Positive integers $a$, $b$ an $n$ satisfy \[ \frac{a}{b}=\frac{a^2+n^2}{b^2+n^2}. \]Prove that $\sqrt{ab}$ is an integer.

Point $M$ is the midpoint of the hypotenuse $AB$ of a right angled triangle $ABC$. Points $P$ and $Q$ lie on segments $AM$ and $MB$ respectively and $PQ=CQ$. Prove that $AP\leq 2\cdot MQ$.

In a badminton tournament there were 16 participants. Each pair of participants played at most one game and there were no draws. After the tournament it turned out that each participant has won a different number of games. Prove that each participant has lost a different number of games.

On side $AB$ of a scalene triangle $ABC$ there are points $M$, $N$ such that $AN=AC$ and $BM=BC$. The line parallel to $BC$ through $M$ and the line parallel to $AC$ through $N$ intersect at $S$. Prove that $\measuredangle{CSM} = \measuredangle{CSN}$.

The numbers $a, b$ satisfy the condition $2a + a^2= 2b + b^2$. Prove that if $a$ is an integer, $b$ is also an integer.

Given is the square $ABCD$. Point $E$ lies on the diagonal $AC$, where $AE> EC$. On the side $AB$, a different point from $B$ has been selected for which $EF = DE$. Prove that $\angle DEF = 90^o$.

Given are positive integers $a, b$ for which $5a + 3b$ is divisible by $a + b$. Prove that $a = b$.

Points $K$ and $L$ are on the sides $BC$ and $CD$, respectively of the parallelogram $ABCD$, such that $AB + BK = AD + DL$. Prove that the bisector of angle $BAD$ is perpendicular to the line $KL$.

Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.

A triangle $ABC$ is given with $AC = BC = 5$. The altitude of this triangle drawn from vertex $A$ has length $4$. Calculate the length of the altitude of $ABC$ drawn from vertex $C$.

The numbers $a, b, c$ satisfy the condition $| a - b | = 2 | b - c | = 3 | c - a |$. Prove that $a = b = c$.

A convex quadrilateral $ABCD$ is given where $\angle DAB =\angle ABC = 120^o$ and $CD = 3$,$BC = 2$, $AB = 1$. Calculate the length of segment $AD$.