Determine all triples $(x, y, z)$ of real numbers satisfying the following system of equations:
$$
\begin{aligned}
2^{\sqrt[3]{x^2}} \cdot 4^{\sqrt[3]{y^2}} \cdot 16^{\sqrt[3]{z^2}} &= 128 \\
(xy^2 + z^4)^2 &= 4 + (xy^2 - z^4)^2.
\end{aligned}
$$
Given a rectangular grid, split into $m \times n$ squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:
* all squares touching the border of the grid should be coloured black.
* No four squares forming a $2 \times 2$-square should be coloured in the same colour.
* No four squares forming a $2 \times 2$-square should be coloured in such a way that only the diagonally touching squares have the same colour.
For which grid sizes $m \times n$ (with $m, n \ge 3$) does there exist a valid colouring?
The side of a triangle is subdivided by the bisector of its opposite angle into two segments of lengths $1$ and $3$. Determine all possible values of the area of that triangle.
In Greifswald, there are three schools called $A$, $B$, and $C$, each of which is attended by at least one student. Among any three students $a$ from $A$, $b$ from $B$, and $c$ from $C$ there are two knowing each other and two others not knowing each other. Prove that either some student from $A$ knows all students from $B$, or some student from $B$ knows all students from $C$, or some student from $C$ knows all students from $A$.
Two persons play the following game with positive integers. The initial number is $2011^{2011}$. Each move consists of subtraction by an integer between $1$ and $2010$ inclusive, or division by $2011$, rounding down when necessary. The player who obtains a non-positive integer wins. Who will win this game: the first player or the second?
Given an integer number $n \ge 3$, determine the maximum value the product of $n$ non-negative real numbers $x_1, x_2, \dots, x_n$ may achieve, subject to
$$
\frac{x_1}{1+x_1} + \frac{x_2}{1+x_2} + \dots + \frac{x_n}{1+x_n} = 1.
$$
Let $p$ and $q$ be real numbers such that the quadratic equation
$$
x^2 + px + q = 0
$$
has two real solutions $x_1$ and $x_2$.
The following two conditions hold:
(i) The numbers $x_1$ and $x_2$ differ by 1.
(ii) The numbers $p$ and $q$ differ by 1.
Show that $p, q, x_1$ and $x_2$ are integers.
In a plane, we are given line $l$, two points $A$ and $B$ neither of which lies on line $l$, and the reflection $A_{1}$ of point $A$ across line $l$. Using only a straightedge, construct the reflection $B_{1}$ of point $B$ across line $l$. Prove that your construction works.
Note: "Using only a straightedge" means that you can perform only the following operations:
(a) Given two points, you can construct the line through them.
(b) Given two intersecting lines, you can construct their intersection point.
(c) You can select (mark) points in the plane that lie on or off objects already drawn in the plane. (The only facts you can use about these points are which lines they are on or not on.)
A sequence $\langle a_n \rangle$ of positive integers is given, such that $a_1 = 1$ and $a_{n+1}$ is the smallest positive integer such that
$$lcm(a_1, a_2, \dots, a_n, a_{n+1}) \gt lcm(a_1, a_2, \dots, a_n).$$
Which numbers are contained in the sequence?
A deck consists of $3n$ cards, $n$ each colored red, green and blue in denominations $1$ through $n$. We choose a subset $S$ of the denominations and deal all cards of the chosen denominations into three equal size hands to players designated red, green and blue in such a way that no player receives a card of her own color. Prove that the number of deals for which the denominations appearing in the red player's hand are $1, 2, \dots, k$ equals $\binom{n}{k}\binom{2k}{k}$. (So it doesn't depend on the size of $S$.)
In a group of $n$ people everyone has at least $k$ friends. Each day every member of the group shares with all his friends all the news that he received in the previous days. Suppose that if some information is revealed to any member of the group, then after some number of days all the members will eventually know the news. Prove that actually all the members know the news after at most $3n/k$ days.
Find all real $a$ such that there exists a function $f : \mathbb{R} \to \mathbb{R}$ satisfying the equality $f(\sin x) + a f(\cos x) = \cos 2x$ for all real $x$. (I. Voronovich)
The incircle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $G$ be a point on the incircle such that $FG$ is a diameter. The lines $EG$ and $FD$ intersect at $H$. Prove that $CH \parallel AB$.
Let $A_0, A_1, \dots, A_5$ be a circular labelling of six distinct points on a circle centred at the point $O$, and let $B_i$ be the midpoint of the segment $A_iA_{i+1}$, $i = 0, 1, \dots, 5$ (indices are reduced modulo $6$). Assume that no opposite sides of the hexagon $A_0A_1\cdots A_5$ are parallel. A line through the point $O$ meets again the circle $B_iOB_{i+3}$ at the point $C_i$, $i = 0, 1, 2$. Let $\ell_i$ be the tangent to the circle $B_iOB_{i+3}$ at the point $C_i$, $i = 0, 1, 2$. The lines $\ell_i$ and $\ell_j$ meet to produce the point $D_k$, where $\{i, j, k\} = \{0, 1, 2\}$. Show that the three circles $B_iOB_{i+3}$ and the circle $D_0D_1D_2$ share a point in the plane.
Is it possible to partition the set of positive integer numbers into two classes, none of which contains an infinite arithmetic sequence (with a positive ratio)?
What if we require the extra condition that, in each class $C$ of the partition, the set of differences
$$
\{ \min \{ n : n \in C \text{ and } n \gt m \} - m : m \in C \}
$$
be bounded?