Given two circles with the radiuses $R$ and $r$, touching each other from the outer side. Consider all the trapezoids, such that its lateral sides touch both circles, and its bases touch different circles. Find the shortest possible lateral side.

Each of $K$ friends simultaneously learns one different item of news. They begin to phone one another to tell them their news. Each conversation lasts exactly one hour, during which time it is possible for two friends to tell each other all of their news. What is the minimum number of hours needed in order for all of the friends to know all of the news? Consider in this problem: (a) $K = 64$. (b) $K = 55$. (c) $K = 100$. (A Andjans, Riga) PS. (a) was the junior problem, (a),(b),(c) the senior one

What is the smallest perimeter of the convex $32$-gon, having all the vertices in the nodes of cross-lined paper with the sides of its squares equal to $1$?

For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$?

Let $ABC$ a right isosceles triangle with hypotenuse $AB=\sqrt2$ . Determine the positions of the points $X,Y,Z$ on the sides $BC,CA,AB$ respectively so that the triangle $XYZ$ is isosceles, right, and with minimum area.

$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?

Find minimal value of $A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$

Let $ABC$ be a triangle. On its sides $AB$ and $BC$ are fixed points $C_1$ and $A_1$, respectively. Find a point $ P$ on the circumscribed circle of triangle $ABC$ such that the distance between the centers of the circumscribed circles of the triangles $APC_1$ and $CPA_1$ is minimal.

* Given $\vartriangle ABC$, divide it into the minimal number of parts so that after being flipped over these parts can constitute the same $\vartriangle ABC$.

For a positive integer $K$, define a sequence, $\{a_n\}$, as following: $a_1 = K$ and $a_{n+1} =a_n -1$ if $a_n$ is even $a_{n+1} =\frac{a_n - 1}{2}$ if $a_n$ is odd , for all $n \ge 1$. Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to $0$.

All the fields of a square $n\times n$ (n>2) table are filled with $+1$ or $-1$ according to the rules: [i]At the beginning $-1$ are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column. [/i] a) What is the minimal b) What is the maximal possible number of $+1$ in the obtained table?

We start with any finite list of distinct positive integers. We may replace any pair $n, n + 1$ (not necessarily adjacent in the list) by the single integer $n-2$, now allowing negatives and repeats in the list. We may also replace any pair $n, n + 4$ by $n - 1$. We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.

Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$. by Elton Bojaxhiu, Albania

Given regular pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $$r_1\le r_2\le r_3\le r_4\le r_5.$$ Find all the positions of the $M$, giving $r_3$ the minimal possible value. Find all the positions of the $M$, giving $r_3$ the maximal possible value.

Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$. Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.

If $x$ and $y$ are positive real numbers such that $(x + \sqrt{x^2 + 1})(y +\sqrt{y^2 + 1}) = 2018$: The minimum possible value of $x + y$ is A. $\frac{2017}{\sqrt{2018}}$ B. $\frac{2018}{\sqrt{2019}}$ C. $\frac{2017}{2\sqrt{2018}}$ D. $\frac{2019}{\sqrt{2018}}$ E. $\sqrt{3}$

Given a $\triangle ABC$, determine which is the circle with the smallest area that contains it.

Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$. She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$. Mary wants to prevent Pat from winning the game. Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.

What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?

Represent $1957$ as the sum of $12$ positive integer summands $a_1, a_2, ... , a_{12}$ for which the number $a_1! \cdot a_2! \cdot a_3! \cdot ... \cdot a_{12}!$ is minimal.

Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an $m x n$ rectangle ($m, n > 1$) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?

Let be a natural number $ n\ge 3. $ Find $$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$ where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained.

There are $21$ towns. Each airline runs direct flights between every pair of towns in a group of five. What is the minimum number of airlines needed to ensure that at least one airline runs direct flights between every pair of towns?

Consider a board consisting of $n\times n$ unit squares where $n \ge 2$. Two cells are called neighbors if they share a horizontal or vertical border. In the beginning, all cells together contain $k$ tokens. Each cell may contain one or several tokens or none. In each turn, choose one of the cells that contains at least one token for each of its neighbors and move one of those to each of its neighbors. The game ends if no such cell exists. (a) Find the minimal $k$ such that the game does not end for any starting configuration and choice of cells during the game. (b) Find the maximal $k$ such that the game ends for any starting configuration and choice of cells during the game. Proposed by Theresia Eisenkölbl

Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$. Find the locus of $H$ and $H'$. If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value. Show that this value does not depend on the position of $L$. Comment on the case $OO'^2 > R^2 + R'^2$.