For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right. Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle. For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation. Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.

Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.

Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then \[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\] [i]A. Golovanov[/i]

In the triangle $ABC$ let $D,E$ and $F$ be the mid-points of the three sides, $X,Y$ and $Z$ the feet of the three altitudes, $H$ the orthocenter, and $P,Q$ and $R$ the mid-points of the line segment joining $H$ to the three vertices. Show that the nine points $D,E,F,P,Q,R,X,Y,Z$ lie on a circle.

A square cake of uniform height is evenly covered with frosting on the top and all four sides. Find a way to cut the cake into five portions such that: (a) All portions contain the same amount of cake. (b) All portions contain the same amount of cake and frosting.

Four grasshoppers sit at the vertices of a square. Every second, one of them jumps over one of the others to the symmetrical point on the other side (if $X$ jumps over $Y$ to the point $X'$, then $X$, $Y$ and $X'$ lie on a straight line and $XY = YX'$). Prove that after several jumps no three grasshoppers can be: (a) on a line parallel to a side of the square, (b) on a straight line. (AK Kovaldzhy)

The squares of a chessboard are numbered in the following way. The upper left corner is numbered 1. The two squares on the next diagonal from top-right to bottom-left are numbered 2 and 3. The three squares on the next diagonal are numbered 4, 5 and 6, and so on. The two squares on the second-to-last diagonal are numbered 62 and 63, and the lower right corner is numbered 64. Peter puts eight pebbles on the squares of the chessboard in such a way that there is exactly one pebble in each column and each row. Then he moves each pebble to a square with a number greater than that of the original square. Can it happen that there is still exactly one pebble in each column and each row? [i](8 points)[/i]

Let ${(a_n)}_{n=0}^{\infty}$ and ${(b_n)}_{n=0}^{\infty}$ be real squences such that $a_0=40$, $b_0=41$ and for all $n\geq 0$ the given equalities hold. $$a_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n}$$ Find the least possible positive integer value of $k$ such that the value of $a_k$ is strictly bigger than $80$.

Find the number of sequences of $2005$ terms with the following properties: (i) No three consecutive terms of the sequence are equal, (ii) Every term equals either $1$ or $-1$, (iii) The sum of all terms of the sequence is at least $666$.

Let $ABCD$ be a cyclie quadrilateral, $\omega$ be it's circumcircle and $M$ be the midpoint of the arc $AB$ of $\omega$ which does not contain the vertices $C$ and $D$. The line that passes through $M$ and the intersection point of segments $AC$ and $BD$, intersects again $\omega$ in $N$. Let $P$ and $Q$ be points in the $CD$ segment such that $\angle AQD = \angle DAP$ and $\angle BPC = \angle CBQ$. Prove that the circumcircle of $NPQ$ and $\omega$ are tangent to each other.

Two identical balls of radius $\sqrt{15}$ and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface $S$ of the ball with the smaller radius.

Let $S = \{1, 2,..., 8\}$. How many ways are there to select two disjoint subsets of $S$?

Show that there are infinitely many odd composite numbers in the sequence $1^1, 1^1 + 2^2, 1^1 + 2^2 + 3^3, 1^1 + 2^2 + 3^3 + 4^4, ...$ .

In triangle $\triangle ABC$ , $I$ is the incenter and $M$ is the midpoint of arc $(BC)$ in the circumcircle of $(ABC)$not containing $A$. Let $X$ be an arbitrary point on the external angle bisector of $A$. Let $BX \cap (BIC) = T$. $Y$ lies on $(AXC)$ , different from $A$ , st $MA=MY$ . Prove that $TC || AY$ (Assume that $X$ is not on $(ABC)$ or $BC$)

What is the tens digit of the sum \[\left(1!\right)^2+\left(2!\right)^2+\left(3!\right)^2+\ldots+\left(2018!\right)^2?\] [i]2018 CCA Math Bonanza Individual Round #1[/i]

Let $\mathcal{P}$ be a regular $10$-gon in the coordinate plane. Mark computes the number of distinct $x$-coordinates that vertices of $\mathcal{P}$ take. Across all possible placements of $\mathcal{P}$ in the plane, compute the sum of all possible answers Mark could get.

Let $a, b, c, p, q, r$ be positive integers with $p, q, r \ge 2$. Denote \[Q=\{(x, y, z)\in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b , 0 \le z \le c \}. \] Initially, some pieces are put on the each point in $Q$, with a total of $M$ pieces. Then, one can perform the following three types of operations repeatedly: (1) Remove $p$ pieces on $(x, y, z)$ and place a piece on $(x-1, y, z)$ ; (2) Remove $q$ pieces on $(x, y, z)$ and place a piece on $(x, y-1, z)$ ; (3) Remove $r$ pieces on $(x, y, z)$ and place a piece on $(x, y, z-1)$. Find the smallest positive integer $M$ such that one can always perform a sequence of operations, making a piece placed on $(0,0,0)$, no matter how the pieces are distributed initially.

A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.) $ \textbf{(A)}\ 21\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 50$

If $a, b, c$ are real numbers such that a$b + bc + ca = 0$, prove the inequality $$2(a^2 + b^2 + c^2)(a^2b^2 + b^2c^2 + c^2a^2) \ge 27a^2b^2c^2$$ When does the equality hold ? Leonard Giugiuc

In how many ways can we color exactly $k$ vertices of an $n$-gon in red such that any $2$ consecutive vertices are not both red. (Vertices are considered to be labeled)

The angles of the triangle $\alpha, \beta, \gamma$ satisfy the inequalities $$\sin \alpha > \cos \beta, \sin \beta > \cos \gamma, \sin \gamma > \cos \alpha. $$Prove that the trαiangle is acute-angled.

Determine all pairs $(a,b)$ of real numbers with $a\leqslant b$ that maximise the integral $$\int_a^b e^{\cos (x)}(380-x-x^2) \mathrm{d} x.$$

A sequence of positive real numbers $a_1, a_2, \dots$ is called $\textit{phine}$ if it satisfies $$a_{n+2}=\frac{a_{n+1}+a_{n-1}}{a_n},$$ for all $n\geq2$. Is there a $\textit{phine}$ sequence such that, for every real number $r$, there is some $n$ for which $a_n>r$?

Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\ $\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\ $\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\ $\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true? \\ \\ [i] (Bogdan Blaga)[/i]