(BUL 1) The set $M=\{1,2,\ldots,2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2,\ldots,M_k$, where $n\ge k^3+k$. Prove that there exist even numbers $2j_1,2j_2,\ldots,2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ ($1\le i\le k$) such that the numbers $2j_1-1,2j_2-1,\ldots,2j_{k+1}-1$ are also in one and the same subset $M_j$ ($1\le j\le k$).
(BUL 4) Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A'\ne S$ and $B'\ne S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.
(CUB 3; IMO 1) Let $n\gt m\ge1$ be natural numbers such that the groups of the last three digits in the decimal representation of $1978^m,1978^n$ coincide. Find the ordered pair $(m,n)$ of such $m,n$ for which $m+n$ is minimal.
(CZS 2) Let $T_1$ be a triangle having $a,b,c$ as lengths of its sides and let $T_2$ be another triangle having $u,v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that
$$16PQ\le a^2(-u^2+v^2+w^2)+b^2(u^2-v^2+w^2)+c^2(u^2+v^2-w^2).$$
When does equality hold?
(GDR 2) For every integer $d\ge1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A=M_1$, $B=M_2\setminus\{2\}$, $C=M_3$. Prove that every $c\in C$ may be written in a unique way as $c=ab$ with $a\in A,b\in B$.
(FRA 2; IMO 5) Let $\varphi:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ be injective. Prove that for all $n$,
$$\sum_{k=1}^n\frac{\varphi(k)}{k^2}\ge\sum_{k=1}^n\frac1k.$$
(FRA 5) We consider three distinct half-lines $Ox,Oy,Oz$ in a plane. Prove the existence and uniqueness of three points $A\in Ox,B\in Oy,C\in Oz$ such that the perimeters of the triangles $OAB,OBC,OCA$ are all equal to a given number $2p\gt 0$.
(GBR 4) Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?
(GBR 5; IMO 3) Let $\{f(n)\}$ be a strictly increasing sequence of positive integers: $0\lt f(1)\lt f(2)\lt f(3)\lt \cdots$. Of the positive integers not belonging to the sequence, the $n$th in order of magnitude is $f(f(n))+1$. Determine $f(240)$.
(NET 1; IMO 6) An international society has its members in $6$ different countries. The list of members contains $1978$ names, numbered $1,2,\ldots,1978$. Prove that there is at least one member whose number is the sum of the numbers of two, not necessarily distinct, of his compatriots.
(SWE 2) A function $f:I\to\mathbb R$, defined on an interval $I$, is called concave if $f(\theta x+(1-\theta)y)\ge\theta f(x)+(1-\theta)f(y)$ for all $x,y\in I$ and $0\le\theta\le1$. Assume that the functions $f_1,\ldots,f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2\cdots f_n)^{1/n}$ is concave.
(USA 1; IMO 4) In a triangle $ABC$ we have $AB=AC$. A circle is tangent internally to the circumcircle of $ABC$ and also to the sides $AB,AC$, at $P,Q$ respectively. Prove that the midpoint of $PQ$ is the center of the incircle of $ABC$.
(USA 6; IMO 2) Given any point $P$ in the interior of a sphere with radius $R$, three mutually perpendicular segments $PA,PB,PC$ are drawn terminating on the sphere and having one common vertex in $P$. Consider the rectangular parallelepiped of which $PA,PB,PC$ are coterminal edges. Find the locus of the point $Q$ that is diagonally opposite $P$ in the parallelepiped when $P$ and the sphere are fixed.
(VIE 2) Prove that it is possible to place $2n(2n+1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1\times2\times(n+1)$ in a cubic box with edge $2n+1$ if and only if $n$ is even or $n=1$.
Remark. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.
(YUG 1) Let $p$ be a prime and $A=\{a_1,\ldots,a_{p-1}\}$ an arbitrary subset of the natural numbers such that none of its elements is divisible by $p$. Define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to $P=\{0,1,\ldots,p-1\}$ as follows:
(i) if $B=\{a_{i_1},\ldots,a_{i_k}\}\subset A$ and $\sum_{j=1}^k a_{i_j}\equiv n\pmod p$, then $f(B)=n$;
(ii) $f(\varnothing)=0$, $\varnothing$ being the empty set.
Prove that for each $n\in P$ there is a subset $B\subset A$ with $f(B)=n$.
(YUG 2) Determine all triples $(a,b,c)$ of positive real numbers such that the system
$$ax+by-cz=0,$$
$$a\sqrt{1-x^2}+b\sqrt{1-y^2}-c\sqrt{1-z^2}=0$$
is compatible in the real numbers, and then find all its real solutions.
(FRA 3) Prove that for any positive integers $x,y,z$ with $xy-z^2=1$ one can find nonnegative integers $a,b,c,d$ such that $x=a^2+b^2$, $y=c^2+d^2$, $z=ac+bd$.
Set $z=(2q)!$ to deduce that for every prime $p=4q+1$, the number $p$ can be represented as the sum of the squares of two integers.