Eleventh International Olympiad, 1969 1969/1. Prove that there are infi nitely many natural numbers a with the following property: the number z = n4+ a is not prime for any natura1 number n. 1969/2. Let a1,a2,anbe real constants, x a real variable, and f(x)=cos(a1+ x) + 1 2 cos(a2+ x) + 1 4 cos(a3+ x) + + 1 2n1 cos(an+ x). Given that f(x1) = f(x2) = 0, prove that x2 x1= m for some integer m. 1969/3. For each value of k = 1,2,3,4,5, fi nd necessary and suffi cient 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. 1969/4. A semicircular arc is drawn on AB as diameter. C is a point on other than A and B, and D is the foot of the perpendicular from C to AB. We consider three circles, 1,2,3, all tangent to the line AB. Of these, 1is inscribed in ABC, while 2and 3are both tangent to CD and to , one on each side of CD. Prove that 1,2and 3have a second tangent in common. 1969/5. Given n 4 points in the plane such that no three are collinear. Prove that there are at least n3 2 convex quadrilaterals whose vertices are four of the given points. 1969/6. Prove that for all real numbers x1,x2,y1,y2,z1,z2, with x1 0, x2 0,x1y1 z2 1 0,x2y2 z2 2 0, the inequality 8 (x1+ x2)(y1+ y2) (z1+ z2)2 1 x1y1 z2 1 + 1 x2y2 z2 2 is satisfi ed. Give necessary and suffi cient conditions for equality. 。