IMO 1985 (XXVI International Mathematical Olympiad)

Circle tangent to sides of cyclic quadrilateral, prove AD+BC=AB

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD+BC=AB$.

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

Circle tangent to sides of cyclic quadrilateral, tests AD+BC=AB

A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that$AD+BC=AB$.

src_imho_1985__Q01

Coloring integers in M with two colors preserving sum condition

Let $n$ and $k$ be given relatively prime natural numbers, $k<n$. Each number in the set $M=\{1,2,\dots ,n-1\}$ is colored either blue or white. It is given that (i) for each $i\in M$, both $i$ and $n-i$ have the same color; (ii) for each $i\in M$, $i\ne k$, both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

Topic: Combinatoria, Teoria dei Numeri Metodo: Invarianti, Congruenze Abilita: Ragionamento geometrico, Riconoscimento di pattern, Lettura attenta Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Coloring integers in M with two colors preserving sum condition

Let $n$ and $k$ be given relatively prime natural numbers, $k<n$. Each number in the set $M=\{1,2,\dots ,n-1\}$ is colored either blue or white. It is given that (i) for each $i\in M$, both $i$ and $n-i$ have the same color; (ii) for each $i\in M$, $i\ne k$, both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

src_imho_1985__Q02

Polynomial with integer coefficients: odd-indexed partial sums are integers

For any polynomial $P(x)=a_0+a_{1}x+\cdots +a_k{x}^k$ with integer coefficients, the number of coefficients which are odd is denoted by $w(P)$. For $i=0,1,\dots$, let $Q_i(x)=(1+x{)}^i$. Prove that if $i_1,i_2,\dots ,i_n$ are integers such that $0\le i_1<i_2<\cdots <i_n$, then $w(Q_{i_1}+Q_{i_2}+\cdots +Q_{i_n})\ge w(Q_{i_n}).$

Topic: Algebra, Combinatoria Metodo: Induzione, Congruenze Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Algebra e Analisi, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Polynomial with integer coefficients: odd-indexed partial sums are integers

For any polynomial $P(x)=a_0+a_{1}x+\cdots +a_k{x}^k$ with integer coefficients, the number of coefficients which are odd is denoted by $w(P)$. For $i=0,1,\dots$, let $Q_i(x)=(1+x{)}^i$. Prove that if $i_1,i_2,\dots ,i_n$ are integers such that $0\le i_1<i_2<\cdots <i_n$, then $w(Q_{i_1}+Q_{i_2}+\cdots +Q_{i_n})\ge w(Q_{i_n}).$

src_imho_1985__Q03

Set M of 1985 distinct positive integers with no subset product a perfect power

Given a set $M$ of 1985 distinct positive integers, none of which has a prime divisor greater than 26. Prove that $M$ contains at least one subset of four distinct elements whose product is the fourth power of an integer.

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

Set M of 1985 distinct positive integers with no subset product a perfect power

Given a set $M$ of 1985 distinct positive integers, none of which has a prime divisor greater than 26. Prove that $M$ contains at least one subset of four distinct elements whose product is the fourth power of an integer.

src_imho_1985__Q04

Circle through vertices A and C meets sides AB and BC at K and N

A circle with center $O$ passes through the vertices $A$ and $C$ of triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$, respectively. The circumscribed circles of the triangles $ABC$ and $KBN$ intersect at exactly two distinct points $B$ and $M$. Prove that angle $OMB$ is a right angle.

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

Circle through vertices A and C meets sides AB and BC at K and N

A circle with center $O$ passes through the vertices $A$ and $C$ of triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$, respectively. The circumscribed circles of the triangles $ABC$ and $KBN$ intersect at exactly two distinct points $B$ and $M$. Prove that angle is a right angle.

src_imho_1985__Q05

Sequence defined by recursion has exactly one initial value in (0,1)

For every real number $x_1$, construct the sequence $x_1,x_2,\dots$ by setting $x_{n+1}=x_n\left(x_n+\frac{1}{n}\right)$ for each $n\ge 1$. Prove that there exists exactly one value of $x_1$ for which $0<x_n<x_{n+1}<1$ for every $n$.

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

*Sequence defined by recursion has exactly one initial value in (0,1) *

For every real number $x_1$, construct the sequence $x_1,x_2,\dots$ by setting $x_{n+1}=x_n\left(x_n+\frac{1}{n}\right)$ for each $n\ge 1$. Prove that there exists exactly one value of $x_1$ for which $0<x_n<x_{n+1}<1$ for every $n$.

src_imho_1985__Q06