IMO 1972 (XIV International Mathematical Olympiad)

Two disjoint subsets of ten two-digit numbers with equal sums

Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.

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

Two disjoint subsets of ten two-digit numbers with equal sums

Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.

src_imho_1972__Q01

Quadrilateral inscribed in circle can be dissected into inscribable quadrilaterals

Prove that if $n\ge 4$, every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle.

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

Quadrilateral inscribed in circle can be dissected into inscribable quadrilaterals

Prove that if $n\ge 4$, every quadrilateral that can be inscribed in a circle can be dissected into $n$ quadrilaterals each of which is inscribable in a circle.

src_imho_1972__Q02

Prove expression involving binomial coefficients is an integer

Let $m$ and $n$ be arbitrary non-negative integers. Prove that $\frac{(2m)!(2n)!}{m!n!(m+n)!}$ is an integer. ($0!=1$.)

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

Prove expression involving binomial coefficients is an integer

Let $m$ and $n$ be arbitrary non-negative integers. Prove that $\frac{(2m)!(2n)!}{m!n!(m+n)!}$ is an integer. ($0!=1$.)

src_imho_1972__Q03

Find all solutions of a system of four inequalities

Find all solutions $(x_1,x_2,x_3,x_4)$ of the system of inequalities

(x_1^2 - x_3 x_1)(x_2^2 - x_3 x_2) &\le 0 \\ (x_2^2 - x_4 x_2)(x_3^2 - x_4 x_3) &\le 0 \\ (x_3^2 - x_1 x_3)(x_4^2 - x_1 x_4) &\le 0 \\ (x_4^2 - x_2 x_4)(x_1^2 - x_2 x_1) &\le 0 \end{aligned}$$ where $x_1, x_2, x_3, x_4$ are positive real numbers.

Topic: Algebra, Disuguaglianze Metodo: Casework, Disuguaglianze, Simmetria Abilita: Manipolazione algebrica, Lettura attenta, Casework accurato Area: Algebra e Analisi Fonte: apri PDF p.1

Find all solutions of a system of four inequalities

Find all solutions $(x_1,x_2,x_3,x_4)$ of the system of inequalities $\begin{array}{rl}(x_1^2-x_3{x}_1)(x_2^2-x_3{x}_2)& \le 0\\ (x_2^2-x_4{x}_2)(x_3^2-x_4{x}_3)& \le 0\\ (x_3^2-x_1{x}_3)(x_4^2-x_1{x}_4)& \le 0\\ (x_4^2-x_2{x}_4)(x_1^2-x_2{x}_1)& \le 0\end{array}$ where $x_1,x_2,x_3,x_4$ are positive real numbers.

src_imho_1972__Q04

Functional equation f(x+y)+f(x-y)=2f(x)g(y) with bounds

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $f(x+y)+f(x-y)=2f(x)g(y)$ for all $x,y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)|\le 1$ for all $x$, then $|g(y)|\le 1$ for all $y$.

Topic: Equazioni funzionali, Algebra Metodo: Backward, Disuguaglianze Abilita: Manipolazione algebrica, Astrazione, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Functional equation f(x+y)+f(x-y)=2f(x)g(y) with bounds

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $f(x+y)+f(x-y)=2f(x)g(y)$ for all $x,y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)|\le 1$ for all $x$, then $|g(y)|\le 1$ for all $y$.

src_imho_1972__Q05

Four distinct parallel planes imply a regular tetrahedron with vertex on each

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

Topic: Geometria solida Metodo: Estremalità, Simmetria Abilita: Ragionamento geometrico, Astrazione, Modellizzazione Area: Geometria Fonte: apri PDF p.1

Four distinct parallel planes imply a regular tetrahedron with vertex on each

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

src_imho_1972__Q06