Find the maximum value of $\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}$ as $x, y$ and $z$ range over the set of all positive real numbers. Justify your answer.

We say that a polygon $P$ is [i]inscribed[/i] in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is [i]circumscribed[/i] to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$. Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.

Find maximum value of $|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of sum $P_A +P_B$. PS. It is a variation of [url=https://artofproblemsolving.com/community/c6h2267998p17621980]JBMO Shortlist 2019 A3 [/url]

Real numbers $x, y, z$ satisfy $$x + xy + xyz = 1, y + yz + xyz = 2, z + xz + xyz = 4.$$ The largest possible value of $xyz$ is $\frac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, $d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d$.

[url=https://artofproblemsolving.com/community/c677808][b]Cyprus IMO TST 2018[/b][/url] [url=https://artofproblemsolving.com/community/c6h1666662p10591751][b]Problem 1.[/b][/url] Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square. [url=https://artofproblemsolving.com/community/c6h1666663p10591753][b]Problem 2.[/b][/url] Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$. [url=https://artofproblemsolving.com/community/c6h1666660p10591747][b]Problem 3.[/b][/url] Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$obtains its minimum value. [url=https://artofproblemsolving.com/community/c6h1666661p10591749][b]Problem 4.[/b][/url] Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$. (a) Prove that $$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence: $$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

Let $a,b,c \ge 0$ and $a+b+c=3$. Prove that: $2(ab+bc+ca)-3abc\ge \sum_{cyc}^{}a\sqrt{\frac{b^2+c^2}{2}}$

(R.Pirkuliev) Prove the inequality \[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}}, \] where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.

Let $x,y,z\in \mathbb{R}^+$ be such that $xy+yz+zx=x+y+z$. Prove the following inequality: $\frac{1}{x^2+y+1}+\frac{1}{y^2+z+1}+\frac{1}{z^2+x+1}\leq 1$.

For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.) Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?

Prove that the inequality $ (x^{k}\minus{}y^{k})^{n}<(x^{n}\minus{}y^{n})^{k}$ holds forall reals $ x>y>0$ and positive integers $ n>k$.

Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that \[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]

Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$, with $a$ is the longest side.

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*} if and only if \[ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). \]

Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set $A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$ $B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$ Prove that $2|A|\geq |B|$.

For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$. Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).

Prove that for all real $x > 0$ holds the inequality $$\sqrt{\frac{1}{3x+1}}+\sqrt{\frac{x}{x+3}}\ge 1.$$ For what values of $x$ does the equality hold?

Find the greatest real number $M$ for which \[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \] for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$

Let $ABC$ be an acute triangle, and $H$ its orthocenter. The distance from $H$ to rays $BC$, $CA$, and $AB$ is denoted by $d_a$, $d_b$, and $d_c$, respectively. Let $R$ be the radius of circumcenter of $\triangle ABC$ and $r$ be the radius of incenter of $\triangle ABC$. Prove the following inequality: $$d_a+d_b+d_c \le \frac{3R^2}{4r}$$.

Given are positive reals $x_1, x_2,..., x_n$ such that $\sum\frac {1}{1+x_i^2}=1$. Find the minimal value of the expression $\frac{\sum x_i}{\sum \frac{1}{x_i}}$ and find when it is achieved.

Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?

Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$? $ \textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256} $

The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other: an $MP$ is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of $MP$s who are satisfied with their salaries?