IMO 1974 (XVI International Mathematical Olympiad)

Card game: who received r counters in last round?

Three players $A$, $B$ and $C$ play the following game. On each of three cards an integer is written. These three numbers $p$, $q$, $r$ satisfy $0<p<q<r$. The three cards are shuffled and one is dealt to each player. Each then receives the number of counters indicated by the card. Then the cards are shuffled again; the counters remaining with the players.

This process (shuffling, dealing, giving out counters) takes place for at least two rounds. After the last round $A$ has 20 counters in all, $B$ has 10 and $C$ has 9. At the last round $B$ received $r$ counters. Who received $r$ counters on the first round?

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

Card game: who received r counters in last round?

Three players $A$, $B$ and $C$ play the following game. On each of three cards an integer is written. These three numbers $p$, $q$, $r$ satisfy $0<p<q<r$. The three cards are shuffled and one is dealt to each player. Each then receives the number of counters indicated by the card. Then the cards are shuffled again; the counters remaining with the players.

This process (shuffling, dealing, giving out counters) takes place for at least two rounds. After the last round $A$ has 20 counters in all, $B$ has 10 and $C$ has 9. At the last round $B$ received $r$ counters. Who received $r$ counters on the first round?

src_imho_1974__Q01

Geometric mean condition via sine inequality in triangle ABC

In the triangle $ABC$, prove that there is a point $D$ on side $AB$ such that $CD$ is the geometric mean of $AD$ and $DB$ if and only if $\sin A\sin B\le {\sin }^2\frac{C}{2}.$

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

Geometric mean condition by sine inequality in triangle ABC

In the $ABC$ triangle, prove that there is a point $D$ on side $AB$ such that $CD$ is the geometric mean of $AD$ and $DB$ if and only if $\sin A\sin B\le {\sin }^2\frac{C}{2}.$

src_imho_1974__Q02

Sum involving binomial coefficients not divisible by 5

Prove that the number ${\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{2n+1}{2k+1}\right)2^{3k}$ is not divisible by 5 for any integer $n\ge 0$.

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

Sum involving binomial coefficients not divisible by 5

Prove that the number ${\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{2n+1}{2k+1}\right)2^{3k}$ is not divisible by 5 for any integer $n\ge 0$.

src_imho_1974__Q03

Chessboard decomposition into rectangles, maximum white squares

Consider decompositions of an $8\times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1<a_2<\cdots <a_p$. Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$, determine all possible sequences $a_1,a_2,\dots ,a_p$.

Topic: Combinatoria Metodo: Casework, Estremalità, Conteggio Abilita: Conteggio sistematico, Casework accurato, Modellizzazione Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Chessboard decomposition into rectangles, maximum white squares

Consider decompositions of an $8\times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1<a_2<\cdots <a_p$. Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$, determine all possible sequences $a_1,a_2,\dots ,a_p$.

src_imho_1974__Q04

Find all possible values of a cyclic sum of four fractions

Determine all possible values of $S=\frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}$ where $a,b,c,d$ are arbitrary positive numbers.

Topic: Algebra, Disuguaglianze Metodo: Disuguaglianze, Estremalità Abilita: Manipolazione algebrica, Stima, Riconoscimento di pattern Area: Algebra e Analisi Fonte: apri PDF p.1

Find all possible values of a cyclic sum of four fractions

Determine all possible values of $S=\frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}$ where $a,b,c,d$ are arbitrary positive numbers.

src_imho_1974__Q05

Bound on integers where polynomial squares equal 1

Let $P$ be a non-constant polynomial with integer coefficients. If $n(P)$ is the number of distinct integers $k$ such that $(P(k){)}^2=1$, prove that $n(P)-\mathrm{deg}(P)\le 2$, where $\mathrm{deg}(P)$ denotes the degree of the polynomial $P$.

Topic: Algebra, Teoria dei Numeri Metodo: Fattorizzazione, Congruenze Abilita: Manipolazione algebrica, Ragionamento geometrico, Astrazione Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Bound on integers where polynomial squares equal 1

Let $P$ be a non-constant polynomial with integer coefficients. If $n(P)$ is the number of distinct integers $k$ such that $(P(k){)}^2=1$, prove that $n(P)-\mathrm{deg}(P)\le 2$, where $\mathrm{deg}(P)$ denotes the degree of the polynomial $P$.

src_imho_1974__Q06