Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$

Find the maximum value of $k$ for which one can choose $k$ integers out of $1,2... ,2n$ so that none of them divides another one.

Let $BC$ be a chord of a circle $(O)$ such that $BC$ is not a diameter. Let $AE$ be the diameter perpendicular to $BC$ such that $A$ belongs to the larger arc $BC$ of $(O)$. Let $D$ be a point on the larger arc $BC$ of $(O)$ which is different from $A$. Suppose that $AD$ intersects $BC$ at $S$ and $DE$ intersects $BC$ at $T$. Let $F$ be the midpoint of $ST$ and $I$ be the second intersection point of the circle $(ODF)$ with the line $BC$. 1. Let the line passing through $I$ and parallel to $OD$ intersect $AD$ and $DE$ at $M$ and $N$, respectively. Find the maximum value of the area of the triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (such that $D \ne A$). 2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)

There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.

In a village, the maximum distance between two houses is $M$ and the minimum distance is $m$. Prove that if the village has $6$ houses, then $\frac{M}{m} \ge \sqrt3$.

A group of $n$ people play a board game with the following rules: 1) In each round of the game exactly $3$ people play 2) The game ends after exactly $n$ rounds 3) Every pair of players has played together at least at one round Find the largest possible value of $n$

We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd$(4, 6, 8)=2$ and gcd $(12, 15)=3$.) Suppose that positive integers $a, b, c$ satisfy the following four conditions: $\bullet$ gcd $(a, b, c)=1$, $\bullet$ gcd $(a, b + c)>1$, $\bullet$ gcd $(b, c + a)>1$, $\bullet$ gcd $(c, a + b)>1$. a) Is it possible that $a + b + c = 2015$? b) Determine the minimum possible value that the sum $a+ b+ c$ can take.

Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$.

Show that for all positive real numbers $a, b, c$, we have that $$\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}$$

Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.

For a given integer $n \ge 2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.

Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that max $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$

Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$

Let $a, b, c$ be nonnegative real numbers satisfying $a + b + c =3$. a) If $c > \frac32$, prove that $3(ab + bc + ca) - 2abc < 7$. b) Find the greatest possible value of $M =3(ab + bc + ca) - 2abc $.

Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$ where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$

a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $ b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $

Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

Let $n \ge 4$ be an integer. Find the maximum value of the area of a $n$-gon which is inscribed in the circle of radius $1$ and has two perpendicular diagonals.

Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$

Let $x, y,z$ be positive real numbers satisfying $2x^2+3y^2+6z^2+12(x+y+z) =108$. Find the maximum value of $x^3y^2z$. Alexandru Gırban

The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?

Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions: $ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$. Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$.