ISL 2002

Let $a_1,a_2,\ldots$ 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 $$\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \text{for all }i,\ j \text{ with } i \neq j.$$ Prove that $c\geq1$ .

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Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$ , where $a,b,c,d$ are integers and $a\ne0$ . 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.

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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

• for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$ , and

• the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$

Prove that $${1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.$$

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Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called full if it satisfies the following condition: for each positive integer $k\geq2$ , if the number $k$ appears in the sequence then so does the number $k-1$ , and moreover the first occurrence of $k-1$ comes before the last occurrence of $k$ . For each $n$ , how many full sequences are there ?

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Let $T$ be the set of ordered triples $(x,y,z)$ , where $x,y,z$ are integers with $0\leq x,y,z\leq9$ . Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$ , and Player $B$ has to discover $A$ 's triple in as few moves as possible. A move consists of the following: $B$ gives $A$ a triple $(a,b,c)$ in $T$ , and $A$ replies by giving $B$ the number $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$ . Find the minimum number of moves that $B$ needs to be sure of determining $A$ 's triple.

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Let $r\geq2$ be a fixed positive integer, and let $F$ be an infinite family of sets, each of size $r$ , no two of which are disjoint. Prove that there exists a set of size $r-1$ that meets each set in $F$ .

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Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1, 2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$ , the number $x_{i+1}$ is one of the numbers $2x_{i}$ , $2x_{i}-1$ , $2x_{i}-n$ , $2x_{i}-n-1$ . Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$ .

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Among a group of 120 people, some pairs are friends. A weak quartet is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?

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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 $$AB+AC\geq4DE.$$

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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 $A_1P$ and $B_1P$ meet $S_2$ again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet at $C$ . Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles $A_1A_2C$ all lie on one fixed circle.

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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 $M(S)/m(S)$ ?

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The incircle $\Omega$ of the acute-angled triangle $ABC$ is tangent to its side $BC$ at a point $K$ . Let $AD$ be an altitude of triangle $ABC$ , and let $M$ be the midpoint of the segment $AD$ . If $N$ is the common point of the circle $\Omega$ and the line $KM$ (distinct from $K$ ), then prove that the incircle $\Omega$ and the circumcircle of triangle $BCN$ are tangent to each other at the point $N$ .

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What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with $$x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?$$

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Is there a positive integer $m$ such that the equation $${1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c}$$ has infinitely many solutions in positive integers $a,b,c$ ?

Let $m,n\geq2$ 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 $\left|{ e}_i\,\right|<m$ for all $i$ , such that $e_1a_1+e_2a_2+\,\ldots\,+e_na_n$ is a multiple of $m^n$ .

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Let two circles $S_{1}$ and $S_{2}$ meet at the points $A$ and $B$ . A line through $A$ meets $S_{1}$ again at $C$ and $S_{2}$ again at $D$ . Let $M$ , $N$ , $K$ be three points 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 $\angle EMF=90^{\circ}$ .

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