A cutting of an $8 \times 8$ chessboard into rectangles is called diversified if the following two conditions hold:
* Each rectangle contains the same number of black and white cells.
* No two rectangles have the same total number of cells.
Find the largest $n$ such that a diversified cutting into $n$ rectangles exists. (Ratko Višak)
Let $x$ and $y$ be positive real numbers such that
$$
2^{x^2} = 16^y \quad \text{and} \quad \log_{\sqrt{2017}} x + \log_{\sqrt{2017}} y \gt 0.
$$
Prove that $y \gt \frac{1}{2}$. (Kristina Ana Škreb)
A number written only with digits $2$ and $3$ is called *happy*. Therefore, the happy numbers are $2$, $3$, $22$, $23$, $32$, $33$, $222$, $223$, $232$, $233$, $322$, $323$, $332$, $333$, ... Determine the $2050$th happy number.
Let $\triangle ABC$ be an acute-angled triangle with $\angle BAC = 75^\circ$. Let $P$ be the midpoint of the side $\overline{BC}$, and let $M$ and $N$ be the feet of the altitudes from vertices $B$ and $C$ respectively. Determine the angle $\angle MPN$.
Let $\triangle ABC$ be an acute-angled triangle and let $BCD$, $ACE$ and $ABF$ be equilateral triangles drawn on the outside of triangle $ABC$. Let $M$ be the midpoint of the segment $\overline{BD}$, and let $O$ be the centre of the triangle $ACE$. Prove that $|AM| : |OF| = \frac{\sqrt{3}}{2}$.
Twenty points with integer coordinates are given in the coordinate plane so that no three of them are collinear. Prove that there exists a triangle with vertices chosen among the given points whose centroid is also a point with integer coordinates.
Sequences $(x_n)$ and $(y_n)$ are given recursively:
$x_1 = 3$, $y_1 = 1$,
$x_{n+1} = 3x_n + y_n$, for all $n \in \mathbb{N}$;
$y_{n+1} = x_n + 3y_n$, for all $n \in \mathbb{N}$.
Prove that $x_{2017}^2 - y_{2017}^2 = 8^{2017}$.
In the rectangle with vertices $A(-1, 0)$, $B(1, 0)$, $C(1, 1)$, $D(-1, 1)$ a point $T$ is randomly chosen. Find the probability that $T$ is closer to $A$ than to the point $P(2, 1)$.
Find the number of nine-digit numbers with digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ and $9$, such that no three consecutive digits equal $123$, $246$ or $678$.
Let $z_1$ and $z_2$ be complex numbers such that $|z_1| = |z_2| = 1$, and let $a$ and $b$ be real numbers such that $a + b = 1$. Prove
$$
|a z_1 + b z_2| \ge \frac{1}{2} |z_1 + z_2|.
$$
Let $A$, $B$ and $C$ be points on the curve (hyperbola) with equation $xy = 1$. Prove that the orthocentre of the triangle $ABC$ also belongs to that curve.
Gargamel caught $N$ Smurfs and distributed them into three bags. After moving Papa Smurf from the first bag to the second, Grouchy Smurf from the second bag to the third, and Smurfette from the third bag to the first one, the average Smurfs' height in the first bag decreased by 8 millimetres, while the average Smurfs' heights in the second and the third bag increased by 5 and 8 millimetres, respectively.
If there were nine Smurfs in the first bag, determine $N$. (Stipe Vidak)
Points $M$ and $N$ lie on the sides $\overline{BC}$ and $\overline{CD}$ (respectively) of the square $ABCD$ so that $\angle BMA = \angle NMC = 60^\circ$. Determine $\angle MAN$. (Ukraine 2013)
Karlo and Lovro play the following game. Karlo cuts the paper of dimensions $9 \times 9$ into rectangles of integer dimensions having at least one side of length $1$. Lovro then chooses a positive integer $k \in \{1, \dots, 9\}$, after which Karlo gives him as many coins as the total area of $1 \times k$ and $k \times 1$ cut rectangles is. Lovro chooses $k$ so that he gets as many coins from Karlo as possible, while Karlo wants to give as few coins to Lovro as possible.
Find the minimum possible number of coins that Karlo gives to Lovro. (Ukraine 2013)