IMO 1969 (XI International Mathematical Olympiad)

Infinitely many natural numbers with non-prime property

Prove that there are infinitely many natural numbers $a$ with the following property: the number $z=n^4+a$ is not prime for any natural number $n$.

Topic: Teoria dei Numeri Metodo: Fattorizzazione, Induzione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Ragionamento geometrico Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Infinitely many natural numbers with non-prime property

Prove that there are infinitely many natural numbers $a$ with the following property: the number $z=n^4+a$ is not prime for any natural number $n$.

src_imho_1969__Q01

Prove x1 minus x2 equals pi over m

Let $a_1,a_2,\dots ,a_n$ be real constants, $x$ a real variable, and $f(x)=\cos (a_1+x)+\frac{1}{2}\cos (a_2+x)+\frac{1}{4}\cos (a_3+x)+\cdots +\frac{1}{2^{n-1}}\cos (a_n+x).$ Given that $f(x_1)=f(x_2)=0$, prove that $x_2-x_1=m\pi$ for some integer $m$.

Topic: Trigonometria, Algebra Metodo: Trigonometria Abilita: Manipolazione algebrica, Lettura attenta, Ragionamento geometrico Area: Geometria, Algebra e Analisi Fonte: apri PDF p.1

Prove x1 minus x2 equals pi over m

Let $a_1,a_2,\dots ,a_n$ be real constants, $x$ a real variable, and $f(x)=\cos (a_1+x)+\frac{1}{2}\cos (a_2+x)+\frac{1}{4}\cos (a_3+x)+\cdots +\frac{1}{2^{n-1}}\cos (a_n+x).$ Given that $f(x_1)=f(x_2)=0$, prove that $x_2-x_1=m\pi$ for some integer $m$.

src_imho_1969__Q02

Necessary and sufficient conditions for tetrahedron with k equal edges

For each value of $k=1,2,3,4,5$, find necessary and sufficient conditions on the number $a>0$ so that there exists a tetrahedron with $k$ edges of length $a$ and the remaining $6-k$ edges of length 1.

Topic: Geometria solida Metodo: Casework, Casi e conteggio Abilita: Casework accurato, Ragionamento geometrico, Modellizzazione Area: Geometria Fonte: apri PDF p.1

Necessary and sufficient conditions for tetrahedron with k equal edges

For each value of $k=1,2,3,4,5$, find necessary and sufficient conditions on the number $a>0$ so that there exists a tetrahedron with $k$ edges of length $a$ and the remaining $6-k$ edges of length 1.

src_imho_1969__Q03

Three tangent circles inscribed in triangle, prove common tangent

A semicircular arc $\gamma$ is drawn on $AB$ as diameter. $C$ is a point on $\gamma$ other than $A$ and $B$, and $D$ is the foot of the perpendicular from $C$ to $AB$. We consider three circles ${\gamma }_1,{\gamma }_2,{\gamma }_3$, all tangent to the line $AB$. Of these, ${\gamma }_1$ is inscribed in $△ABC$, while ${\gamma }_2$ and ${\gamma }_3$ are both tangent to $CD$ and to $\gamma$, one on each side of $CD$. Prove that ${\gamma }_1,{\gamma }_2$, and ${\gamma }_3$ have a second tangent in common.

Topic: Geometria piana Metodo: Trigonometria, Coordinate Abilita: Ragionamento geometrico, Modellizzazione, Manipolazione algebrica Area: Geometria Fonte: apri PDF p.1

Three tangent circles inscribed in triangles, common tangent proofs

A semicircular arc $\gamma$ is drawn on $AB$ as diameter. $C$ is a point on $\gamma$ other than $A$ and $B$, and $D$ is the foot of the perpendicular from $C$ to $AB$. We consider three circles ${\gamma }_1,{\gamma }_2,{\gamma }_3$, all tangent to the line $AB$. Of these, ${\gamma }_1$ is inscribed in $△ABC$, while ${\gamma }_2$ and ${\gamma }_3$ are both tangent to $CD$ and to $\gamma$, one on each side of $CD$. Prove that${\gamma }_1,{\gamma }_2$, and${\gamma }_3$have a second tangent in common.

src_imho_1969__Q04

n>4 points, no three collinear, at least C(n,2) convex quadrilaterals

Given $n>4$ points in the plane such that no three are collinear. Prove that there are at least $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ convex quadrilaterals whose vertices are four of the given points.

Topic: Combinatoria, Geometria piana Metodo: Conteggio, Doppio conteggio, Estremalità Abilita: Conteggio sistematico, Ragionamento geometrico, Astrazione Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

n>4 points, not three hill, at least C(n,2) convex quadrilaterals

Given $n>4$ points in the plane such that no three are collinear. Prove that there are at least $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ convex quadrilaterals whose vertices are four of the given points.

src_imho_1969__Q05

Inequality for real numbers, find equality conditions

Prove that for all real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ with $x_1>0$, $x_2>0$, $x_1{y}_1-z_1^2>0$, $x_2{y}_2-z_2^2>0$, the inequality $\frac{8}{(x_1+x_2)(y_1+y_2)-(z_1+z_2{)}^2}\le \frac{1}{x_1{y}_1-z_1^2}+\frac{1}{x_2{y}_2-z_2^2}$ is satisfied. Give necessary and sufficient conditions for equality.

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

Inequality for real numbers, find equality conditions

Prove that for all real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ with $x_1>0$, $x_2>0$, $x_1{y}_1-z_1^2>0$, $x_2{y}_2-z_2^2>0$, the inequality $\frac{8}{(x_1+x_2)(y_1+y_2)-(z_1+z_2{)}^2}\le \frac{1}{x_1{y}_1-z_1^2}+\frac{1}{x_2{y}_2-z_2^2}$ is satisfied. Give necessary and sufficient conditions for equality.

src_imho_1969__Q06