Find all positive integers n with the following property: For every positive divisor $d$ of $n$, $d+1$ divides $n+1$.

Let $ABC$ be an acute triangle. Circle $k$ with diameter $AB$ intersects $AC$ and $BC$ again at $M$ and $N$ respectively. The tangents to $k$ at $M$ and $N$ meet at point $P$. Given that $CP=MN$, determine $\angle ACB$.

If $x,y,z$ are nonnegative numbers with $x+y+z=3$, prove that $$\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.$$

There are $c$ red, $p$ blue, and $b$ white balls on a table. Two players $A$ and $B$ play a game by alternately making moves. In every move, a player takes two or three balls from the table. Player $A$ begins. A player wins if after his/her move at least one of the three colors no longer exists among the balls remaining on the table. For which values of $c,p,b$ does player $A$ have a winning strategy?

Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.

Every square of a $3\times3$ board is assigned a sign $+$ or $-$. In every move, one square is selected and the signs are changed in the selected square and all the neighboring squares (two squares are neighboring if they have a common side). Is it true that, no matter how the signs were initially distributed, one can obtain a table in which all signs are $-$ after finitely many moves?

In a triangle $ABC$, $D$ is the orthogonal projection of the incenter $I$ onto $BC$. Line $DI$ meets the incircle again at $E$. Line $AE$ intersects side $BC$ at point $F$. Suppose that the segment IO is parallel to $BC$, where $O$ is the circumcenter of $\triangle ABC$. If $R$ is the circumradius and $r$ the inradius of the triangle, prove that $EF=2(R-2r)$.

Inside a circle $k$ of radius $R$ some round spots are made. The area of each spot is $1$. Every radius of circle $k$, as well as every circle concentric with $k$, meets in no more than one spot. Prove that the total area of all the spots is less than $$\pi\sqrt R+\frac12R\sqrt R.$$

If $x,y,z$ are positive numbers, prove that $$\frac x{\sqrt{y+z}}+\frac y{\sqrt{z+x}}+\frac z{\sqrt{x+y}}\ge\sqrt{\frac32(x+y+z)}.$$

Suppose that in a convex hexagon, each of the three lines connecting the midpoints of two opposite sides divides the hexagon into two parts of equal area. Prove that these three lines intersect in a point.

Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and $$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.

On each cell of a $2005\times2005$ chessboard, there is a marker. In each move, we are allowed to remove a marker that is a neighbor to an even number of markers (but at least one). Two markers are considered neighboring if their cells share a vertex. (a) Find the least number $n$ of markers that we can end up with on the chessboard. (b) If we end up with this minimum number $n$ of markers, prove that no two of them will be neighboring.