IMO 1969 — Official Contest Problems

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

Let $a_1,a_2,\ldots,a_n$ be real constants and
$$y(x)=\cos(a_1+x)+\frac{\cos(a_2+x)}2+\frac{\cos(a_3+x)}{2^2}+\cdots+\frac{\cos(a_n+x)}{2^{n-1}}.$$
If $x_1,x_2$ are real and $y(x_1)=y(x_2)=0$, prove that $x_1-x_2=m\pi$ for some integer $m$.

Find conditions on the positive real number $a$ such that there exists a tetrahedron $k$ of whose edges $(k=1,2,3,4,5)$ have length $a$, and the other $6-k$ edges have length $1$.

Let $AB$ be a diameter of a circle $\gamma$. A point $C$ different from $A$ and $B$ is on the circle $\gamma$. Let $D$ be the projection of the point $C$ onto the line $AB$. Consider three other circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ with the common tangent $AB$: $\gamma_1$ inscribed in the triangle $ABC$, and $\gamma_2$ and $\gamma_3$ tangent to both (the segment) $CD$ and $\gamma$. Prove that $\gamma_1$, $\gamma_2$, and $\gamma_3$ have two common tangents.

Given $n$ points in the plane such that no three of them are collinear, prove that one can find at least $\binom{n-3}{2}$ convex quadrilaterals with their vertices at these points.

Under the conditions $x_1,x_2>0$, $x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove the inequality
$$\frac8{(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le\frac1{x_1y_1-z_1^2}+\frac1{x_2y_2-z_2^2}.$$