IMO 1969

Prove that there are infinitely many positive integers $m$ , such that $n^4+m$ is not prime for any positive integer $n$ .

Let $f(x)=\cos(a_1+x)+{1\over2}\cos(a_2+x)+{1\over4}\cos(a_3+x)+\ldots+{1\over2^{n-1}}\cos(a_n+x)$ , where $a_i$ are real constants and $x$ is a real variable. If $f(x_1)=f(x_2)=0$ , prove that $x_1-x_2$ is a multiple of $\pi$ .

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

$C$ is a point on the semicircle diameter $AB$ , between $A$ and $B$ . $D$ is the foot of the perpendicular from $C$ to $AB$ . The circle $K_1$ is the incircle of $ABC$ , the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$ .

Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.

Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$ , and $x_2y_2>z_2^2$ , prove that: $${8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}.$$ Give necessary and sufficient conditions for equality.