An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

Let $n,k$, $1\le k\le n$ be fixed integers. Alice has $n$ cards in a row, where the card has position $i$ has the label $i+k$ (or $i+k-n$ if $i+k>n$). Alice starts by colouring each card either red or blue. Afterwards, she is allowed to make several moves, where each move consists of choosing two cards of different colours and swapping them. Find the minimum number of moves she has to make (given that she chooses the colouring optimally) to put the cards in order (i.e. card $i$ is at position $i$). NOTE: edited from original phrasing, which was ambiguous.

In a country with $n$ towns, all the direct flights are of double destinations (back and forth). There are $r>2014$ rootes between different pairs of towns, that include no more than one intermediate stop (direction of each root matters). Find the minimum possible value of $n$ and the minimum possible $r$ for that value of $n$.

A switch has two inputs $1, 2$ and two outputs $1, 2$. It either connects $1$ to $1$ and $2$ to $2$, or $1$ to $2$ and $2$ to 1. If you have three inputs $1, 2, 3$ and three outputs $1, 2, 3$, then you can use three switches, the first across $1$ and $2$, then the second across $2$ and $3$, and finally the third across $1$ and $2$. It is easy to check that this allows the output to be any permutation of the inputs and that at least three switches are required to achieve this. What is the minimum number of switches required for $4$ inputs, so that by suitably setting the switches the output can be any permutation of the inputs?

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 $ABC$ 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.

Find the smallest natural number $n$ for which $sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}$ .

A cinema has its seats arranged in $n$ rows $\times m$ columns. It sold mn tickets but sold some seats more than once. The usher managed to allocate seats so that every ticket holder was in the correct row or column. Show that he could have allocated seats so that every ticket holder was in the correct row or column and at least one person was in the correct seat. What is the maximum $k$ such that he could have always put every ticket holder in the correct row or column and at least $k$ people in the correct seat?

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

$ABC$ is a triangle with $\angle A = 90^o$. For a point $D$ on the side $BC$, the feet of the perpendiculars to $AB$ and $AC$ are $E$ and$ F$. For which point $D$ is $ EF$ a minimum?

For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.

Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.

The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.

A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?

Determine for which $m$ there exist exactly $2^{15}$ subsets $X$ of $\{1,2,...,47\}$ with the following property: $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m > 47$.

What is the smallest possible value of $x^2+2y^2-x-2y-xy$?

Let two positive integers $x, y$ satisfy the condition $44 /( x^2 + y^2)$. Determine the smallest value of $T = x^3 + y^3$.

Positive integers $x, y, z$ satisfy $xy + z = 160$. Compute the smallest possible value of $x + yz$.

A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N$ this will be possible, regardless of the number of tasks on Olympiad and the student successes?

Let $a$ and $b$ be two positive reals such that the following inequality \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$. [i]Proposed by Fajar Yuliawan[/i]

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

In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.

Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$

Convex quadrilateral $ABCD$ is given. Lines $BC$ and $AD$ intersect at point $O$, with $B$ lying on the segment $OC$, and $A$ on the segment $OD$. $I$ is the center of the circle inscribed in the $OAB$ triangle, $J$ is the center of the circle exscribed in the triangle $OCD$ touching the side of $CD$ and the extensions of the other two sides. The perpendicular from the midpoint of the segment $IJ$ on the lines $BC$ and $AD$ intersect the corresponding sides of the quadrilateral (not the extension) at points $X$ and $Y$. Prove that the segment $XY$ divides the perimeter of the quadrilateral$ABCD$ in half, and from all segments with this property and ends on $BC$ and $AD$, segment $XY$ has the smallest length.

The three sides and the area of a triangle are integers. What is the smallest value of the area of this triangle?