IMO 1969

Problem 1. Prove that there are infinitely many natural numbers $a$ with the following property: the number $z=n^4+a$ is not prime for any natura1 number $n$.

Problem 2. Let $a_1,a_2,\cdots ,a_n$ be real constants, $x$ a real variable, and

Given that $f(x_1)=f(x_2)=0$, prove that $x_2-x_1=m\pi$ for some integer $m$.

Problem 3. 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 $1$.

Problem 4. 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 $\Delta ABC$, 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.

Problem 5. 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 points.

Problem 6. 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

is satisfied. Give necessary and sufficient conditions for equality.