Prove that for all positive real numbers $a, b, c$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1$ the following inequality holds: $$3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1$$

Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that $$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$holds for all real numbers $x; y$

All integers are written on an axis in an increasing order. A grasshopper starts its journey at $x=0$. During each jump, the grasshopper can jump either to the right or the left, and additionally the length of its $n$-th jump is exactly $n^2$ units long. Prove that the grasshopper can reach any integer from its initial position.

Let $P(x)$ be a polynomial with degree $n$ and real coefficients. For all $0 \le y \le 1$ holds $\mid p(y) \mid \le 1$. Prove that $p(-\frac{1}{n}) \le 2^{n+1} -1$

There are $2019$ students sitting around circular table. Initially each of them have one candy. Teacher is allowed to pick one student, who has at least one can candy, and this student can decide, whether he gives his candy to his neighbour on the right or on the left. Prove that no matter what students teacher picks during the process, students can always ensure that any point of time no student has more than $2$ candies.

A grandpa has a finite number of boxes in his attic. Each box is a straight rectangular prism with integer edge lengths. For every box its width is greater or equal to its height and its length is greater or equal to its width. A box can be put inside another box if and only if all of its dimensions are respectively smaller than the other one's. You can put two or more boxes in a bigger box only if the smaller boxes are all already inside one of the boxes. The grandpa decided to put the boxes in each other so that there would be a minimal number of visible boxes in the attic (boxes that have not been put inside another). He decided to use the following algorithm: at each step he finds the longest sequence of boxes so that the first can be put in the second, the second can be put in the third, etc., and then he puts them inside each other in the aforementioned order. The grandpa used the algorithm until no box could be put inside another. It is known that at each step the longest sequence of boxes was unique, e.g., at no moment were there two different sequences with the same length. The grandpa now claims that he has the minimal possible number of visible boxes in his attic. Is the claim necessarily true?

Two sequences $b_i$, $c_i$, $0 \le i \le 100$ contain positive integers, except $c_0=0$ and $b_{100}=0$. Some towns in Graphland are connected with roads, and each road connects exactly two towns and is precisely $1$ km long. Towns, which are connected by a road or a sequence of roads, are called neighbours. The length of the shortest path between two towns $X$ and $Y$ is denoted as distance. It is known that the greatest distance between two towns in Graphland is $100$ km. Also the following property holds for every pair $X$ and $Y$ of towns (not necessarily distinct): if the distance between $X$ and $Y$ is exactly $k$ km, then $Y$ has exactly $b_k$ neighbours that are at the distance $k+1$ from $X$, and exactly $c_k$ neighbours that are at the distance $k-1$ from $X$. Prove that $$\frac{b_0b_1 \cdot \cdot \cdot b_{99}}{c_1c_2 \cdot \cdot \cdot c_{100}}$$is a positive integer.

A $20 \times 20$ rectangular grid has been given. It is known that one of the grid's unit squares contains a hidden treasure. To find the treasure, we have been given an opportunity to order several scientific studies at the same time, results of which will be known only after some time. For each study we must choose one $1 \times 4$ rectangle, and the study will tell whether the rectangle contains the treasure. The $1 \times 4$ rectangle can be either horizontal or vertical, and it can extend over a side of the $20 \times 20$ grid, coming back in at the opposite side (you can think of the $20 \times 20$ grid as a torus - the opposite sides are connected). What is the minimal amount of studies that have to ordered for us to precisely determine the unit square containing the treasure?

Let $ABCD$ be a rhombus with the condition $\angle ABC > 90^o$. The circle $\Gamma_B$ with center at $B$ goes through $C$, and the circle $\Gamma_C$ with center at $C$ goes through $B$. Denote by $E$ one of the intersection points of $\Gamma_B$ and $\Gamma_C$. The line $ED$ intersects intersects $\Gamma_B$ again at $F$. Find the value of $\angle AFB$.

Let $\triangle ABC$ be an acute angled triangle with orthocenter $H$ and let $M$ be a midpoint of $BC$. Circle with diameter $AH$ is $\omega_1$ and circle with center $M$ is $\omega_2$. If $\omega_2$ is tangent to circumcircle of $\triangle ABC$, then prove that circles $\omega_1$ and $\omega_2$ are tangent to each other.

Let $A_1A_2...A_{2018}$ be regular $2018$-gon. Radius of it's circumcircle is $R$. Prove that: $$A_1A_{1008}-A_1A_{1006}+A_1A_{1004}-A_1A_{1002} + ... + A_1A_4 -A_1A_2=R$$

Let $AX$, $AY$ be tangents to circle $\omega$ from point $A$. Le $B$, $C$ be points inside $AX$ and $AY$ respectively, such that perimeter of $\triangle ABC$ is equal to length of $AX$. $D$ is reflection of $A$ over $BC$. Prove that circumcircle $\triangle BDC$ and $\omega$ are tangent to each other.

Let $s(k)$ denotes sum of digits of positive integer $k$. Prove that there are infinitely many positive integers $n$, which are not divisible by $10$ and satisfies: $$s(n^2) < s(n) - 5$$

Let $m$ be a positive integer and $p$ be a prime, such that $m^2 - 2$ is divisible by $p$. Suppose that there exists positive integer $a$ such that $a^2+m-2$ is divisible by $p$. Prove that there exists positive integer $b$ such that $b^2- m -2$ is divisible by $p$.

Determine all tuples of integers $(a,b,c)$ such that: $$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$

Determine all tuples of positive integers $(x, y, z, t)$ such that: $$ xyz = t!$$$$ (x+1)(y+1)(z+1) = (t+1)!$$holds simultaneously.