Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]

In $ABC$ with $\angle C = 60^{\circ}$, prove that \[\frac{c}{a} + \frac{c}{b} \ge2.\]

Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.

Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy: (a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$. (b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$. Find the general term of $\{ a_{n}\}$.

The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

Let $a,b,c>0$ and $abc=1$ . Prove that $$ \sqrt{2(1+a^2)(1+b^2)(1+c^2)}\ge 1+a+b+c.$$

Solve the inequality $$\log_{\frac12} x\ge 16^x$$

Prove that : $\frac{1}{(\log_{bc} a)^n}+\frac{1}{(\log_{ac} b)^n}+\frac{1}{(\log_{bc} a)^n}\geq 3\cdot2^{n}$ where $a,b,c>1$ and $n$ is natural number.

Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ [i]Proposed by Milan Haiman[/i]

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

On the coordinate plane, draw a set of points $M(x,y)$, the coordinates of which satisfy the inequality $$\log_{x+y} (x^2+y^2) \le 1.$$

Prove that for any real numbers $ x_1, x_2, x_3, \ldots, x_n $ the inequality is true $$ x_1x_2x_3\ldots x_n \leq \frac{x_1^2}{2} + \frac{x_2^4}{4} + \frac{x_3^8}{8} + \ldots + \frac{x_n^{2^ n}}{2^n} + \frac{1}{2^n}$$

Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $1 + ab + bc + cd + da + ac + bd > a + b + c + d$.

Let $a$, $b$ and $c$ be positive real numbers and $a^3+b^3+c^3=3$. Prove $$\frac{1}{3-2a}+\frac{1}{3-2b}+\frac{1}{3-2c}\geq 3$$

Fix reals $x, y,z > 0$ such that $x + y + z = \sqrt[5]{x} + \sqrt[5]{y} +\sqrt[5]{z}$ . Prove that $x^x y^y z^z \ge 1$. (Paolo Leonetti)

Suppose $x, y, z, t$ are real numbers such that $\begin{cases} |x + y + z -t |\le 1 \\ |y + z + t - x|\le 1 \\ |z + t + x - y|\le 1 \\ |t + x + y - z|\le 1 \end{cases}$ Prove that $x^2 + y^2 + z^2 + t^2 \le 1$.

Prove that for every integer $n > 2$, $$(1\cdot 2 \cdot 3 \cdot ... \cdot n)^2 > n^n.$$

For $ a,b,c,d $ positive, prove: $$ \frac{2a}{a^2+bc} +\frac{2b}{b^2+cd} +\frac{2c}{c^2+da} +\frac{2d}{d^2+ab}\le \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} $$ [i]Dan Popescu[/i]

For a natural number $n$, let $w(n)$ denote the number of (positive) prime divisors of $n$. Find the smallest positive integer $k$ such that $2^{w(n)} \le k \sqrt[4]{ n}$ for each $n \in N$.

1. Prove the following inequality for positive reals $a_1,a_2...,a_n$ and $b_1,b_2...,b_n$: $(\sum a_i)(\sum b_i)\geq (\sum a_i+b_i)(\sum\frac{a_ib_i}{a_i+b_i})$

[url=https://artofproblemsolving.com/community/c678145][b]Czech-Polish-Slovak Match 2018[/b][/url] [b]Austria, 24 - 27 June 2018[/b] [url=http://artofproblemsolving.com/community/c6h1667029p10595005][b]Problem 1.[/b][/url] Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i] [url=http://artofproblemsolving.com/community/c6h1667030p10595011][b]Problem 2.[/b][/url] Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$. [i]Proposed by Patrik Bak, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667031p10595016][b]Problem 3.[/b][/url] There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns. [i]Proposed by Peter Novotný, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667033p10595021][b]Problem 4.[/b][/url] Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i] [url=http://artofproblemsolving.com/community/c6h1667034p10595023][b]Problem 5.[/b][/url] In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points. [i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667036p10595032][b]Problem 6.[/b][/url] We say that a positive integer $n$ is [i]fantastic[/i] if there exist positive rational numbers $a$ and $b$ such that $$ n = a + \frac 1a + b + \frac 1b.$$ [b](a)[/b] Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic. [b](b)[/b] Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic. [i]Proposed by Walther Janous, Austria[/i]

Positive real numbers $a$,$b$,$c$ are given such that $abc=1$.Prove that $$2(a+b+c)+\frac{9}{(ab+bc+ca)^2}\geq7.$$

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.