Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$. ([i]Serbia[/i])

$Abdulqodir$ cut out $2024$ congruent regular $n-$gons from a sheet of paper and placed these $n-$gons on the table such that some parts of each of these $n-$gons may be covered by others. We say that a vertex of one of the afore-mentioned $n-$gons is $visible$ if it is not in the interior of another $n-$gon that is placed on top of it. For any $n>2$ determine the minimum possible number of visible vertices. \\ Proposed by David Hrushka, Slovakia

Three real numbers satisfy the following statements: (1) the square of their sum equals to the sum their squares. (2) the product of the first two numbers is equal to the square of the third number. Find these numbers.

Prove that, if there exist natural numbers $a_1,a_2…a_{2017}$ for which the product $(a_1^{2017}+a_2 )(a_2^{2017}+a_3 )…(a_{2016}^{2017}+a_{2017})(a_{2017}^{2017}+a_1)$ is a $k$-th power of a prime number, then $k=2017$ or $k\geq 2017.2018$.

There are a number of arcs on the edge of a circular disk. Each pair of arcs has the least one point in common. Show that on the circle you can choose two diametrical opposites points such that each arc contains at least one of these two points.

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

Each detail of the “Young Solderer” instructor is a bracket in the shape of the letter $\Pi$, consisting of three single segments. Is it possible from the parts of this constructor are soldered together, a complete wire frame of the cube $2 \times 2 \times 2$, divided into $1 \times 1 \times 1$ cubes? (The frame consists of 27 points, connected by single segments; any two adjacent points must be connected by exactly one piece of wire.) [hide]=original wording]Каждая деталько нструктора ''Юный паяльщик'' — это скобка в виде буквы П, остоящая из трех единичных отрезков. Можно ли издеталей этого конструктора спаятьполный роволочный каркас куба 2 × × 2 × 2, разбитого на кубики 1 × 1 × 1? (Каркас состоит из 27 точек,соединенных единичными отрезками; любые две соседние точки должны бытьсоединены ровно одним проволочным отрезком.)[/hide]

Suppose you started at the origin on the number line in a coin-flipping game. Every time you flip a heads, you move forward one step, otherwise you move back one step. However, there are walls at positions $8$ and $-8$; if you are at these positions and your coin flip dictates that you should move past them, instead you must stay. What is the expected number of coin flips needed to have visited both walls?

Does there exist a positive integer that is a power of $2$ and we get another power of $2$ by swapping its digits? Justify your answer.

In a certain year the price of gasoline rose by $ 20\%$ during January, fell by $ 20\%$ during February, rose by $ 25\%$ during March, and fell by $ x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $ x$? $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 17\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 25\qquad \textbf{(E)}\ 35$

On the circumcircle of triangle $ABC$, point $P$ is taken in such a way that the perpendicular drawn by the point $P$ to the line $AC$ cuts the circle also at the point $Q$, the perpendicular drawn by the point $Q$ to the line $AB$ cuts the circle also at point R and the perpendicular drawn by point $R$ to the line BC cuts the circle also at the point $P$. Let $O$ be the center of this circle. Prove that $\angle POC = 90^o$ .

Given a convex polygon with vertices at lattice points on a plane containing origin $O$. Let $V_1$ be the set of vectors going from $O$ to the vertices of the polygon, and $V_2$ be the set of vectors going from $O$ to the lattice points that lie inside or on the boundary of the polygon (thus, $V_1$ is contained in $V_2$.) Two grasshoppers jump on the whole plane: each jump of the first grasshopper shift its position by a vector from the set $V_1$, and the second by the set $V_2$. Prove that there exists positive integer $c$ that the following statement is true: if both grasshoppers can jump from $O$ to some point $A$ and the second grasshopper needs $n$ jumps to do it, then the first grasshopper can use at most $n+c$ jumps to do so.

The center of a circle and nine randomly selected points on this circle are colored in red. Every pair of those points is connected by a line segment, and every point of intersection of two line segments inside the circle is colored in red. What is the largest possible number of red points? A. $235$ B. $245$ C. $250$ D. $220$ E. $265$

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a function having the following property: For any two points $A$ and $B$ in $\mathbb{R}^2$, the distance between $A$ and $B$ is the same as the distance between the points $f(A)$ and $f(B)$. Denote the unique straight line passing through $A$ and $B$ by $l(A,B)$ (a) Suppose that $C,D$ are two fixed points in $\mathbb{R}^2$. If $X$ is a point on the line $l(C,D)$, then show that $f(X)$ is a point on the line $l(f(C),f(D))$. (b) Consider two more point $E$ and $F$ in $\mathbb{R}^2$ and suppose that $l(E,F)$ intersects $l(C,D)$ at an angle $\alpha$. Show that $l(f(C),f(D))$ intersects $l(f(E),f(F))$ at an angle $\alpha$. What happens if the two lines $l(C,D)$ and $l(E,F)$ do not intersect? Justify your answer.

Let $a, b, c >1$ be positive real numbers such that $a^{\log_b c}=27, b^{\log_c a}=81,$ and $c^{\log_a b}=243$. Then the value of $\log_3{abc}$ can be written as $\sqrt{x}+\sqrt{y}+\sqrt{z}$ for positive integers $x,y,$ and $z$. Find $x+y+z$. [i]Proposed by [b]AOPS12142015[/b][/i]

Find all functions $f : \mathbb{Q}[x] \to \mathbb{Q}[x]$ such that two following conditions holds : $$\forall P , Q \in \mathbb{Q}[x] : f(P+Q)=f(P)+f(Q)$$ $$\forall P \in \mathbb{Q}[x] : gcd(P , f(P))=1 \iff$$ $P$ is square-free. Which a square-free polynomial with rational coefficients is a polynomial such that there doesn't exist square of a non-constant polynomial with rational coefficients that divides it. [i]Proposed by Sina Azizedin[/i]

Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$

Every cell of a $3\times 3$ board is coloured either by red or blue. Find the number of all colorings in which there are no $2\times 2$ squares in which all cells are red.

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$. Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set. [i]Dan Schwarz, Romania[/i]

A racing tournament has $12$ stages and $n$ participants. After each stage, all participants, depending on their place $k$, receive points $a_k$ (numbers $a_k$ are natural numbers and $a_1 > a_2 >... > a_n$). At what smallest $n$ can the organizer of the tournament choose numbers $a_1$, $...$ , $a_n$ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place?

The angles in triangle $ABC$ are such that $\angle A$, $\angle B$, $\angle C$ form an arithmetic progression in that order. Find the measure of $\angle B$, in degrees.

Let $a, b, c$ be nonnegative integers such that $a \le b \le c, 2b \neq a + c$ and $\frac{a+b+c}{3}$ is an integer. Is it possible to find three nonnegative integers $d, e$, and $f$ such that $d \le e \le f, f \neq c$, and such that $a^2+b^2+c^2 = d^2 + e^2 + f^2$?

Prove that for the arbitrary positive $a_1, a_2, ... , a_n$ the following inequality is held $$\frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_4}+....+\frac{a_{n-1}}{a_n+a_1}+\frac{a_n}{a_1+a_2}>\frac{n}{4}$$

There are $30$ apparently equal balls, $15$ of which have the weight $a$ and the remaining $15$ have the weight $b$ with $a \ne b$. The balls are to be partitioned into two groups of $15$, according to their weight. An assistant partitioned them into two groups, and we wish to check if this partition is correct. How can we check that with as few weighings as possible?

In how many ways every unit square of a $2018$ x $2018$ board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.