Given real numbers $x$ and $y$, such that $$x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 \le 0 .$$Prove that $x \ge - \frac16$

It is known about the function $f : R \to R$ that $\bullet$ $f(x) > f(y)$ for all real $x > y$ $\bullet$ $f(x) > x$ for all real $x$ $\bullet$ $f(2x - f (x)) = x$ for all real $x$. Prove that $f(x) = x + f(0)$ for all real numbers $x$.

Prove that there does not exist a polynomial $P (x)$ with integer coefficients and a natural number $m$ such that $$x^m + x + 2 = P(P(x))$$holds for all integers $x$.

Can you draw some diagonals in a convex $2014$-gon so that they do not intersect, the whole $2014$-gon is divided into triangles and each vertex belongs to an odd number of these triangles?

$BE$ is the altitude of acute triangle $ABC$. The line $\ell$ touches the circumscribed circle of the triangle $ABC$ at point $B$. A perpendicular $CF$ is drawn from $C$ on line $\ell$. Prove that the lines $EF$ and $AB$ are parallel.

$AM$ is the median of triangle $ABC$. A perpendicular $CC_1$ is drawn from point $C$ on the bisector of angle $\angle CMA$, a perpendicular $BB_1$ is drawn from point $B$ on the bisector of angle $\angle BMA$. Prove that line $AM$ intersects segment $B_1C_1$ at its midpoint.

Two circle $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$, point $P$ is not on the line $AB$. Line $AP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $K$ and $L$ respectively, line $BP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles $KMP$ and $LNP$ are $O_1$ and $O_2$ respectively. Prove that $O_1O_2$ is perpendicular to $AB$.

Given a fixed rational number $q$. Let's call a number $x$ charismatic if we can find a natural number $n$ and integers $a_1, a_2,.., a_n$ such that $$x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .$$i) Prove that one can find a $q$ such that all positive rational numbers are charismatic. ii) Is it true that for all $q$, if the number $x$ is charismatic, then $x + 1$ is also charismatic?

Two players play the following game on a square of $N \times N$ squares. They color one square in turn so that no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of $N$ does the first player have a winning strategy?

Is it true that for all natural $n$, it is always possible to give each of the $n$ children a hat painted in one of $100$ colors so that if a girl is known to more than $2015$ boys, then not all of these boys have hats of the same color, and, if a boy is acquainted with more than $2015$ girls, don't all these girls have hats of the same color? original wordingVai tiesa, ka visiem naturāliem n vienmēr iespējams katram no n bērniem iedot pa cepurei, kas nokrāsota vienā no 100 krāsām tā, ka, ja kāda meitene ir pazīstama ar vairāk nekā 2015 zēniem, tad ne visiem šiem zēniem cepures ir vienā krāsā, un, ja kāds zēns ir pazīštams ar vairāk nekā 2015 meitenēm, tad ne visām šīm meitenēm cepures ir vienā krāsā?

Let us call a figure on a sheet of squares an arbitrary finite set of connected squares, i.e. a set of squares in which it is possible to go from any square to any other by walking only on the squares of this figure. Prove that for every natural n there exists such a figure on the sheet of squares that it can be cut into "corners" (Fig. 1) exactly in $F_n$ ways, where $F_n$ s the $n$-th Fibonacci number (in the series of Fibonacci numbers $F_1 = F_2 = 1$ and for each $i > 1$ holds $F_{i+2} = F_{i+1} + F_i$). For example, a rectangle of $2 \times 3$ squares can be cut at the corners in exactly two ways (Fig. $2$).

For real positive numbers $a, b, c$, the equality $abc = 1$ holds. Prove that $$\frac{a^{2014}}{1 + 2 bc}+\frac{b^{2014}}{1 + 2ac}+\frac{c^{2014}}{1 + 2ab} \ge \frac{3}{ab+bc+ca}.$$

Are there positive real numbers $a$ and $b$ such that $[an+b]$ is prime for all natural values of $n$ ? $[x]$ denotes the integer part of the number $x$, the largest integer that does not exceed $x$.

Let $S(a)$ denote the sum of the digits of the number $a$. Given a natural $R$ can one find a natural $n$ such that $\frac{S (n^2)}{S (n)}= R$?

Let $w (n)$ denote the number of different prime numbers by which $n$ is divisible. Prove that there are infinitely many natural numbers $n$ such that $w(n) < w(n + 1) < w(n + 2)$.

Points $X$ , $Y$, $Z$ lie on a line $k$ in this order. Let $\omega_1$, $\omega_2$, $\omega_3$ be three circles of diameters $XZ$, $XY$ , $YZ$ , respectively. Line $\ell$ passing through point $Y$ intersects $\omega_1$ at points $A$ and $D$, $\omega_2$ at $B$ and $\omega_3$ at $C$ in such manner that points $A, B, Y, X, D$ lie on $\ell$ in this order. Prove that $AB =CD$.