IMO 1968 (Tenth International Olympiad)

Unique triangle with consecutive integer sides and angle double

Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

Topic: Geometria piana, Trigonometria Metodo: congruenze, Tecniche trigonometriche Area: Geometria Fonte: apri PDF p.1

Unique triangle with consecutive integer sides and angle double

Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

src_imo_1968_all__Q01

Find naturals where digit product equals x^2-10x-22

Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2-10x-22$.

Topic: Teoria dei Numeri Metodo: Analisi per casi, congruenze Abilita: Casework accurato Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

*Find naturals where digit product equals x^2-10x-22 *

Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2-10x-22$.

src_imo_1968_all__Q02

Number of solutions of cyclic quadratic system via discriminant

Consider the system of equations $a{x}_1^2+b{x}_1+c=x_2$ $a{x}_2^2+b{x}_2+c=x_3$ $\cdots$ $a{x}_{n-1}^2+b{x}_{n-1}+c=x_n$ $a{x}_n^2+b{x}_n+c=x_1,$ with unknowns $x_1,x_2,\dots ,x_n$, where $a,b,c$ are real and $a\ne 0$. Let $\Delta =(b-1{)}^2-4ac$. Prove that for this system

(a) if $\Delta <0$, there is no solution,

(b) if $\Delta =0$, there is exactly one solution,

(c) if $\Delta >0$, there is more than one solution.

Topic: Algebra Metodo: Analisi per casi, Sfruttamento della simmetria Abilita: Casework accurato Area: Algebra e Analisi Fonte: apri PDF p.1

Number of solutions of cyclic quadratic system via discriminant

Consider the system of equations $a{x}_1^2+b{x}_1+c=x_2$ $a{x}_2^2+b{x}_2+c=x_3$ $\cdots$ $a{x}_{n-1}^2+b{x}_{n-1}+c=x_n$ $a{x}_n^2+b{x}_n+c=x_1,$ with unknowns $x_1,x_2,\dots ,x_n$, where $a,b,c$ are real and $a\ne 0$. Let $\Delta =(b-1{)}^2-4ac$. Prove that for this system

(a) if $\Delta <0$, there is no solution,

(b) if $\Delta =0$, there is exactly one solution,

(c) if $\Delta >0$, there is more than one solution.

src_imo_1968_all__Q03

Every tetrahedron has a vertex whose edges form a triangle

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

Topic: Geometria solida Metodo: Principio di estremalita Area: Geometria Fonte: apri PDF p.1

Every tetrahedron has a vertex whose edges form a triangle

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle.

src_imo_1968_all__Q04

Prove functional equation implies f periodic; give example

Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$, the equation $f(x+a)=\frac{1}{2}+\sqrt{f(x)-[f(x){]}^2}$ holds for all $x$.

(a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that $f(x+b)=f(x)$ for all $x$).

(b) For $a=1$, give an example of a non-constant function with the required properties.

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

Prove functional equation implies f periodic; give example

Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$, the equation $f(x+a)=\frac{1}{2}+\sqrt{f(x)-[f(x){]}^2}$ holds for all $x$.

(a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that $f(x+b)=f(x)$ for all $x$).

For $a=1$, give an example of a non-constant function with the required properties.

src_imo_1968_all__Q05

Evaluate infinite sum of floor((n+2^k)/2^(k+1))

For every natural number $n$, evaluate the sum ${\sum }_{k=0}^{\infty }\lfloor \frac{n+2^k}{2^{k+1}}\rfloor =\lfloor \frac{n+1}{2}\rfloor +\lfloor \frac{n+2}{4}\rfloor +\cdots +\lfloor \frac{n+2^k}{2^{k+1}}\rfloor +\cdots$ (The symbol $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.)

Topic: Teoria dei Numeri Metodo: Somme telescopiche Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.2

*Evaluate infinite sums of floor(((n+2^k)/2^(k+1)) *

For every natural number $n$, evaluate the sum ${\sum }_{k=0}^{\infty }\lfloor \frac{n+2^k}{2^{k+1}}\rfloor =\lfloor \frac{n+1}{2}\rfloor +\lfloor \frac{n+2}{4}\rfloor +\cdots +\lfloor \frac{n+2^k}{2^{k+1}}\rfloor +\cdots$ (The symbol $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.)

src_imo_1968_all__Q06