Question bank IMO Shortlist 1999

N1 (TWN), IMO4. Find all pairs of positive integers $(x,p)$ such that $p$ is a prime, $x\le2p$, and $x^{p-1}$ is a divisor of $(p-1)^x+1$.

N2 (ARM). Prove that every positive rational number can be represented in the form $\frac{a^3+b^3}{c^3+d^3}$, where $a,b,c,d$ are positive integers.

N3 (RUS). Prove that there exist two strictly increasing sequences $(a_n)$ and $(b_n)$ such that $a_n(a_n+1)$ divides $b_n^2+1$ for every natural number $n$.

N4 (FRA). Denote by $S$ the set of all primes $p$ such that the decimal representation of $1/p$ has its fundamental period divisible by $3$. For every $p\in S$ such that $1/p$ has its fundamental period $3r$ one may write $1/p=0.a_1a_2\ldots a_{3r}a_1a_2\ldots a_{3r}\ldots$, where $r=r(p)$; for every $p\in S$ and every integer $k\ge1$ define $f(k,p)$ by
$$f(k,p)=a_k+a_{k+r(p)}+a_{k+2r(p)}.$$
(a) Prove that $S$ is infinite.
(b) Find the highest value of $f(k,p)$ for $k\ge1$ and $p\in S$.

N5 (ARM). Let $n,k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there exists a positive integer $m$ that is divisible by $n$ and the sum of whose digits in decimal representation is $k$.

N6 (BLR). Prove that for every real number $M$ there exists an infinite arithmetic progression such that:
(i) each term is a positive integer and the common difference is not divisible by $10$;
(ii) the sum of the digits of each term (in decimal representation) exceeds $M$.

G1 (ARM). Let $ABC$ be a triangle and $M$ an interior point. Prove that
$$\min\{MA,MB,MC\}+MA+MB+MC\lt AB+AC+BC.$$

G2 (JAP). A circle is called a separator for a set of five points in a plane if it passes through three of these points, it contains a fourth point in its interior, and the fifth point is outside the circle.
Prove that every set of five points such that no three are collinear and no four are concyclic has exactly four separators.

G3 (EST), IMO1. A set $S$ of points in space will be called completely symmetric if it has at least three elements and satisfies the following condition: For every two distinct points $A,B$ from $S$ the perpendicular bisector of the segment $AB$ is an axis of symmetry for $S$.
Prove that if a completely symmetric set is finite, then it consists of the vertices of either a regular polygon, a regular tetrahedron, or a regular octahedron.

G4 (GBR). For a triangle $T=ABC$ we take the point $X$ on the side $(AB)$ such that $AX/XB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY=2YX$, and, if possible, the point $Z$ on the ray $(CA$ such that $\angle CXZ=180^\circ-\angle ABC$. We denote by $\Sigma$ the set of all triangles $T$ for which $\angle XYZ=45^\circ$.
Prove that all the triangles from $\Sigma$ are similar and find the measure of their smallest angle.

G5 (FRA). Let $ABC$ be a triangle, $\Omega$ its incircle and $\Omega_a,\Omega_b,\Omega_c$ three circles three circles orthogonal to $\Omega$ passing through $B$ and $C$, $A$ and $C$, and $A$ and $B$ respectively. The circles $\Omega_a,\Omega_b$ meet again in $C'$; in the same way we obtain the points $B'$ and $A'$. Prove that the radius of the circumcircle of $A'B'C'$ is half the radius of $\Omega$.

G6 (RUS), IMO5. Two circles $\Omega_1$ and $\Omega_2$ touch internally the circle $\Omega$ in $M$ and $N$, and the center of $\Omega_2$ is on $\Omega_1$. The common chord of the circles $\Omega_1$ and $\Omega_2$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersect $\Omega_1$ in $C$ and $D$. Prove that $\Omega_2$ is tangent to $CD$.

G7 (ARM). The point $M$ inside the convex quadrilateral $ABCD$ is such that
$$MA=MC,\quad\angle AMB=\angle MAD+\angle MCD,\quad\angle CMD=\angle MCB+\angle MAB.$$
Prove that $AB\cdot CM=BC\cdot MD$ and $BM\cdot AD=MA\cdot CD$.

G8 (RUS) Points $A,B,C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs. Let $X$ be a variable point on the arc $AB$, and let $O_1,O_2$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_1O_2$ intersects $\Omega$ in a fixed point.

A1 (POL), IMO2. Let $n\ge2$ be a fixed integer. Find the least constant $C$ such that the inequality
$$\sum_{i\lt j}x_ix_j(x_i^2+x_j^2)\le C\left(\sum_i x_i\right)^4$$
holds for every $x_1,\ldots,x_n\ge0$ (the sum on the left consists of $\binom n2$ summands). For this constant $C$, characterize the instances of equality.

A2 (RUS). The numbers from $1$ to $n^2$ are randomly arranged in the cells of a $n\times n$ square ($n\ge2$). For any pair of numbers situated in the same row or in the same column, the ratio of the greater number to the smaller one is calculated.
Let us call the characteristic of the arrangement the smallest of these $n^2(n-1)$ fractions. What is the highest possible value of the characteristic?

A3 (FIN). A game is played by $n$ girls ($n\ge2$), everybody having a ball. Each of the $\binom n2$ pairs of players, in an arbitrary order, exchange the balls they have at that moment. The game is called nice if at the end nobody has her own ball, and it is called tiresome if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.

A4 (BLR). Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.

A5 (JAP), IMO6. Find all the functions $f:\mathbb R\to\mathbb R$ that satisfy
$$f(x-f(y))=f(f(y))+xf(y)+f(x)-1$$
for all $x,y\in\mathbb R$.

A6 (SWE). For $n\ge3$ and $a_1\le a_2\le\cdots\le a_n$ given real numbers we have the following instructions:
(1) place the numbers in some order in a circle;
(2) delete one of the numbers from the circle;
(3) if just two numbers are remaining in the circle, let $S$ be the sum of these two numbers. Otherwise, if there are more than two numbers in the circle, replace $(x_1,x_2,x_3,\ldots,x_{p-1},x_p)$ with $(x_1+x_2,x_2+x_3,\ldots,x_{p-1}+x_p,x_p+x_1)$. Afterwards, start again with step (2).
Show that the largest sum $S$ that can result in this way is given by the formula
$$S_{\max}=\sum_{k=2}^n\binom{n-2}{\lfloor k/2\rfloor-1}a_k.$$