IMO 1998 (XXXIX International Mathematical Olympiad)

Cyclic quadrilateral iff two triangles have equal areas

In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. Suppose that the point $P$, where the perpendicular bisectors of $AB$ and $DC$ meet, is inside $ABCD$. Prove that $ABCD$ is a cyclic quadrilateral if and only if the triangles $ABP$ and $CDP$ have equal areas.

Topic: Geometria piana Metodo: Simmetria Abilita: Ragionamento geometrico, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Cyclic quadrilateral if two triangles have equal areas

In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. Suppose that the point $P$, where the perpendicular bisectors of $AB$ and $DC$ meet, is inside $ABCD$. Prove that $ABCD$ is a cyclic quadrilateral if and only if the triangles $ABP$ and $CDP$ have equal areas.

src_imho_1998__Q01

Judges rating contestants: prove k/a >= (b-1)/(2b)

In a competition, there are $a$ contestants and $b$ judges, where $b\ge 3$ is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose $k$ is a number such that, for any two judges, their ratings coincide for at most $k$ contestants. Prove that $k/a\ge (b-1)/(2b)$.

Topic: Combinatoria, Disuguaglianze Metodo: Doppio conteggio, Principio dei cassetti Abilita: Conteggio sistematico, Manipolazione algebrica, Modellizzazione Area: Combinatoria, Logica e Probabilita, Algebra e Analisi Fonte: apri PDF p.1

Judges rating contestants: evidence k/a >= (b-1)/(2b)

In a competition, there are $a$ contestants and $b$ judges, where $b\ge 3$ is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose $k$ is a number such that, for any two judges, their ratings coincide for at most $k$ contestants. Prove that $k/a\ge (b-1)/(2b)$.

src_imho_1998__Q02

Find all positive integers k with d(n^2)/d(n)=k for some n

For any positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$ (including $1$ and $n$ itself). Determine all positive integers $k$ such that $d(n^2)/d(n)=k$ for some $n$.

Topic: Teoria dei Numeri Metodo: Fattorizzazione, Casework Abilita: Manipolazione algebrica, Riconoscimento di pattern, Conteggio sistematico Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find the positive integers k with d(n^2)/d(n)=k for some n

For any positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$ (including $1$ and $n$ itself). Determine the positive integers $k$ such that $d(n^2)/d(n)=k$ for some $n$.

src_imho_1998__Q03

Find all pairs (a,b) of positive integers with given divisibility

Determine all pairs $(a,b)$ of positive integers such that $a{b}^2+b+7$ divides $a^{2}b+a+b$.

Topic: Teoria dei Numeri Metodo: Fattorizzazione, Congruenze, Casework Abilita: Manipolazione algebrica, Casework accurato, Lettura attenta Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find all pairs (a,b) of positive integers with given divisibility

Determine the pairs $(a,b)$ of positive integers such that $a{b}^2+b+7$ divides $a^{2}b+a+b$.

src_imho_1998__Q04

Incircle touch points construction: angle RIS is acute

Let $I$ be the incenter of triangle $ABC$. Let the incircle of $ABC$ touch the sides $BC$, $CA$, and $AB$ at $K$, $L$, and $M$, respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$, respectively. Prove that angle $RIS$ is acute.

Topic: Geometria piana Metodo: Trigonometria, Simmetria Abilita: Ragionamento geometrico, Manipolazione algebrica, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Incircle touch points construction: angle RIS is acute

Let $I$ be the incenter of triangle $ABC$. Let the incircle of $ABC$ touch the sides $BC$, $CA$, and $AB$ at $K$, $L$, and $M$, respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$, respectively. Prove that angle $RIS$ is sharp.

src_imho_1998__Q05

Functional equation on positive integers; find least f(1998)

Consider all functions $f$ from the set $N$ of all positive integers into itself satisfying $f(t^{2}f(s))=s(f(t){)}^2$ for all $s$ and $t$ in $N$. Determine the least possible value of $f(1998)$.

Topic: Insiemi e funzioni, Teoria dei Numeri Metodo: Fattorizzazione, Casework Abilita: Manipolazione algebrica, Riconoscimento di pattern, Astrazione Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

*Functional equation on positive integers; find at least f(1998) *

Consider the functions $f$ from the set $N$ of all positive integers in itself satisfying $f(t^{2}f(s))=s(f(t){)}^2$ for all $s$ and $t$ in $N$. Determine the least possible value of $f(1998)$.

src_imho_1998__Q06