IMO 2001 (XLII International Mathematical Olympiad)

Acute triangle with circumcentre; angle inequality implies angle sum bound

Let $ABC$ be an acute-angled triangle with circumcentre $O$. Let $P$ on $BC$ be the foot of the altitude from $A$.

Suppose that $\angle BCA\ge \angle ABC+30^{\circ }$.

Prove that $\angle CAB+\angle COP<90^{\circ }$.

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

Acute triangle with circumcentre; angle inequality implies angle sum bound

Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $P$ on $BC$ be the foot of the altitude from $A$.

Suppose that $\angle BCA\ge \angle ABC+30^{\circ }$.

Prove that $\angle CAB+\angle COP<90^{\circ }$.

src_imho_2001__Q01

Inequality with three positive reals and square roots

Prove that $\frac{a}{\sqrt{a^2+8bc}}+\frac{b}{\sqrt{b^2+8ca}}+\frac{c}{\sqrt{c^2+8ab}}\ge 1$ for all positive real numbers $a$, $b$ and $c$.

Topic: Disuguaglianze Metodo: Disuguaglianze Abilita: Manipolazione algebrica, Stima, Ragionamento geometrico Area: Algebra e Analisi Fonte: apri PDF p.1

Inequality with three positive reals and square roots

Prove that $\frac{a}{\sqrt{a^2+8bc}}+\frac{b}{\sqrt{b^2+8ca}}+\frac{c}{\sqrt{c^2+8ab}}\ge 1$ for all positive real numbers $a$, $b$ and $c$.

src_imho_2001__Q02

Twenty-one girls and boys; pigeonhole gives a problem solved by 3 of each

Twenty-one girls and twenty-one boys took part in a mathematical contest.

• Each contestant solved at most six problems.
• For each girl and each boy, at least one problem was solved by both of them.

Prove that there was a problem that was solved by at least three girls and at least three boys.

Topic: Combinatoria Metodo: Principio dei cassetti, Doppio conteggio, Casework Abilita: Conteggio sistematico, Ragionamento geometrico, Modellizzazione, Lettura attenta Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Twenty-one girls and boys; pigeonhole gives a problem solved by 3 of each

Twenty-one girls and twenty-one boys took part in a mathematical contest.

• Each contestant solved at most six problems. For every girl and every boy, at least one problem was solved by both of them.

Prove that there was a problem that was solved by at least three girls and at least three boys.

src_imho_2001__Q03

Permutations of 1..n with weighted sums; n! divides difference of two sums

Let $n$ be an odd integer greater than $1$, and let $k_1,k_2,\dots ,k_n$ be given integers. For each of the $n!$ permutations $a=(a_1,a_2,\dots ,a_n)$ of $1,2,\dots ,n$, let $S(a)={\sum }_{i=1}^n{k}_i{a}_i.$ Prove that there are two permutations $b$ and $c$, $b\ne c$, such that $n!$ is a divisor of $S(b)-S(c)$.

Topic: Combinatoria, Teoria dei Numeri Metodo: Principio dei cassetti, Congruenze, Conteggio Abilita: Conteggio sistematico, Manipolazione algebrica, Modellizzazione, Riconoscimento di pattern Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Permutations of 1..n with weighted sums; n divides difference of two sums*

Let $n$ be an odd integer greater than $1$, and let $k_1,k_2,\dots ,k_n$ be given integers. For each of the $n!$ permutations $a=(a_1,a_2,\dots ,a_n)$ of $1,2,\dots ,n$, let $S(a)={\sum }_{i=1}^n{k}_i{a}_i.$ Prove that there are two permutations $b$ and $c$, $b\ne c$, such that $n!$ is a divisor of $S(b)-S(c)$.

src_imho_2001__Q04

Triangle with two angle bisectors and a length condition; find all possible angles

In a triangle $ABC$, let $AP$ bisect $\angle BAC$, with $P$ on $BC$, and let $BQ$ bisect $\angle ABC$, with $Q$ on $CA$.

It is known that $\angle BAC=60^{\circ }$ and that $AB+BP=AQ+QB$.

What are the possible angles of triangle $ABC$?

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

Triangle with two angle bisectors and a length condition; find all possible angles

In a $ABC$ triangle, let $AP$ bisect $\angle BAC$, with $P$ on $BC$, and let $BQ$ bisect $\angle ABC$, with $Q$ on $CA$.

It is known that $\angle BAC=60^{\circ }$ and that $AB+BP=AQ+QB$.

What are the possible angles of triangle $ABC$?

src_imho_2001__Q05

Integers a>b>c>d>0 with a product condition; ab+cd is not prime

Let $a$, $b$, $c$, $d$ be integers with $a>b>c>d>0$. Suppose that $ac+bd=(b+d+a-c)(b+d-a+c).$ Prove that $ab+cd$ is not prime.

Topic: Teoria dei Numeri, Algebra Metodo: Fattorizzazione, Congruenze, Simmetria Abilita: Manipolazione algebrica, Riconoscimento di pattern, Astrazione, Lettura attenta Area: Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Integers a>b>c>d>0 with a product condition; ab+cd is not prime

Let $a$, $b$, $c$, $d$ be integers with $a>b>c>d>0$. Suppose that $ac+bd=(b+d+a-c)(b+d-a+c).$ Prove that $ab+cd$ is not prime.

src_imho_2001__Q06