Let $a,b,c>0$ and $abc=1$ . Prove that $$ \sqrt{2(1+a^2)(1+b^2)(1+c^2)}\ge 1+a+b+c.$$

Find all prime numbers $p$ such that $\frac{3^{p-1} - 1}{p}$ is a perfect square.

On the table, there are $1024$ marbles and two students, $A$ and $B$, alternatively take a positive number of marble(s). The student $A$ goes first, $B$ goes after that and so on. On the first move, $A$ takes $k$ marbles with $1 < k < 1024$. On the moves after that, $A$ and $B$ are not allowed to take more than $k$ marbles or $0$ marbles. The student that takes the last marble(s) from the table wins. Find all values of $k$ the student $A$ should choose to make sure that there is a strategy for him to win the game.

Let $ABC$ be an acute, non isosceles triangle and $(O)$ be its circumcircle (with center $O$). Denote by $G$ the centroid of the triangle $ABC$, by $H$ the foot of the altitude from $A$ onto the side $BC$ and by $I$ the midpoint of $AH$. The line $IG$ intersects $BC$ at $K$. 1. Prove that $CK = BH$. 2. The ray $(GH$ intersects $(O)$ at L. Denote by $T$ the circumcenter of the triangle $BHL$. Prove that $AO$ and $BT$ intersect on the circle $(O)$.

Given a polynomial $f(x) = x^4+ax^3+bx^2+cx$. It is known that each of the equations $f(x) = 1$ and $f(x) = 2$ has four real roots (not necessarily distinct). Prove that if the roots of the first equation satisfy the equality $x_1 + x_2 = x_3 + x_4$, then the same equation holds for the roots of the second equation

A positive integer $k > 1$ is called nice if for any pair $(m, n)$ of positive integers satisfying the condition $kn + m | km + n$ we have $n | m$. 1. Prove that $5$ is a nice number. 2. Find all the nice numbers.

Let $(O)$ be a circle, and $BC$ be a chord of $(O)$ such that $BC$ is not a diameter. Let $A$ be a point on the larger arc $BC$ of $(O)$, and let $E, F$ be the feet of the perpendiculars from $B$ and $C$ to $AC$ and $AB$, respectively. 1. Prove that the tangents to $(AEF)$ at $E$ and $F$ intersect at a fixed point $M$ when $A$ moves on the larger arc $BC$ of $(O)$. 2. Let $T$ be the intersection of $EF$ and $BC$, and let $H$ be the orthocenter of $ABC$. Prove that $TH$ is perpendicular to $AM$.

Find the number of ways one can put numbers $1$ or $2$ in each cell of an $8\times 8$ chessboard in such a way that the sum of the numbers in each column and in each row is an odd number. (Two ways are considered different if the number in some cell in the first way is different from the number in the cell situated in the corresponding position in the second way)

Let $a,b,c>0$ and $a^2+b^2+c^2=3$ . Prove that $$ \frac{a(a-b^2)}{a+b^2}+\frac{b(b-c^2)}{b+c^2}+\frac{c(c-a^2)}{c+a^2}\ge 0.$$

Find all pairs of positive integers $(p; q) $such that both the equations $x^2- px + q = 0 $ and $ x^2 -qx + p = 0 $ have integral solutions.

Let $BC$ be a chord of a circle $(O)$ such that $BC$ is not a diameter. Let $AE$ be the diameter perpendicular to $BC$ such that $A$ belongs to the larger arc $BC$ of $(O)$. Let $D$ be a point on the larger arc $BC$ of $(O)$ which is different from $A$. Suppose that $AD$ intersects $BC$ at $S$ and $DE$ intersects $BC$ at $T$. Let $F$ be the midpoint of $ST$ and $I$ be the second intersection point of the circle $(ODF)$ with the line $BC$. 1. Let the line passing through $I$ and parallel to $OD$ intersect $AD$ and $DE$ at $M$ and $N$, respectively. Find the maximum value of the area of the triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (such that $D \ne A$). 2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$

Consider a set $S$ of $200$ points on the plane such that $100$ points are the vertices of a convex polygon $A$ and the other $100$ points are in the interior of the polygon. Moreover, no three of the given points are collinear. A triangulation is a way to partition the interior of the polygon $A$ into triangles by drawing the edges between some two points of S such that any two edges do not intersect in the interior, and each point in $S$ is the vertex of at least one triangle. 1. Prove that the number of edges does not depend on the triangulation. 2. Show that for any triangulation, one can color each triangle by one of three given colors such that any two adjacent triangles have different colors.

For each pair of positive integers $(x, y)$ a nonnegative integer $x\Delta y$ is defined. It is known that for all positive integers $a$ and $b$ the following equalities hold: i. $(a + b)\Delta b = a\Delta b + 1$. ii. $(a\Delta b) \cdot (b\Delta a) = 0$. Find the values of the expressions $2016\Delta 121$ and $2016\Delta 144$.

Let $ABC$ be a triangle inscribed in circle $(O)$ such that points $B, C$ are fixed, while $A$ moves on major arc $BC$ of $(O)$. The tangents through $B$ and $C$ to $(O)$ intersect at $P$. The circle with diameter $OP$ intersects $AC$ and $AB$ at $D$ and $E$, respectively. Prove that $DE$ is tangent to a fixed circle whose radius is half the radius of $(O)$.

Find all pairs of primes $(p, q)$ such that $p^3 - q^5 = (p + q)^2$ .

Let $S = \{-17, -16, ..., 16, 17\}$. We call a subset $T$ of $S$ a good set if $-x \in T$ for all $x \in T$ and if $x, y, z \in T (x, y, z$ may be equal) then $x + y + z \ne 0$. Find the largest number of elements in a good set.

Let $a,b,c>0$ and $a+b+c=6$ . Prove that $$ \frac{1}{a^2b+16}+\frac{1}{b^2c+16}+\frac{1}{c^2a+16} \ge \frac{1}{8}.$$

Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.

Let $ABC$ be a triangle inscribed in the circle $(O)$, with orthocenter $H$. Let d be an arbitrary line which passes through $H$ and intersects $(O)$ at $P$ and $Q$. Draw diameter $AA'$ of circle $(O)$. Lines $A'P$ and $A'Q$ meet $BC$ at $K$ and $L$, respectively. Prove that $O, K, L$ and $A'$ are concyclic.

A chessboard has 64 cells painted black and white in the usual way. A bishop path is a sequence of distinct cells such that two consecutive cells have exactly one common point. At least how many bishop paths can the set of all white cells be divided into?