Let $x, y, z$ be positive real numbers satisfying $x + y + z =\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}$. Prove that $$\frac{3}{2}\leq\frac{3}{\sqrt[3]{xyz}(\sqrt[3]{xyz}+1)}\leq\frac{1}{x(y+1)}+\frac{1}{y(z+1)}+\frac{1}{z(x+1)}.$$

Let $p_{1}, p_{2},...,p_{n}$, where $n>2$, be the first $n$ prime numbers. Prove that $\frac{1}{p_{1}^2}+\frac{1}{p_{2}^2}+...+\frac{1}{p_{n}^2}+\frac{1}{p_{1}p_{2}...p_{n}}<\frac{1}{2}$

A positive rational number is written on the blackboard. Every minute Vasya replaces the number $ r $ written on the board with $ \sqrt {r + 1} $. Prove that someday he will get an irrational number.

Sixteen dots are arranged in a four by four grid as shown. The distance between any two dots in the grid is the minimum number of horizontal and vertical steps along the grid lines it takes to get from one dot to the other. For example, two adjacent dots are a distance 1 apart, and two dots at opposite corners of the grid are a distance 6 apart. The mean distance between two distinct dots in the grid is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$. [center][img]https://i.snag.gy/c1tB7z.jpg[/img][/center]

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$.

Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$

A bank charges $ \$6$ for a loan of $ \$120$. The borrower receives $ \$114$ and repays the loan in $ 12$ installments of $ \$10$ a month. The interest rate is approximately: $ \textbf{(A)}\ 5 \% \qquad \textbf{(B)}\ 6 \% \qquad \textbf{(C)}\ 7 \% \qquad \textbf{(D)}\ 9\% \qquad \textbf{(E)}\ 15 \%$

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $0 \leq \alpha , \beta , \gamma \leq \frac{\pi}{2}$, such that $\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma = 1$. Prove that $2 \leq (1 + \cos ^{2} \alpha ) ^{2} \sin^{4} \alpha + (1 + \cos ^{2} \beta ) ^{2} \sin ^{4} \beta + (1 + \cos ^{2} \gamma ) ^{2} \sin ^{4} \gamma \leq (1 + \cos ^{2} \alpha )(1 + \cos ^{2} \beta)(1 + \cos ^{2} \gamma ).$

It is known that the reflection of the orthocenter of a triangle $ABC$ about its circumcenter lies on $BC$. Let $A_1$ be the foot of the altitude from $A$. Prove that $A_1$ lies on the circle passing through the midpoints of the altitudes of $ABC$.

$A$ circle $\gamma$ is drawn and let $AB$ be a diameter. The point $C$ on $\gamma$ is the midpoint of the line segment $BD$. The line segments $AC$ and $DO$, where $O$ is the center of $\gamma$, intersect at $P$. Prove that there is a point $E$ on $AB$ such that $P$ is on the circle with diameter $AE.$

Incircle of triangle $ABC$ is tangent to sides $BC,CA,AB$ at $D,E,F$,respectively.Points $P,Q$ are inside angle $BAC$ such that $FP=FB,FP||AC$ and $EQ=EC,EQ||AB$.Prove that $P,Q,D$ are collinear.

Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing. [i](Poland) Andrzej Komisarski and Marcin Kuczma[/i]

Let $ \{a_n\}_{n\geq 1}$ be a sequence of real numbers such that $ |a_{n\plus{}1}\minus{}a_n|\leq 1$, for all positive integers $ n$. Let $ \{b_n\}_{n\geq 1}$ be the sequence defined by \[ b_n \equal{} \frac { a_1\plus{} a_2 \plus{} \cdots \plus{}a_n} {n}.\] Prove that $ |b_{n\plus{}1}\minus{}b_n | \leq \frac 12$, for all positive integers $ n$.

Suppose that $ ABC$ is an acute triangle such that $ \tan A$, $ \tan B$, $ \tan C$ are the three roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0$, where $ q\neq 1$. Show that $ p \le \minus{} 3\sqrt 3$ and $ q > 1$.

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]