Let $k$ be a positive integer. Prove that: (a) If $k=m+2mn+n$ for some positive integers $m,n$, then $2k+1$ is composite. (b) If $2k+1$ is composite, then there exist positive integers $m,n$ such that $k=m+2mn+n$.

Let $a$ be an integer and $p$ a prime number that divides both $5a-1$ and $a-10$. Show that $p$ also divides $a-3$.

Let $MN$ be a chord of a circle with diameter $AB$, and let $A'$ and $B'$ be the orthogonal projections of $A$ and $B$ onto $MN$. Prove that $MA'=B'N$.

Janez wants to make an $m\times n$ grid (consisting of unit squares) using equal elements of the form $\llcorner$, where each leg of an element has the unit length. No two elements can overlap. For which values of $m$ and $n$ can Janez do the task?

Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.

Points $M,N,P,Q$ are taken on the sides $AB,BC,CD,DA$ respectively of a square $ABCD$ such that $AM=BN=CP=DQ=\frac1nAB$. Find the ratio of the area of the square determined by the lines $MN,NP,PQ,QM$ to the ratio of $ABCD$.

Let $C$ and $D$ be different points on the semicircle with diameter $AB$. The lines $AC$ and $BD$ intersect at $E$, and the lines $AD$ and $BC$ intersect at $F$. Prove that the midpoints $X,Y,Z$ of the segments $AB,CD,EF$ respectively are collinear.

Prove that among any $1001$ numbers taken from the numbers $1,2,\ldots,1997$ there exist two with the difference $4$.

Suppose that $m,n$ are integers greater than $1$ such that $m+n-1$ divides $m^2+n^2-1$. Prove that $m+n-1$ cannot be a prime number.

Determine all positive integers $n$ for which there exists a polynomial $p(x)$ of degree $n$ with integer coefficients such that it takes the value $n$ in $n$ distinct integer points and takes the value $0$ at point $0$.

In a convex quadrilateral $ABCD$ we have $\angle ADB=\angle ACD$ and $AC=CD=DB$. If the diagonals $AC$ and $BD$ intersect at $X$, prove that $\frac{CX}{BX}-\frac{AX}{DX}=1$.

In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims: (i) Less than $12$ employees have a more difficult work; (ii) At least $30$ employees take a higher salary. Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.

Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

The Fibonacci sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\in\mathbb N$. (a) Show that $f_{1005}$ is divisible by $10$. (b) Show that $f_{1005}$ is not divisible by $100$.

Two disjoint circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively lie on the same side of a line $p$ and touch the line at $A_1$ and $A_2$ respectively. The segment $O_1O_2$ intersects $k_1$ at $B_1$ and $k_2$ at $B_2$. Prove that $A_1B_1\perp A_2B_2$.

The expression $*3^5*3^4*3^3*3^2*3*1$ is given. Ana and Branka alternately change the signs $*$ to $+$ or $-$ (one time each turn). Can Branka, who plays second, do this so as to obtain an expression whose value is divisible by $7$?