Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

The following facts are known in a mathematical contest: [list] (a) The number of problems tested was $n\ge 4$ (b) Each problem was solved by exactly four contestants. (c) For each pair of problems, there is exactly one contestant who solved both problems [/list] Assuming the number of contestants is greater than or equal to $4n$, find the minimum value of $n$ for which there always exists a contestant who solved all the problems.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Twenty-five points are given on the plane. Among any three of them, one can choose two less than one inch apart. Prove that there are 13 points among them which lie in a circle of radius 1.

(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k + 1)^2)$ whose product is a perfect square. (b) Given an integer $n > 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k + 1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$. [i]AMM Magazine[/i]

There is a sequence with $a(2)=0$, $a(3)=1$ and $a(n)=a\left(\left\lfloor\dfrac n2\right\rfloor\right)+a\left(\left\lceil\dfrac n2\right\rceil\right)$ for $n\geq 4$. Find $a(2014)$. [Note that $\left\lfloor\dfrac n2\right\rfloor$ and $\left\lceil\dfrac n2\right\rceil$ denote the floor function (largest integer $\leq\tfrac n2$) and the ceiling function (smallest integer $\geq\tfrac n2$), respectively.]

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

Prove for any $M>2$, there exists an increasing sequence of positive integers $a_1<a_2<\ldots $ satisfying: 1) $a_i>M^i$ for any $i$; 2) There exists a positive integer $m$ and $b_1,b_2,\ldots ,b_m\in\left\{ -1,1\right\}$, satisfying $n=a_1b_1+a_2b_2+\ldots +a_mb_m$ if and only if $n\in\mathbb{Z}/ \{0\}$.

Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)

A function $f(S)$ assigns to each nine-element subset of $S$ of the set $\{1,2,\ldots, 20\}$ a whole number from $1$ to $20$. Prove that regardless of how the function $f$ is chosen, there will be a ten-element subset $T\subset\{1,2,\ldots, 20\}$ such that $f(T - \{k\})\neq k$ for all $k\in T$.

Let $n$ be a positive integer and let $S$ be a set of $2^n+1$ elements. Let $f$ be a function from the set of two-element subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.

A network is a simple directed graph such that each edge $ e$ has two intger lower and upper capacities $ 0\leq c_l(e)\leq c_u(e)$. A circular flow on this graph is a function such that: 1) For each edge $ e$, $ c_l(e)\leq f(e)\leq c_u(e)$. 2) For each vertex $ v$: \[ \sum_{e\in v^\plus{}}f(e)\equal{}\sum_{e\in v^\minus{}}f(e)\] a) Prove that this graph has a circular flow, if and only if for each partition $ X,Y$ of vertices of the network we have: \[ \sum_{\begin{array}{c}{e\equal{}xy}\\{x\in X,y\in Y}\end{array}} c_l(e)\leq \sum_{\begin{array}{c}{e\equal{}yx}\\{y\in Y,x\in X}\end{array}} c_l(e)\] b) Suppose that $ f$ is a circular flow in this network. Prove that there exists a circular flow $ g$ in this network such that $ g(e)\equal{}\lfloor f(e)\rfloor$ or $ g(e)\equal{}\lceil f(e)\rceil$ for each edge $ e$.

Let $n$ be a positive integer. Determine the size of the largest subset of $\{ -n, -n+1, \dots, n-1, n\}$ which does not contain three elements $a$, $b$, $c$ (not necessarily distinct) satisfying $a+b+c=0$.

Ben is throwing darts at a circular target with diameter 10. Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target. Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target. What is the expected number of points that Ben can earn from throwing a single dart? (Note that $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$.)

Let be a real number $ a. $ For any real number $ p $ and natural number $ k, $ let be the set $$ A_k(p)=\{ px\in\mathbb{Z}\mid k=\lceil x \rceil \} . $$ Find all real numbers $ b $ such that $ \# A_n(a)=\# A_n(b) , $ for any natural number $ n. $ $ \# $ [i]denotes the cardinal.[/i] [i]Eugen Păltânea[/i]

Let $n$ be a fixed natural number. [b]a)[/b] Find all solutions to the following equation : \[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \] [b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) : \[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]

The integers from $1$ to $999999$ are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even. In which group is the sum of the elements greater?

Let $R$ be the set of points $(x, y)$ such that $x$ and $y$ are positive, $x + y$ is at most 2013, and \[ \lceil x \rceil \lfloor y \rfloor = \lfloor x \rfloor \lceil y \rceil. \] Compute the area of set $R$. Recall that $\lfloor a \rfloor$ is the greatest integer that is less than or equal to $a$, and $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.

Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.

A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace? [i]Proposed by A. Golovanov[/i]

Given a positive integer $n\ge 3$, colour each cell of an $n\times n$ square array with one of $\lfloor (n+2)^2/3\rfloor$ colours, each colour being used at least once. Prove that there is some $1\times 3$ or $3\times 1$ rectangular subarray whose three cells are coloured with three different colours. [i](Russia) Ilya Bogdanov, Grigory Chelnokov, Dmitry Khramtsov[/i]

In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. $k$ roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of $k.$