IMO 1976 (Eighteenth International Olympiad)

find other diagonal lengths

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 Metodo: Tecniche trigonometriche Area: Geometria Fonte: apri PDF p.1

find other diagonal lengths

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_imo_1976_all__Q01

Iterated polynomial P_n(x)=x has real distinct roots

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 Metodo: Induzione, Ricorsione Area: Algebra e Analisi Fonte: apri PDF p.1

Iterated polynomial P_n(x)=x has real distinct roots

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_imo_1976_all__Q02

Box dimensions where volume-2 cubes fill exactly 40%

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: Analisi per casi Area: Aritmetica e Teoria dei Numeri, Geometria Fonte: apri PDF p.1

Box dimensions where volume-2 cubes fill exactly 40%

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_imo_1976_all__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: Algebra Metodo: Disuguaglianze classiche Area: 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_imo_1976_all__Q04

System with +-1,0 coefficients has bounded nonzero 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 Metodo: Principio dei cassetti Area: Algebra e Analisi Fonte: apri PDF p.1

System with +-1,0 coefficients has bounded nonzero 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_imo_1976_all__Q05

Prove closed form for floor of recurrence sequence u_n

A 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: successioni Metodo: Induzione, Ricorsione Area: Algebra e Analisi Fonte: apri PDF p.1

Prove closed form for floor of recurrence sequence u_n

A 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_imo_1976_all__Q06