The positive integers $a, b, c$ are such that $$gcd \,\,\, (a, b, c) = 1,$$ $$gcd \,\,\,(a, b + c) > 1,$$ $$gcd \,\,\,(b, c + a) > 1,$$ $$gcd \,\,\,(c, a + b) > 1.$$ Determine the smallest possible value of $a + b + c$. Clarification: gcd stands for greatest common divisor.

Let $x$ be a real number in the interval $\left(0,\dfrac{\pi}{2}\right)$ such that $\dfrac{1}{\sin x \cos x}+2\cot 2x=\dfrac{1}{2}$. Evaluate $\dfrac{1}{\sin x \cos x}-2\cot 2x$.

A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$. Prove that the inequality $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.

Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero. ([i]Hint[/i]: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)

There are several counters of various colours and sizes. No two of them have, simultaneously, the same colour and the same size. On each counter $F$ two numbers are written. One of them is the number of counters that have the same colour as $F$ but a different size than $F$. The other number is the number of counters that have the same size as $F$ but a different colour. It is known that each of the $101$ numbers $0,1,\ldots,100$ is written at least once. Determine the smallest number of counters for which this is possible.

An isosceles right triangle is inscribed in a circle of radius 5, thereby separating the circle into four regions. Compute the sum of the areas of the two smallest regions.

Prove that, for every tetrahedron $ABCD$, there exists a unique point $P$ in the interior of the tetrahedron such that the tetrahedra $PABC,PABD,PACD,PBCD$ have equal volumes.

For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$. Prove that $xyz\geq 3600$

Region $ABCDEFGHIJ$ consists of $13$ equal squares and is inscribed in rectangle $PQRS$ with $A$ on $\overline{PQ}$, $B$ on $\overline{QR}$, $E$ on $\overline{RS}$, and $H$ on $\overline{SP}$, as shown in the figure on the right. Given that $PQ=28$ and $QR=26$, determine, with proof, the area of region $ABCDEFGHIJ$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(12)); pair P=(0,0), Q=(0,28), R=(26,28), S=(26,0), B=(3,28); draw(P--Q--R--S--cycle); picture p = new picture; draw(p, (0,0)--(3,0)^^(0,-1)--(3,-1)^^(0,-2)--(5,-2)^^(0,-3)--(5,-3)^^(2,-4)--(3,-4)^^(2,-5)--(3,-5)); draw(p, (0,0)--(0,-3)^^(1,0)--(1,-3)^^(2,0)--(2,-5)^^(3,0)--(3,-5)^^(4,-2)--(4,-3)^^(5,-2)--(5,-3)); transform t = shift(B) * rotate(-aSin(1/26^.5)) * scale(26^.5); add(t*p); label("$P$",P,SW); label("$Q$",Q,NW); label("$R$",R,NE); label("$S$",S,SE); label("$A$",t*(0,-3),W); label("$B$",B,N); label("$C$",t*(3,0),plain.ENE); label("$D$",t*(3,-2),NE); label("$E$",t*(5,-2),plain.E); label("$F$",t*(5,-3),plain.SW); label("$G$",t*(3,-3),(0.81,-1.3)); label("$H$",t*(3,-5),plain.S); label("$I$",t*(2,-5),NW); label("$J$",t*(2,-3),SW);[/asy]

Let $ ABCD $ be a cyclic quadrilateral,$ M $ midpoint of $ AC $ and $ N $ midpoint of $ BD. $ If $ \angle AMB =\angle AMD, $ prove that $ \angle ANB =\angle BNC. $

Let $\omega_1$, $\omega_2$, $\omega_3$ are three externally tangent circles, with $\omega_1$ and $\omega_2$ tangent at $A$. Choose points $B$ and $C$ on $\omega_1$ so that lines $AB$ and $AC$ are tangent to $\omega_3$. Suppose the line $BC$ intersect $\omega_3$ at two distinct points, and $X$ is the intersection further away to $B$ and $C$ than the other one. Prove that one of the tangent lines of $\omega_2$ passing through $X$, is also tangent to an excircle of triangle $ABC$. [i]Proposed by Ivan Chan Kai Chin[/i]

Prove that if real numbers $a,b,c$ satisfy the equality $$\frac{a}{m+2}+\frac{b}{m+1}+\frac{c}{m}= 0$$ for some positive number $m$, then the equation $ax^2 + bx + c = 0$ has a root between $0$ and $1$.

Find the angle between two common sections of the page $2x+y-z=0$ and the cone $4x^2-y^2+3z^2=0.$

Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?

Let $P$ and $Q$ be the points on the diagonal $BD$ of a quadrilateral $ABCD$ such that $BP = PQ = QD$. Let $AP$ and $BC$ meet at $E$, and let $AQ$ meet $DC$ at $F$. (a) Prove that if $ABCD$ is a parallelogram, then $E$ and $F$ are the midpoints of the corresponding sides. (b) Prove the converse of (a).

Consider a sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ and a primitivable function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]a)[/b] Prove that $ f $ is monotonic and continuous if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+x_n \right)\geqslant f(x) $$ is true. [b]b)[/b] Show that $ f $ is convex if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+2x_n \right) +f(x)\geqslant 2f\left( x+x_n \right) $$ is true. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

Let $f$ and $g$ be linear functions such that $f(g(2021))-g(f(2021)) = 20$. Compute $f(g(2022))- g(f(2022))$. (Note: A function h is linear if $h(x) = ax + b$ for all real numbers $x$.)

The centre $O$ of the circumcircle of $\vartriangle ABC$ lies inside the triangle. Perpendiculars are drawn rom $O$ on the sides. When produced beyond the sides they meet the circumcircle at points $K, M$ and $P$. Prove that $\overrightarrow{OK} + \overrightarrow{OM} + \overrightarrow{OP} = \overrightarrow{OI}$, where $I$ is the centre of the inscribed circle of $\vartriangle ABC$. (V Galperin, Moscow)

Let $m$ be a number containing only $0$ and $6$ as its digits.Show that $m$ can't be a perfect square.

The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.

Prove that for each positive integer $k$ there exists a number base $b$ along with $k$ triples of Fibonacci numbers $(F_u,F_v,F_w)$ such that when they are written in base $b$, their concatenation is also a Fibonacci number written in base $b$. (Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.) [i]Proposed by Serbia[/i]

Let $a, b,c$ be positive real numbers such that $ab + bc + ca = 1$. Prove that $$a\sqrt{b^2 + c^2 + bc} + b\sqrt{c^2 + a^2 + ca} + c\sqrt{a^2 + b^2 + ab} \ge \sqrt3$$

Two players take turns placing an unused number from {1, 2, 3, 4, 5, 6, 7, 8} into one of the empty squares in the array to the right. The game ends once all the squares are filled. The first player wins if the product of the numbers in the top row is greater. The second player wins if the product of the numbers in the bottom row is greater. If both players play with perfect strategy, who wins this game? [asy] unitsize(32); int[][] a = { {1, 2, 3, 4}, {5, 6, 7, 8}}; for (int i = 0; i < 4; ++i) { for (int j = 0; j < 2; ++j) { draw((i, -j)--(i+1, -j)--(i+1, -j-1)--(i, -j-1)--cycle); if (a[j][i] > 0) label(string(a[j][i]), (i+0.5, -j-0.5), fontsize(16pt)); } } [/asy]

A square $ABCD$ is given. A line passing through $A$ intersects $CD$ at $Q$. Draw a line parallel to $AQ$ that intersects the boundary of the square at points $M$ and $N$ such that the area of the quadrilateral $AMNQ$ is maximal.

Is it possible to fill a $3 \times 3$ grid with each of the numbers $1,2,\ldots,9$ once each such that the sum of any two numbers sharing a side is prime?