(AUS 3) The integer 9 can be written as a sum of two consecutive integers: $9=4+5$. Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $9=4+5=2+3+4$. Is there an integer that can be written as a sum of 1990 consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly 1990 ways?
(CAN 1) Given $n$ countries with three representatives each, $m$ committees $A(1),A(2),\ldots,A(m)$ are called a cycle if
(i) each committee has $n$ members, one from each country;
(ii) no two committees have the same membership;
(iii) for $i=1,2,\ldots,m$, committee $A(i)$ and committee $A(i+1)$ have no member in common, where $A(m+1)$ denotes $A(1)$;
(iv) if $1\lt |i-j|\lt m-1$, then committees $A(i)$ and $A(j)$ have at least one member in common.
Is it possible to have a cycle of 1990 committees with 11 countries?
(CZS 1) IMO2 On a circle, $2n-1$ $(n\ge3)$ different points are given. Find the minimal natural number $N$ with the property that whenever $N$ of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly $n$ of the given $2n-1$ points.
(CZS 2) Assume that the set of all positive integers is decomposed into $r$ (disjoint) subsets $A_1\cup A_2\cup\cdots A_r=\mathbb N$. Prove that one of them, say $A_i$, has the following property: There exists a positive $m$ such that for any $k$ one can find numbers $a_1,a_2,\ldots,a_k$ in $A_i$ with $0\lt a_{j+1}-a_j\le m$ $(1\le j\le k-1)$.
(FRA 1) Given $\triangle ABC$ with no side equal to another side, let $G$, $K$, and $H$ be its centroid, incenter, and orthocenter, respectively. Prove that $\angle GKH\gt 90^\circ$.
(FRG 2) IMO5 Two players $A$ and $B$ play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number $n_0\gt 1$ is given. Knowing $n_{2k}$, player $A$ may choose any $n_{2k+1}\in\mathbb N$ such that
$$n_{2k}\le n_{2k+1}\le n_{2k}^2.$$
Then player $B$ chooses a number $n_{2k+2}\in\mathbb N$ such that
$$\frac{n_{2k+1}}{n_{2k+2}}=p^r,$$
where $p$ is a prime number and $r\in\mathbb N$.
It is stipulated that player $A$ wins the game if he (she) succeeds in choosing the number 1990, and player $B$ wins if he (she) succeeds in choosing 1. For which natural numbers $n_0$ can player $A$ manage to win the game, for which $n_0$ can player $B$ manage to win, and for which $n_0$ can players $A$ and $B$ each force a tie?
(GRE 2) Let $f(0)=f(1)=0$ and
$$f(n+2)=4^{n+2}f(n+1)-16^{n+1}f(n)+n\cdot2^{n^2},\qquad n=0,1,2,3,\ldots.$$
Show that the numbers $f(1989),f(1990),f(1991)$ are divisible by 13.
(HUN 1) For a given positive integer $k$ denote the square of the sum of its digits by $f_1(k)$ and let $f_{n+1}(k)=f_1(f_n(k))$. Determine the value of $f_{1991}(2^{1990})$.
(HUN 3) The incenter of the triangle $ABC$ is $K$. The midpoint of $AB$ is $C_1$ and that of $AC$ is $B_1$. The lines $C_1K$ and $AC$ meet at $B_2$, the lines $B_1K$ and $AB$ at $C_2$. If the areas of the triangles $AB_2C_2$ and $ABC$ are equal, what is the measure of angle $\angle CAB$?
(ICE 2) A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.
(IND 3') IMO1 Given a circle with two chords $AB$, $CD$ that meet at $E$, let $M$ be a point of chord $AB$ other than $E$. Draw the circle through $D$, $E$, and $M$. The tangent line to the circle $DEM$ at $E$ meets the lines $BC$, $AC$ at $F$, $G$, respectively. Given $\frac{AM}{AB}=\lambda$, find $\frac{GE}{EF}$.
(IRE 1) Let $ABC$ be a triangle and $L$ the line through $C$ parallel to the side $AB$. Let the internal bisector of the angle at $A$ meet the side $BC$ at $D$ and the line $L$ at $E$ and let the internal bisector of the angle at $B$ meet the side $AC$ at $F$ and the line $L$ at $G$. If $GF=DE$, prove that $AC=BC$.
(IRE 2) An eccentric mathematician has a ladder with $n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $a$ rungs of the ladder, and when he descends, each step he takes covers $b$ rungs of the ladder, where $a$ and $b$ are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of $n$, expressed in terms of $a$ and $b$.
(JAP 2) In the coordinate plane a rectangle with vertices $(0,0)$, $(m,0)$, $(0,n)$, $(m,n)$ is given where both $m$ and $n$ are odd integers. The rectangle is partitioned into triangles in such a way that
(i) each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form $x=j$ or $y=k$, where $j$ and $k$ are integers, and the altitude on this side has length 1;
(ii) each “bad” side (i.e., a side of any triangle in the partition that is not a “good” one) is a common side of two triangles in the partition.
Prove that there exist at least two triangles in the partition each of which has two good sides.
(MEX 2) Determine for which positive integers $k$ the set
$$X=\{1990,1990+1,1990+2,\ldots,1990+k\}$$
can be partitioned into two disjoint subsets $A$ and $B$ such that the sum of the elements of $A$ is equal to the sum of the elements of $B$.
(NET 1) IMO6 Is there a 1990-gon with the following properties (i) and (ii)?
(i) All angles are equal;
(ii) The lengths of the 1990 sides are a permutation of the numbers $1^2,2^2,\ldots,1989^2,1990^2$.
(NET 3) Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $A$ be the beginning vertex and $B$ be the end vertex. Let there be $p\times q\times r$ cubes on the string $(p,q,r\ge1)$.
(a) Determine for which values of $p$, $q$, and $r$ it is possible to build a block with dimensions $p$, $q$, and $r$. Give reasons for your answers.
(b) The same question as (a) with the extra condition that $A=B$.
(NOR) Let $a,b$ be natural numbers with $1\le a\le b$, and $M=\left[\frac{a+b}{2}\right]$. Define the function $f:\mathbb Z\to\mathbb Z$ by
$$f(n)=\begin{cases}n+a,&n\lt M,\\n-b,&n\ge M.\end{cases}$$
Let $f^1(n)=f(n)$, $f^{i+1}(n)=f(f^i(n))$, $i=1,2,\ldots$. Find the smallest natural number $k$ such that $f^k(0)=0$.
(POL 1) Let $P$ be a point inside a regular tetrahedron $T$ of unit volume. The four planes passing through $P$ and parallel to the faces of $T$ partition $T$ into 14 pieces. Let $f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $f(P)$ as $P$ varies over $T$.
(POL 3) Prove that every integer $k$ greater than 1 has a multiple that is less than $k^4$ and can be written in the decimal system with at most four different digits.