A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node $A$ to the node $B$ along the shortest possible trajectory. Prove that the half of his way he moved in one direction.

In the diagram below $a, b, c, d, e, f, g, h, i, j$ are distinct positive integers and each (except $a, e, h$ and $j$) is the sum of the two numbers to the left and above. For example, $b = a + e, f = e + h, i = h + j$. What is the smallest possible value of $d$? j h i e f g a b c d

Let $f(x) = \sin \frac{x}{3}+ \cos \frac{3x}{10}$ for all real $x$. Find the least natural number $n$ such that $f(n\pi + x)= f(x)$ for all real $x$.

[b]a)[/b] Show that the expression $ x^3-5x^2+8x-4 $ is nonegative, for every $ x\in [1,\infty ) . $ [b]b)[/b] Determine $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $

Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$ for any positive integer $ n $ not less than 2 . Find the minimum of $m.$

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?

Let $m,n$ be positive integers with $1 \le n < m$. A box is locked with several padlocks which must all be opened to open the box, and which all have different keys. The keys are distributed among $m$ people. Suppose that among these people, no $n$ can open the box, but any $n+1$ can open it. Find the smallest possible number $l$ of locks and then the total number of keys for which this is possible.

One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.

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 $n$.

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.

The figure below is cut along the lines into polygons (which need not be convex). No polygon contains a $2 \times 2$ square. What is the smallest possible number of polygons? [missing figure]

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.

Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.

a) Find the minimal value of the polynomial $$P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$$ b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.

The smallest value of the function $f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|$ where $x \in [-1, 1] $ is: (A): $\frac{1}{2012}$, (B): $\frac{1}{2013}$, (C): $\frac{1}{2014}$, (D): $\frac{1}{2015}$, (E): None of the above.

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