IMO 1969 (Eleventh International Olympiad)

Infinitely many a making n^4+a never prime

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: manipolazione algebrica Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Infinitely many a making n^4+a never prime

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_imo_1969_all__Q01

Common zeros of weighted cosine sum differ by multiple of pi

Let $a_1,a_2,\dots ,a_n$ be real constants, $x$ be 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 $x_2-x_1$ is a rational multiple of $\pi$.

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

Common zeros of weighted cosines sum differ by multiple of pi

Let $a_1,a_2,\dots ,a_n$ be real constants, $x$ be 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 $x_2-x_1$ is a rational multiple of $\pi$.

src_imo_1969_all__Q02

Conditions on a for tetrahedron with k edges length a

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

Topic: Geometria solida Metodo: Analisi per casi Abilita: Casework accurato Area: Geometria Fonte: apri PDF p.1

Conditions on a for tetrahedron with k edges length a

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

src_imo_1969_all__Q03

Three circles tangent to AB share a second 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 midpoint of arc $AC$. Let $M$ be the foot of the perpendicular from $D$ to line $BC$. Prove that $BM=MA+MC$.

Topic: Geometria piana Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Three circles tangent to AB share a second 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 midpoint of arc $AC$. Let $M$ be the foot of the perpendicular from $D$ to line $BC$. Prove that $BM=MA+MC$.

src_imo_1969_all__Q04

At least (n-3 choose 2) convex quadrilaterals from n points

Given $n>4$ points in the plane such that no three are collinear. Prove that there are at least $\left(\genfrac{}{}{0ex}{}{n-3}{2}\right)$ convex quadrilaterals whose vertices are four of the given $n$ points.

Topic: Combinatoria, Geometria piana Metodo: Conteggio combinatorio Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

At least (n-3 choose 2) convex quadrilaterals from n points

Given $n>4$ points in the plane such that no three are collinear. Prove that there are at least $\left(\genfrac{}{}{0ex}{}{n-3}{2}\right)$ convex quadrilaterals whose vertices are four of the given $n$ points.

src_imo_1969_all__Q05

Prove quadratic-form inequality with 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 following inequality is satisfied: $\frac{8}{(x_1+x_2)(y_1+y_2)-(z_1+z_2{)}^2}\le \frac{1}{x_1{y}_1-z_1^2}+\frac{1}{x_2{y}_2-z_2^2}.$

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

Prove quadratic-form inequality with 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 following inequality is satisfied: $\frac{8}{(x_1+x_2)(y_1+y_2)-(z_1+z_2{)}^2}\le \frac{1}{x_1{y}_1-z_1^2}+\frac{1}{x_2{y}_2-z_2^2}.$

src_imo_1969_all__Q06