Equilateral triangles $ABK,BCL,CDM,DAN$ are constructed inside the square $ABCD$. Prove that the midpoints of the four segments $KL,LM,MN,NK$ and the midpoints of the eight segments $AK,BK,BL,CL,CM,DM,DN,AN$ are the twelve vertices of a regular dodecagon.
In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
Let $n$ be a given integer greater than $2$, and let $V_n$ be the set of integers $1+kn$, where $k=1,2,\ldots$. A number $m\in V_n$ is called indecomposable in $V_n$ if there do not exist numbers $p,q\in V_n$ such that $pq=m$. Prove that there exists a number $r\in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions that differ only in order of the elements of $V_n$ will be considered the same.)
Let $a,b,A,B$ be given constant real numbers and
$$f(x)=1-a\cos x-b\sin x-A\cos2x-B\sin2x.$$
Prove that if $f(x)\ge0$ for all real $x$, then
$$a^2+b^2\le2\qquad A^2+B^2\le1.$$
Let $a$ and $b$ be natural numbers and let $q$ and $r$ be the quotient and remainder respectively when $a^2+b^2$ is divided by $a+b$. Determine the numbers $a$ and $b$ if $q^2+r=1977$.
Let $f:\mathbb N\to\mathbb N$ be a function that satisfies the inequality $f(n+1)>f(f(n))$ for all $n\in\mathbb N$. Prove that $f(n)=n$ for all natural numbers $n$.