Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$: $p(5x)^2-3=p(5x^2+1)$ such that: $a) p(0)\neq 0$ $b) p(0)=0$

A $2^{2014} + 1$ by $2^{2014} + 1$ grid has some black squares filled. The filled black squares form one or more snakes on the plane, each of whose heads splits at some points but never comes back together. In other words, for every positive integer $n$ greater than $2$, there do not exist pairwise distinct black squares $s_1$, $s_2$, \dots, $s_n$ such that $s_i$ and $s_{i+1}$ share an edge for $i=1,2, \dots, n$ (here $s_{n+1}=s_1$). What is the maximum possible number of filled black squares? [i]Proposed by David Yang[/i]

Given a set $X$ and a function $f: X \rightarrow X$, for each $x \in X$ we define $f^1(x)=f(x)$ and, for each $j \ge 1$, $f^{j+1}(x)=f(f^j(x))$. We say that $a \in X$ is a fixed point of $f$ if $f(a)=a$. For each $x \in \mathbb{R}$, let $\pi (x)$ be the quantity of positive primes lesser or equal to $x$. Given an positive integer $n$, we say that $f: \{1,2, \dots, n\} \rightarrow \{1,2, \dots, n\}$ is [i]catracha[/i] if $f^{f(k)}(k)=k$, for every $k=1, 2, \dots n$. Prove that: (a) If $f$ is catracha, $f$ has at least $\pi (n) -\pi (\sqrt{n}) +1$ fixed points. (b) If $n \ge 36$, there exists a catracha function $f$ with exactly $ \pi (n) -\pi (\sqrt{n}) + 1$ fixed points.

For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.

Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$. Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$. [i]Proposed by Maria Monks Gillespie[/i]

Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$. Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.

Given positive integer $m,n$, color the points of the regular $(2m+2n)$-gon in black and white, $2m$ in black and $2n$ in white. The [i]coloring distance[/i] $d(B,C) $ of two black points $B,C$ is defined as the smaller number of white points in the two paths linking the two black points. The [i]coloring distance[/i] $d(W,X) $ of two white points $W,X$ is defined as the smaller number of black points in the two paths linking the two white points. We define the matching of black points $\mathcal{B}$ : label the $2m$ black points with $A_1,\cdots,A_m,B_1,\cdots,B_m$ satisfying no $A_iB_i$ intersects inside the gon. We define the matching of white points $\mathcal{W}$ : label the $2n$ white points with $C_1,\cdots,C_n,D_1,\cdots,D_n$ satisfying no $C_iD_i$ intersects inside the gon. We define $P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j) $. Prove that: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$

Let $ p$ be a prime number, $ p\not \equal{} 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 \plus{} b^3$. Prove that $ p^2\mid a \plus{} b$ or $ p^3\mid a^3 \plus{} b^3$.

Let $ f: \mathbb{R}\to\mathbb{N}$ be a function which satisfies $ f\left(x \plus{} \dfrac{1}{f(y)}\right) \equal{} f\left(y \plus{} \dfrac{1}{f(x)}\right)$ for all $ x$, $ y\in\mathbb{R}$. Prove that there is a positive integer which is not a value of $ f$. [i]Proposed by Žymantas Darbėnas (Zymantas Darbenas), Lithuania[/i]

Let $ S\equal{}\{1,2,\ldots,n\}$ be a set, where $ n\geq 6$ is an integer. Prove that $ S$ is the reunion of 3 pairwise disjoint subsets, with the same number of elements and the same sum of their elements, if and only if $ n$ is a multiple of 3.

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

A rectangle $ABCD$ is given whose sides have lengths $3$ and $2n$, where $n$ is a natural number. Denote by $U(n)$ the number of ways in which one can cut the rectangle into rectangles of side lengths $1$ and $2$. $(a)$ Prove that \[U(n + 1)+U(n -1) = 4U(n);\] $(b)$ Prove that \[U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].\]