The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of $9: 1$ and two triangles of the starting triangle with a difference of areas of $48$ cm$^2$. Find the original triangle sidelengths.

Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.

Rein solved a test on mathematics that consisted of questions on algebra, geometry and logic. After checking the results, it occurred that Rein had answered correctly $50\%$ of questions on algebra, $70\%$ of questions on geometry and $80\%$ of questions on logic. Thereby, Rein had answered correctly altogether $62\%$ of questions on algebra and logic, and altogether $74\%$ of questions on geometry and logic. What was the percentage of correctly answered questions throughout all the test by Rein?

Find all pairs of real numbers $(x, y)$ that satisfy the equation $(x + y)^2 = (x + 3) (y - 3)$.

How many positive integers less than $10,000$ have an even number of even digits and an odd number of odd digits ? (Assume no number starts with zero.)

Seven brothers bought a round pizza and cut it $12$ piece as shown in the figure. Of the six elder brothers, each took one piece of the shape of an equilateral triangle, the remaining $6$ edge pieces by the older brothers did not want, was given to the youngest brother. Did the youngest brother get it more or less a seal than his every older brother?

Let $a, b$ and $c$ be real numbers such that $\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1$. Prove that $\frac{a^2}{b + c}+\frac{b^2}{c + a}+\frac{c^2}{a + b}= 0$.

How many such four-digit natural numbers divisible by $7$ exist such when changing the first and last number we also get a four-digit divisible by $7$?

Represent the number $\sqrt[3]{1342\sqrt{167}+2005}$ in the form where it contains only addition, subtraction, multiplication, division and square roots.

A $5\times 5$ board is covered by eight hooks (a three unit square figure, shown in the picture) so that one unit square remains free. Determine all squares of the board that can remain free after such covering.

Real numbers $x$ and $y$ satisfy the system of equalities $$\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases}$$Prove that $\cos 2x = \cos 2y$.

Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.

A post service of some country uses carriers to transport the mail, each carrier’s task is to bring the mail from one city to a neighbouring city. It is known that it is possible to send mail from any city to the capital $P$ . For any two cities $A$ and $B$, call $B$ more important than $A$, if every possible route of mail from $A$ to the capital $P$ goes through $B$. a) Prove that, for any three different cities $A, B$, and $C$, if $B$ is more important than $A$ and $C$ is more important than $B$, then $C$ is more important than $A$. b) Prove that, for any three different cities $A, B$, and $C$, if both B and C are more important than $A$, then either $C$ is more important than $B$ or $B$ is more important than $C$.

In a fixed plane, consider a convex quadrilateral $ABCD$. Choose a point $O$ in the plane and let $K, L, M$, and $N$ be the circumcentres of triangles $AOB, BOC, COD$, and $DOA$, respectively. Prove that there exists exactly one point $O$ in the plane such that $KLMN$ is a parallelogram.

Does there exist an integer $n > 1$ such that $2^{2^n-1} -7$ is not a perfect square?

Punches in the buses of a certain bus company always cut exactly six holes into the ticket. The possible locations of the holes form a $3 \times 3$ table as shown in the figure. Mr. Freerider wants to put together a collection of tickets such that, for any combination of punch holes, he would have a ticket with the same combination in his collection. The ticket can be viewed both from the front and from the back. Find the smallest number of tickets in such a collection.

Consider a convex $n$-gon in the plane with $n$ being odd. Prove that if one may find a point in the plane from which all the sides of the $n$-gon are viewed at equal angles, then this point is unique. (We say that segment $AB$ is viewed at angle $\gamma$ from point $O$ iff $\angle AOB =\gamma$ .)

A string having a small loop in one end is set over a horizontal pipe so that the ends hang loosely. After that, the other end is put through the loop, pulled as far as possible from the pipe and fixed in that position whereby this end of the string is farther from the pipe than the loop. Let $\alpha$ be the angle by which the string turns at the point where it passes through the loop (see picture). Find $\alpha$.

A sequence of natural numbers $a_1, a_2, a_3,..$ is called periodic modulo $n$ if there exists a positive integer $k$ such that, for any positive integer $i$, the terms $a_i$ and $a_{i+k}$ are equal modulo $n$. Does there exist a strictly increasing sequence of natural numbers that a) is not periodic modulo finitely many positive integers and is periodic modulo all the other positive integers? b) is not periodic modulo infinitely many positive integers and is periodic modulo infinitely many positive integers?

A crymble is a solid consisting of four white and one black unit cubes as shown in the picture. Find the side length of the smallest cube that can be exactly filled up with crymbles.