(AUS 6) Let $f$ be a function that satisfies the following conditions:
(i) If $x\gt y$ and $f(y)-y\ge v\ge f(x)-x$, then $f(z)=v+z$, for some number $z$ between $x$ and $y$.
(ii) The equation $f(x)=0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions;
(iii) $f(0)=1$.
(iv) $f(1987)\le1988$.
(v) $f(x)f(y)=f(xf(y)+yf(x)-xy)$.
Find $f(1987)$.
(USA 3) At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_1,C_2,\ldots,C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n\ge4$, then $k\ge2n$.
(FIN 3) Does there exist a second-degree polynomial $p(x,y)$ in two variables such that every nonnegative integer $n$ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of nonnegative integers?
(FRA 5) Let $ABCDEFGH$ be a parallelepiped with $AE\parallel BF\parallel CG\parallel DH$. Prove the inequality
$$AF+AH+AC\le AB+AD+AE+AG.$$
In what cases does equality hold?
(GBR 1) Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively.
(GRE 4) Show that if $a,b,c$ are the lengths of the sides of a triangle and if $2S=a+b+c$, then
$$\frac{a^n}{b+c}+\frac{b^n}{c+a}+\frac{c^n}{a+b}\ge\left(\frac23\right)^{n-2}S^{n-1},\qquad n\ge1.$$
(NET 1) Given five real numbers $u_0,u_1,u_2,u_3,u_4$, prove that it is always possible to find five real numbers $v_0,v_1,v_2,v_3,v_4$ that satisfy the following conditions:
(i) $u_i-v_i\in\mathbb N$.
(ii) $\sum_{0\le i\lt j\le4}(v_i-v_j)^2\lt 4$.
(HUN 1) (a) Let $(m,k)=1$. Prove that there exist integers $a_1,a_2,\ldots,a_m$ and $b_1,b_2,\ldots,b_k$ such that each product $a_ib_j$ ($i=1,2,\ldots,m$; $j=1,2,\ldots,k$) gives a different residue when divided by $mk$.
(b) Let $(m,k)\gt 1$. Prove that for any integers $a_1,a_2,\ldots,a_m$ and $b_1,b_2,\ldots,b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i,j)\ne(s,t)$) that give the same residue when divided by $mk$.
(HUN 2) Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M\cap\lambda$ is finite and nonempty?
(ICE 3) Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?
(POL 1) Find the number of partitions of the set $\{1,2,\ldots,n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied:
(i) After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
(ii) If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even.
(POL 5) Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B',C'$; $A',B,C'$; and $A',B',C$ are collinear.
(GDR 2) (IMO5) Is it possible to put $1987$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a nondegenerate triangle with rational area?
(FRG 2) (IMO3) Suppose $x_1,x_2,\ldots,x_n$ are real numbers with $x_1^2+x_2^2+\cdots+x_n^2=1$. Prove that for any integer $k\gt 1$ there are integers $e_i$ not all $0$ and with $|e_i|\lt k$ such that
$$|e_1x_1+e_2x_2+\cdots+e_nx_n|\le\frac{(k-1)\sqrt n}{k^n-1}.$$
(FRG 3) (IMO1) Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k)$. Prove:
(a) $\sum_{k=0}^n kp_n(k)=n!$;
(b) $\sum_{k=0}^n(k-1)^2p_n(k)=n!$.
(ROM 1) Prove that there exists a four-coloring of the set $M=\{1,2,\ldots,1987\}$ such that any arithmetic progression with $10$ terms in the set $M$ is not monochromatic.
Alternative formulation. Let $M=\{1,2,\ldots,1987\}$. Prove that there is a function $f:M\to\{1,2,3,4\}$ that is not constant on every set of $10$ terms from $M$ that form an arithmetic progression.
(ROM 4) For any integer $r\ge1$, determine the smallest integer $h(r)\ge1$ such that for any partition of the set $\{1,2,\ldots,h(r)\}$ into $r$ classes, there are integers $a\ge0$, $1\le x\le y$, such that $a+x,a+y,a+x+y$ belong to the same class.
(USS 2) Let $\alpha,\beta,\gamma$ be positive real numbers such that $\alpha+\beta+\gamma\lt \pi$, $\alpha+\beta\gt \gamma$, $\beta+\gamma\gt \alpha$, $\gamma+\alpha\gt \beta$. Prove that with the segments of lengths $\sin\alpha,\sin\beta,\sin\gamma$ we can construct a triangle and that its area is not greater than
$$\frac18(\sin2\alpha+\sin2\beta+\sin2\gamma).$$
(USS 3) (IMO6) Let $f(x)=x^2+x+p$, $p\in\mathbb N$. Prove that if the numbers $f(0),f(1),\ldots,f([\sqrt{p/3}])$ are primes, then all the numbers $f(0),f(1),\ldots,f(p-2)$ are primes.