A1 (GBR 3), IMO 1. The function $f(n)$ is defined for all positive integers $n$ and takes on nonnegative integer values. Also, for all $m,n$,
$f(m+n)-f(m)-f(n)=0$ or $1$;
$f(2)=0$, $f(3)>0$, and $f(9999)=3333$.
Determine $f(1982)$.
A2 (YUG 1). Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that
$$\operatorname{area}(K\cap Q_i)=\frac14\operatorname{area}K\quad(i=1,2,3,4,),$$
where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than 4.
A3 (USS 4), IMO 3. Consider the infinite sequences $\{x_n\}$ of positive real numbers with the following properties:
$x_0=1$ and for all $i\ge0$, $x_{i+1}\le x_i$.
(a) Prove that for every such sequence there is an $n\ge1$ such that $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\ge3.999$.
(b) Find such a sequence for which $\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\cdots+\frac{x_{n-1}^2}{x_n}\lt 4$ for all $n$.
A4 (BUL 2). Determine all real values of the parameter $a$ for which the equation
$$16x^4-ax^3+(2a+17)x^2-ax+16=0$$
has exactly four distinct real roots that form a geometric progression.
A5 (NET 2), IMO 5. Let $A_1A_2A_3A_4A_5A_6$ be a regular hexagon. Each of its diagonals $A_{i-1}A_{i+1}$ is divided into the same ratio $\frac\lambda{1-\lambda}$, where $0\lt \lambda\lt 1$, by a point $B_i$ in such a way that $A_i,B_i,B_{i+2}$ are collinear $(i\equiv1,\ldots,6\pmod6)$. Compute $\lambda$.
A6 (VIE 1), IMO 6. Let $S$ be a square with sides of length 100 and let $L$ be a path within $S$ that does not meet itself and that is composed of linear segments $A_0A_1,A_1A_2,\ldots,A_{n-1}A_n$ with $A_0\ne A_n$. Suppose that for every point $P$ of the boundary of $S$ there is a point of $L$ at a distance from $P$ not greater than $\frac12$. Prove that there are two points $X$ and $Y$ in $L$ such that the distance between $X$ and $Y$ is not greater than 1 and the length of that part of $L$ that lies between $X$ and $Y$ is not smaller than 198.
B1 (CAN 2). Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient 1 and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0))$.
B2 (POL 4). A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.
B3 (GBR 1). Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC=\angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL=DM$.
B4 (BRA 1). A box contains $p$ white balls and $q$ black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until the last two balls are removed from the box and one last ball is put in. What is the probability that this last ball is white?
B5 (CAN 3). (a) Find the rearrangement $\{a_1,\ldots,a_n\}$ of $\{1,2,\ldots,n\}$ that maximizes
$$a_1a_2+a_2a_3+\cdots+a_na_1=Q.$$
(b) Find the rearrangement that minimizes $Q$.
B6 (FIN 3). Four distinct circles $C,C_1,C_2,C_3$ and a line $L$ are given in the plane such that $C$ and $L$ are disjoint and each of the circles $C_1,C_2,C_3$ touches the other two, as well as $C$ and $L$. Assuming the radius of $C$ to be 1, determine the distance between its center and $L$.
C1 (NET 1), IMO 2. A scalene triangle $A_1A_2A_3$ is given with sides $a_1,a_2,a_3$ ($a_i$ is the side opposite to $A_i$). For all $i=1,2,3$, $M_i$ is the midpoint of side $a_i$, $T_i$ is the point where the incircle touches side $a_i$, and the reflection of $T_i$ in the interior bisector of $A_i$ yields the point $S_i$. Prove that the lines $M_1S_1,M_2S_2,M_3S_3$ are concurrent.
C2 (AUS 4). Let $ABCD$ be a convex plane quadrilateral and let $A_1$ denote the circumcenter of $\triangle BCD$. Define $B_1,C_1,D_1$ in a corresponding way.
(a) Prove that either all of $A_1,B_1,C_1,D_1$ coincide in one point, or they are all distinct. Assuming the latter case, show that $A_1,C_1$ are on opposite sides of the line $B_1D_1$, and similarly, $B_1,D_1$ are on opposite sides of the line $A_1C_1$. (This establishes the convexity of the quadrilateral $A_1B_1C_1D_1$.)
(b) Denote by $A_2$ the circumcenter of $B_1C_1D_1$, and define $B_2,C_2,D_2$ in an analogous way. Show that the quadrilateral $A_2B_2C_2D_2$ is similar to the quadrilateral $ABCD$.
C4 (GBR 2), IMO 4. Prove that if $n$ is a positive integer such that the equation $x^3-3xy^2+y^3=n$ has a solution in integers $(x,y)$, then it has at least three such solutions. Show that the equation has no solution in integers when $n=2891$.
C5 (USS 5). The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $\angle CAB=\angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.
C6 (FRA 2). Let $O$ be a point of three-dimensional space and let $l_1,l_2,l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1,P_2,P_3$ the intersection of $S_M$ with the straight lines $l_1,l_2,l_3$, respectively, where we put $P_i\ne O$ if $l_i$ meets $S_M$ at two distinct points and $P_i=O$ otherwise $(i=1,2,3)$. What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?
C7 (CZS 3). Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x\in M,y\in M$ with $x\lt y$, there exists an element $z\in M$ such that $x\lt z\lt y$.
C8 (TUN 3). Let $ABCD$ be a convex quadrilateral and draw regular triangles $ABM,CDP,BCN,ADQ$, the first two outward and the other two inward. Prove that $MN=AC$. What can be said about the quadrilateral $MNPQ$?