Let $n$ be a natural number. Solve the equation $$||\cdots|||x-1|-2|-3|-\ldots-(n-1)|-n|=0.$$

Given are real numbers $a<b<c<d$. Determine all permutations $p,q,r,s$ of the numbers $a,b,c,d$ for which the value of the sum $$(p-q)^2+(q-r)^2+(r-s)^2+(s-p)^2$$is minimal.

A chord divides the interior of a circle $k$ into two parts. Variable circles $k_1$ and $k_2$ are inscribed in these two parts, touching the chord at the same point. Show that the ratio of the radii of circles $k_1$ and $k_2$ is constant, i.e. independent of the tangency point with the chord.

An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)

In a regular hexagon $ABCDEF$ with center $O$, points $M$ and $N$ are the midpoints of the sides $CD$ and $DE$, and $L$ the intersection point of $AM$ and $BN$. Prove that: (a) $ABL$ and $DMLN$ have equal areas; (b) $\angle ALD=\angle OLN=60^\circ$; (c) $\angle OLD=90^\circ$.

For any different positive numbers $a,b,c$ prove the inequality $$a^ab^bc^c>a^bb^cc^a.$$

The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$.

In the plane are given $1997$ points. Show that among the pairwise distances between these points, there are at least $32$ different values.

Integers $x,y,z$ and $a,b,c$ satisfy $$x^2+y^2=a^2,\enspace y^2+z^2=b^2\enspace z^2+x^2=c^2.$$Prove that the product $xyz$ is divisible by (a) $5$, and (b) $55$.

Prove that for every real number $x$ and positive integer $n$ $$|\cos x|+|\cos2x|+|\cos2^2x|+\ldots+|\cos2^nx|\ge\frac n{2\sqrt2}.$$

The areas of the faces $ABD,ACD,BCD,BCA$ of a tetrahedron $ABCD$ are $S_1,S_2,Q_1,Q_2$, respectively. The angle between the faces $ABD$ and $ACD$ equals $\alpha$, and the angle between $BCD$ and $BCA$ is $\beta$. Prove that $$S_1^2+S_2^2-2S_1S_2\cos\alpha=Q_1^2+Q_2^2-2Q_1Q_2\cos\beta.$$

On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.

Find the last four digits of each of the numbers $3^{1000}$ and $3^{1997}$.

Consider a circle $k$ and a point $K$ in the plane. For any two distinct points $P$ and $Q$ on $k$, denote by $k'$ the circle through $P,Q$ and $K$. The tangent to $k'$ at $K$ meets the line $PQ$ at point $M$. Describe the locus of the points $M$ when $P$ and $Q$ assume all possible positions.

Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and $$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$(a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$. (b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$. (c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.

Let $k$ be a natural number. Determine the number of non-congruent triangles with the vertices at vertices of a given regular $6k$-gon.