IMO 1968 (X International Mathematical Olympiad)

Unique triangle with consecutive integer sides, one angle double another

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: Trigonometria, Casework Abilita: Manipolazione algebrica, Ragionamento geometrico, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Unique triangle with consecutive integer sides, one angle double another

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_imho_1968__Q01

Find all real x 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: Casework, Fattorizzazione Abilita: Casework accurato, Manipolazione algebrica, Lettura attenta Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Find the real x 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_imho_1968__Q02

System of equations: analyze solutions based on discriminant Delta

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

(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, Insiemi e funzioni Metodo: Invarianti, Casework Abilita: Manipolazione algebrica, Ragionamento geometrico, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

System of equations: analyze solutions based on discriminant Delta

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

(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_imho_1968__Q03

Tetrahedron with a vertex whose three edges are triangle sides

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: Estremalità, Casework Abilita: Ragionamento geometrico, Lettura attenta, Astrazione Area: Geometria Fonte: apri PDF p.1

Tetrahedron with a vertex whose three edges are triangle sides

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_imho_1968__Q04

Real-valued function satisfying f(x+a) = 1/2 + sqrt(f(x) - f(x)^2)

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: Insiemi e funzioni, Equazioni funzionali Metodo: Invarianti, Ricorsione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

*Real-valued function satisfying f(x+a) = 1/2 + sqrt(f(x) - f(x) ^2) *

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_imho_1968__Q05

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

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 $[x]$ denotes the greatest integer not exceeding $x$.)

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

(Evaluate infinite sums of floor n+2^k)/2^(k+1)) for the natural n

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 $[x]$ denotes the greatest integer not exceeding $x$.)

src_imho_1968__Q06