IMO 1972 (Fourteenth International Olympiad)

Ten two-digit numbers have two disjoint equal-sum subsets

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 Metodo: Principio dei cassetti Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Ten two-digit numbers have two disjoint equal-sum subsets

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_imo_1972_all__Q01

Dissect cyclic quadrilateral into n cyclic 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 Area: Geometria Fonte: apri PDF p.1

Dissect cyclic quadrilateral into n cyclic 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_imo_1972_all__Q02

Prove (2m)!(2n)!/(m!n!(m+n)!) 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 Metodo: Conteggio combinatorio Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Prove (2m)!(2n)!/(m!n!(m+n)!) 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_imo_1972_all__Q03

Solve cyclic system of product inequalities, positive reals

Find all solutions $(x_1,x_2,x_3,x_4,x_5)$ of the system of inequalities

(x_1^2 - x_3 x_5)(x_2^2 - x_3 x_5) \leq 0 \\ (x_2^2 - x_4 x_1)(x_3^2 - x_4 x_1) \leq 0 \\ (x_3^2 - x_5 x_2)(x_4^2 - x_5 x_2) \leq 0 \\ (x_4^2 - x_1 x_3)(x_5^2 - x_1 x_3) \leq 0 \\ (x_5^2 - x_2 x_4)(x_1^2 - x_2 x_4) \leq 0 \end{cases}$$ where $x_1, x_2, x_3, x_4, x_5$ are positive real numbers.

Topic: Algebra Metodo: Disuguaglianze classiche Area: Algebra e Analisi Fonte: apri PDF p.1

Solve cyclic system of product inequalities, positive reals

Find all solutions $(x_1,x_2,x_3,x_4,x_5)$ of the system of inequalities

$\{\begin{array}{l}(x_1^2-x_3{x}_5)(x_2^2-x_3{x}_5)\le 0\\ (x_2^2-x_4{x}_1)(x_3^2-x_4{x}_1)\le 0\\ (x_3^2-x_5{x}_2)(x_4^2-x_5{x}_2)\le 0\\ (x_4^2-x_1{x}_3)(x_5^2-x_1{x}_3)\le 0\\ (x_5^2-x_2{x}_4)(x_1^2-x_2{x}_4)\le 0\end{array}$

where $x_1,x_2,x_3,x_4,x_5$ are positive real numbers.

src_imo_1972_all__Q04

Bounded functional equation implies |g(y)|<=1

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: successioni Area: Algebra e Analisi Fonte: apri PDF p.1

So, what does this mean?

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_imo_1972_all__Q05

Regular tetrahedron with vertex on each of four parallel planes

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

Topic: Geometria solida Metodo: Metodo delle coordinate Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Regular tetrahedron with vertex on each of four parallel planes

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

src_imo_1972_all__Q06