IMO 1976 (XVIII International Mathematical Olympiad)

Convex quadrilateral diagonals and sides sum conditions

In a plane convex quadrilateral of area 32, the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths of the other diagonal.

Topic: Geometria piana, Disuguaglianze Metodo: Estremalità, Disuguaglianze Abilita: Ragionamento geometrico, Manipolazione algebrica, Lettura attenta Area: Geometria, Algebra e Analisi Fonte: apri PDF p.1

Convex quadrilateral diagonals and sides sum conditions

In a convex quadrilateral plane of area 32, the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths of the other diagonal.

src_imho_1976__Q01

Iterated polynomial roots all real and distinct

Let $P_1(x)=x^2-2$ and $P_j(x)=P_1(P_{j-1}(x))$ for $j=2,3,\dots$ Show that, for any positive integer $n$, the roots of the equation $P_n(x)=x$ are real and distinct.

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

Iterated polynomial roots to real and distinct

Let $P_1(x)=x^2-2$ and $P_j(x)=P_1(P_{j-1}(x))$ for $j=2,3,\dots$ Show that, for any positive integer $n$, the roots of the equation $P_n(x)=x$ are real and distinct.

src_imho_1976__Q02

Rectangular box filled with unit cubes, 40% of edges parallel

A rectangular box can be filled completely with unit cubes. If one places as many cubes as possible, each with volume 2, in the box, so that their edges are parallel to the edges of the box, one can fill exactly 40% of the box. Determine the possible dimensions of all such boxes.

Topic: Geometria solida, Teoria dei Numeri Metodo: Casework, Disuguaglianze Abilita: Modellizzazione, Manipolazione algebrica, Lettura attenta, Conteggio sistematico Area: Geometria, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Rectangular box filled with unit cubes, 40% of edges parallel

A rectangular box can be filled completely with unit cubes. If one places as many cubes as possible, each with volume 2, in the box, so that their edges are parallel to the edges of the box, one can fill exactly 40% of the box. Determine the possible dimensions of all such boxes.

src_imho_1976__Q03

Largest product of positive integers summing to 1976

Determine, with proof, the largest number which is the product of positive integers whose sum is 1976.

Topic: Teoria dei Numeri, Disuguaglianze Metodo: Estremalità, Disuguaglianze, Casework Abilita: Manipolazione algebrica, Ragionamento geometrico, Stima, Lettura attenta Area: Aritmetica e Teoria dei Numeri, Algebra e Analisi Fonte: apri PDF p.1

Largest product of positive integers summing to 1976

Determine, with proof, the largest number which is the product of positive integers whose sum is 1976.

src_imho_1976__Q04

Linear system with coefficients in {-1,0,1} has integer solution

Consider the system of $p$ equations in $q=2p$ unknowns $x_1,x_2,\dots ,x_q$: $a_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1q}{x}_q=0$ $a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2q}{x}_q=0$ $⋮$ $a_{p1}{x}_1+a_{p2}{x}_2+\cdots +a_{pq}{x}_q=0$ with every coefficient $a_{ij}$ member of the set $\{-1,0,1\}$. Prove that the system has a solution $(x_1,x_2,\dots ,x_q)$ such that (a) all $x_j$ ($j=1,2,\dots ,q$) are integers, (b) there is at least one value of $j$ for which $x_j\ne 0$, (c) $|x_j|\le q$ ($j=1,2,\dots ,q$).

Topic: Algebra, Combinatoria, Teoria dei Numeri Metodo: Principio dei cassetti, Conteggio, Induzione Abilita: Modellizzazione, Manipolazione algebrica, Conteggio sistematico, Lettura attenta Area: Algebra e Analisi, Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Linear system with coefficients in {-1,0,1} has integer solution

Consider the system of $p$ equations in $q=2p$ unknowns $x_1,x_2,\dots ,x_q$: $a_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1q}{x}_q=0$ $a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2q}{x}_q=0$ $⋮$ $a_{p1}{x}_1+a_{p2}{x}_2+\cdots +a_{pq}{x}_q=0$ with every coefficient $a_{ij}$ member of the set $\{-1,0,1\}$. Prove that the system has a solution $(x_1,x_2,\dots ,x_q)$ such that (a) all $x_j$ ($j=1,2,\dots ,q$) are integers, (b) there is at least one value of $j$ for which $x_j\ne 0$, (c) $|x_j|\le q$ ($j=1,2,\dots ,q$).

src_imho_1976__Q05

Sequence u_n defined by recurrence; prove floor equals 2^{…}

The sequence $\{u_n\}$ is defined by $u_0=2,u_1=\frac{5}{2},u_{n+1}=u_n(u_{n-1}^2-2)-u_1\text{for }n=1,2,\dots$ Prove that for positive integers $n$, $[u_n]=2^{[2^n-(-1{)}^n]/3}$ where $[x]$ denotes the greatest integer $\le x$.

Topic: Algebra, Teoria dei Numeri Metodo: Induzione, Ricorsione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Ragionamento geometrico, Lettura attenta Area: Algebra e Analisi, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Sequence u_n defined by recurrence; proof floor equals 2^{…}

The sequence $\{u_n\}$ is defined by $u_0=2,u_1=\frac{5}{2},u_{n+1}=u_n(u_{n-1}^2-2)-u_1\text{for }n=1,2,\dots$ Prove that for positive integers $n$, $[u_n]=2^{[2^n-(-1{)}^n]/3}$ where $[x]$ denotes the greatest integer $\le x$.

src_imho_1976__Q06