Question bank IMO Shortlist 2002

What is the smallest positive integer $t$ such that there exist integers $x_{1}, x_{2}, \ldots, x_{t}$ with

$$ x_{1}^{3}+x_{2}^{3}+\ldots+x_{t}^{3}=2002^{2002}? $$

Let $ n \geqslant 2 $ be a positive integer, with divisors $ 1 = d_1 \lt d_2 \lt \ldots \lt d_k = n $. Prove that $ d_1d_2 + d_2d_3 + \ldots + d_{k-1}d_k $ is always less than $ n^2 $, and determine when it is a divisor of $ n^2 $.

Let $p_{1}, p_{2}, \ldots, p_{n}$ be distinct primes greater than 3. Show that $2^{p_{1}p_{2} \cdots p_{n}} + 1$ has at least $4^{n}$ divisors.

Is there a positive integer $m$ such that the equation

$$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=\frac{m}{a+b+c} $$

has infinitely many solutions in positive integers $a$, $b$, $c$?

Let $m, n \geq 2$ be positive integers, and let $a_1, a_2, \ldots, a_n$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_1, e_2, \ldots, e_n$, not all zero, with $|e_i| \lt m$ for all $i$, such that $e_1 a_1 + e_2 a_2 + \ldots + e_n a_n$ is a multiple of $m^n$.

Find all pairs of positive integers $m, n \geq 3$ for which there exist infinitely many positive integers $a$ such that

$$\frac{a^m+a-1}{a^n+a^2-1}$$

is itself an integer.

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB = \angle BFC = \angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that

$$ AB\;+\;AC\;\geq\;4DE. $$

The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB \lt 120^{\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF$.

Circles $ S_{1} $ and $ S_{2} $ intersect at points P and Q. Distinct points $ A_{1} $ and $ B_{1} $ (not at P or Q) are selected on $ S_{1} $. The lines $ A_{1}P $ and $ B_{1}P $ meet $ S_{2} $ again at $ A_{2} $ and $ B_{2} $ respectively, and the lines $ A_{1}B_{1} $ and $ A_{2}B_{2} $ meet at C. Prove that, as $ A_{1} $ and $ B_{1} $ vary, the circumcentres of triangles $ A_{1}A_{2}C $ all lie on one fixed circle.

For any set S of five points in the plane, no three of which are collinear, let $ M(S) $ and $ m(S) $ denote the greatest and smallest areas, respectively, of triangles determined by three points from S. What is the minimum possible value of $ M(S)/m(S) $?

Let $n \geq 3$ be a positive integer. Let $C_{1}, C_{2}, C_{3}, \ldots, C_{n}$ be unit circles in the plane, with centres $O_{1}, O_{2}, O_{3}, \ldots, O_{n}$ respectively. If no line meets more than two of the circles, prove that

$$ \sum_{1\leq i \lt j\leq n}\frac{1}{O_{i} O_{j}}\leq\frac{(n-1)\pi}{4}. $$

The incircle $\Omega$ of the acute-angled triangle $ABC$ is tangent to $BC$ at $K$. Let $AD$ be an altitude of triangle $ABC$ and let $M$ be the midpoint of $AD$. If $N$ is the other common point of $\Omega$ and $KM$, prove that $\Omega$ and the circumcircle of triangle $BCN$ are tangent at $N$.

Let $S_1$ and $S_2$ be circles meeting at the points $A$ and $B$. A line through $A$ meets $S_1$ at $C$ and $S_2$ at $D$. Points $M, N, K$ lie on the line segments $CD, BC, BD$ respectively, with $MN$ parallel to $BD$ and $MK$ parallel to $BC$. Let $E$ and $F$ be points on those arcs $BC$ of $S_1$ and $BD$ of $S_2$ respectively that do not contain $A$. Given that $EN$ is perpendicular to $BC$ and $FK$ is perpendicular to $BD$ prove that $\angle EMF = 90^\circ$.

Find all functions f from the reals to the reals such that

$$ f(f(x)+y)=2x+f(f(y)-x) $$

for all real x, y.

Let $a_1$, $a_2$, ... be an infinite sequence of real numbers, for which there exists a real number $c$ with $0 \leq a_i \leq c$ for all $i$, such that

$$ \left|a_{i}-a_{j}\right|\geq\frac{1}{i+j}\quad\text{for all }i,j\text{ with }i\neq j. $$

Prove that $ c \geq 1 $.

Let $P$ be a cubic polynomial given by $P(x) = ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are integers and $a \ne 0$. Suppose that $xP(x) = yP(y)$ for infinitely many pairs $x, y$ of integers with $x \ne y$. Prove that the equation $P(x) = 0$ has an integer root.

Find all functions f from the reals to the reals such that

$$ \left(f\left(x\right)+f\left(z\right)\right)\left(f\left(y\right)+f\left(t\right)\right)=f\left(x y-z t\right)+f\left(x t+y z\right) $$

for all real x, y, z, t.

Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by

$$ a~=~\sqrt[3]{n},\qquad b~=~\frac{1}{a~-~[a]},\qquad c~=~\frac{1}{b~-~[b]}, $$

where $[x]$ denotes the integer part of $x$.

Prove that there are infinitely many such integers $n$ with the property that there exist integers $r$, $s$, $t$, not all zero, such that $ra + sb + tc = 0$.

Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1, \ldots, b_n$ and $c_1, \ldots, c_n$ such that

(i) for each $i$ the set $b_i A + c_i = \{b_i a + c_i : a \in A\}$ is a subset of $A$, and

(ii) the sets $b_{i}A + c_{i}$ and $b_{j}A + c_{j}$ are disjoint whenever $i \neq j$. Prove that

$$ \frac{1}{b_{1}}+\cdots+\frac{1}{b_{n}}\leq1. $$