Let $p_1,p_2,...,p_n$ be $n\ge 2$ fixed positive real numbers. Let $x_1,x_2,...,x_n$ be nonnegative real numbers such that $$x_1p_1+x_2p_2+...+x_np_n=1.$$Determine the (a) maximal; (b) minimal possible value of $x_1^2+x_2^2+...+x_n^2$.

Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations: $$\begin{cases} y(x+y)^2=2\\ 8y(x^3-y^3) = 13. \end{cases}$$

Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that satisfies $$\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2$$for all real $x$. Prove for all real $x$: (a) $f(x)\ge 4$; (b) $f(x)\ge 7.$

Alice and Bob play a game on a numbered row of $n \ge 5$ squares. At the beginning a pebble is put on the first square and then the players make consecutive moves; Alice starts. During a move a player is allowed to choose one of the following: move the pebble one square forward; move the pebble four squares forward; move the pebble two squares backwards. All of the possible moves are only allowed if the pebble stays within the borders of the square row. The player who moves the pebble to the last square (a.k.a $n\text{-th}$) wins. Determine for which values of $n$ each of the players has a winning strategy.

Let $ABCD$ be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that: every vertex that lies on the side $AB$ has been colored in either the $1.$ or $2.$ color; every vertex that lies on the side $BC$ has been colored in either the $2.$ or $3.$ color; every vertex that lies on the side $CD$ has been colored in either the $3.$ or $4.$ color; every vertex that lies on the side $DA$ has been colored in either the $4.$ or $1.$ color; no two neighboring vertices have been colored in $1.$ and $3.$ color; no two neighboring vertices have been colored in $2.$ and $4.$ color. Notice that the constraints imply that vertex $A$ has been colored in $1.$ color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).

Let $n \ge 3$ points be given in the plane, no three of which lie on the same line. Determine whether it is always possible to draw an $n$-gon whose vertices are the given points and whose sides do not intersect. Remark. The $n$-gon can be concave.

Let natural $n \ge 2$ be given. Let Laura be a student in a class of more than $n+2$ students, all of which participated in an olympiad and solved some problems. Additionally, it is known that: for every pair of students there is exactly one problem that was solved by both students; for every pair of problems there is exactly one student who solved both of them; one specific problem was solved by Laura and exactly $n$ other students. Determine the number of students in Laura's class.

Acute triangle $\triangle ABC$ with $AB<AC$, circumcircle $\Gamma$ and circumcenter $O$ is given. Midpoint of side $AB$ is $D$. Point $E$ is chosen on side $AC$ so that $BE=CE$. Circumcircle of triangle $BDE$ intersects $\Gamma$ at point $F$ (different from point $B$). Point $K$ is chosen on line $AO$ satisfying $BK \perp AO$ (points $A$ and $K$ lie in different half-planes with respect to line $BE$). Prove that the intersection of lines $DF$ and $CK$ lies on $\Gamma$.

Let $ABC$ be an obtuse triangle with obtuse angle $\angle B$ and altitudes $AD, BE, CF$. Let $T$ and $S$ be the midpoints of $AD$ and $CF$, respectively. Let $M$ and $N$ and be the symmetric images of $T$ with respect to lines $BE$ and $BD$, respectively. Prove that $S$ lies on the circumcircle of triangle $BMN$.

Let $ABC$ be a triangle with angles $\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ$. Let $P$ be a point on the bisector of $\angle BAC$ satisfying $\angle BPC =130^\circ$. Let $PX, PY, PZ$ be the perpendiculars drawn from $P$ to the sides $BC, AC, AB$, respectively. Prove that the following equation with segment lengths is satisfied $$AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.$$

Let $ABCD$ be a parallelogram. Let $X$ and $Y$ be arbitrary points on sides $BC$ and $CD$, respectively. Segments $BY$ and $DX$ intersect at $P$. Prove that the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$.

Determine whether there exists a prime $q$ so that for any prime $p$ the number $$\sqrt[3]{p^2+q}$$is never an integer.

Let $a_1,a_2,...$ be a sequence of positive integers with $a_1=2$. For each $n \ge 1$, $a_{n+1}$ is the biggest prime divisor of $a_1a_2...a_n+1$. Prove that the sequence does not contain numbers $5$ and $11$.

Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations $$\begin{cases} x+y+z=n\\ xyz = 2t^3. \end{cases}$$

Call a natural number simple if it is not divisible by any square of a prime number (in other words it is square-free). Prove that there are infinitely many positive integers $n$ such that both $n$ and $n+1$ are simple.