(GBR 3) IMO5. Find, with proof, all functions $f$ defined on the nonnegative real numbers and taking nonnegative real values such that
(i) $f[xf(y)]f(y)=f(x+y)$,
(ii) $f(2)=0$ but $f(x)\ne0$ for $0\le x\lt 2$.
(SWE 2) Let $f(x)=x^n$ where $n$ is a fixed positive integer and $x=1,2,\ldots$. Is the decimal expansion $a=0.f(1)f(2)f(3)\ldots$ rational for any value of $n$?
The decimal expansion of $a$ is defined as follows: If $f(x)=d_1(x)d_2(x)\ldots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a=0.1d_1(2)d_2(2)\ldots d_{r(2)}(2)d_1(3)\ldots d_{r(3)}(3)d_1(4)\ldots$.
(USA 3) Let $A$, $B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is 86 meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y$.
(CZS 3) Let $n$ be a positive integer and let $p$ be a prime number, $p>3$. Find at least $3(n+1)$ [easier version: $2(n+1)$] sequences of positive integers $x,y,z$ satisfying
$$xyz=p^n(x+y+z)$$
that do not differ only by permutation.
(FRG 1) IMO1. The set $S=\{2,5,13\}$ has the property that for every $a,b\in S$, $a\ne b$, the number $ab-1$ is a perfect square. Show that for every positive integer $d$ not in $S$, the set $S\cup\{d\}$ does not have the above property.
(FRA 5) Let real numbers $x_1,x_2,\ldots,x_n$ satisfy $0\lt x_1\lt x_2\lt \cdots\lt x_n\lt 1$ and set $x_0=0$, $x_{n+1}=1$. Suppose that these numbers satisfy the following system of equations:
$$\sum_{\substack{j=0\\j\ne i}}^{n+1}\frac1{x_i-x_j}=0\quad(i=1,2,\ldots,n).\tag{1}$$
Prove that $x_{n+1-i}=1-x_i$ for $i=1,2,\ldots,n$.
(USA 1) From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1,\ldots,q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that (i) each pair of consecutive teams on the list have one member in common and (ii) the chain of teams on the list are in rank order.
Alternative formulation. Given a graph with $n$ vertices and $q$ edges numbered $1,\ldots,q$, show that there exists a chain of $m$ edges, $m\ge2q/n$, each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.
(GDR 1) IMO6. Prove or disprove: Given a finite set of points with integer coordinates in the plane, it is possible to color some of these points red and the remaining ones white in such a way that for any straight line $L$ parallel to one of the coordinate axes, the number of red colored points and the number of white colored points on $L$ differ by at most 1.
(HUN 2) Three persons $A,B,C$ are playing the following game: A $k$-element subset of the set $\{1,\ldots,1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to 1986. The winner is $A$, $B$ or $C$, respectively, if the sum of the chosen numbers leaves a remainder of 0, 1, or 2 when divided by 3. For what values of $k$ is this game a fair one? (A game is fair if the three outcomes are equally probable.)
(BUL 1) Let $f(n)$ be the least number of distinct points in the plane such that for each $k=1,2,\ldots,n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n)$.
Simplified version. Show that $f(n)=\left[\frac{n+1}2\right]\left[\frac{n+2}2\right]$ ($[x]$ denoting the greatest integer not exceeding $x$).
(GDR 3) IMO3. To each vertex $P_i$ ($i=1,\ldots,5$) of a pentagon an integer $x_i$ is assigned, the sum $s=\sum x_i$ being positive. The following operation is allowed, provided at least one of the $x_i$'s is negative: Choose a negative $x_i$, replace it by $-x_i$, and add the former value of $x_i$ to the integers assigned to the two neighboring vertices of $P_i$ (the remaining two integers are left unchanged).
This operation is to be performed repeatedly until all negative integers disappear. Decide whether this procedure must eventually terminate.
(FRG 3) A particle moves from $(0,0)$ to $(n,n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n,y)$, $y\lt n$, it stays there if a head comes up and at $(x,n)$, $x\lt n$, it stays there if a tail comes up. Let $k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n,n)$.
(IRE 1) The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E,F$, respectively, and $X,Y,Z$ are the midpoints of $EF,FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.
(NET 2) Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B',C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T(ABCD)=A'B'C'D'$. Let us define $A''B''C''D''=T(A'B'C'D')=T(T(ABCD))$.
(a) Prove that $ABCD$ and $A''B''C''D''$ are similar.
(b) The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.
(ISR 1) IMO4. Let $A,B$ be adjacent vertices of a regular $n$-gon in the plane and let $O$ be its center. Now let the triangle $ABO$ glide around the polygon in such a way that the points $A$ and $B$ move along the whole circumference of the polygon. Describe the figure traced by the vertex $O$.
(CHN 3) IMO2. Let $A,B,C$ be fixed points in the plane. A man starts from a certain point $P_0$ and walks directly to $A$. At $A$ he turns his direction by $60^\circ$ to the left and walks to $P_1$ such that $P_0A=AP_1$. After he does the same action 1986 times successively around the points $A,B,C,A,B,C,\ldots$, he returns to the starting point. Prove that $\triangle ABC$ is equilateral and that the vertices $A,B,C$ are arranged counterclockwise.
(TUR 1) Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DYAZ,DZBX,DXCY$ are circumscribable, so is the third.
(BUL 2) A tetrahedron $ABCD$ is given such that $AD=BC=a$; $AC=BD=b$; $AB\cdot CD=c^2$. Let $f(P)=AP+BP+CP+DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P)$.