshopper jumps on the real line, starting at point $ 0 $ and jumping $ n $ to the right with lengths $ a_1 $, $ a_2 $, ..., $ a_n $, in some order (each length occupies exactly once), ending your tour at the $ 2010 $ point. Find all the possible values $ n $ of jumps that the grasshopper could have made.

Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that $f(m) - f(n) = f(m-n)10^n \forall m>n \in \mathbb{N}$. Additionally, gcd$(f(k),f(k+1)) = 1 \forall k \in \mathbb{N}$. Show that if $a,b$ are coprime natural numbers, that is, gcd$(a,b) = 1$ then $f(a),f(b)$ are also coprime.

An acute-angled triangle $ABC$ has altitudes $AD, BE$ and $CF$. Let $Q$ be an interior point of the segment $AD$, and let the circumcircles of the triangles $QDF$ and $QDE$ meet the line $BC$ again at points $X$ and $Y$ , respectively. Prove that $BX = CY$ .

Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.

How many positive integers which are less or equal with $2013$ such that $3$ or $5$ divide the number.

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) \equal{} | S_1 \minus{} S_2 |$. a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd. b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$. c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$.

$x, y, z$ are positive reals such that $x \leq 1$. Prove that $$xy+y+2z \geq 4 \sqrt{xyz}$$

Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.

$ABCD$ is a cyclic quadrilateral, and $\angle BAC = \angle DAC$. $\astrosun I_1$ and $\astrosun I_2$ are the incircles of $\triangle ABD$ and $\triangle ADC$ respectively. Prove that one of the common external tangents of $\astrosun I_1$ and $\astrosun I_2$ is parallel to $BD$

Let A,B,C be three collinear points and a circle T(A,r). If M and N are two diametrical opposite variable points on T, Find locus geometrical of the intersection BM and CN.

A polynomial $P(x)$ of degree $3$ has three distinct real roots. Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$.

Let $ ABC$ be an acute triangle, and let $ D$ and $ E$ be the feet of the altitudes drawn from vertexes $ A$ and $ B$, respectively. Show that if, \[ Area[BDE]\le Area[DEA]\le Area[EAB]\le Area[ABD]\] then, the triangle is isosceles.

Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.

Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is $\leq d$