International Mathematical Olympiad · 1969

Problems

Problem 1Prove that there are infinitely many natural numbers $a$ such that $n^4+a$ is not prime for every natural number $n$ .Solutions: 1
Problem 2Let $a_1,a_2,\ldots,a_n$ be real constants and let $x$ be real. Define $f(x)=\cos(a_1+x)+\frac12\cos(a_2+x)+\cdots+\frac1{2^{n-1}}\cos(a_n+x)$ . Given that $f(x_1)=f(x_2)=0$ , prove that $x_2-x_1=m\pi$ for some integer $m$ .Solutions: 1
Problem 3For each $k=1,2,3,4,5$ , find necessary and sufficient conditions on $a>0$ for the existence of a tetrahedron with $k$ edges of length $a$ and the remaining $6-k$ edges of length $1$ .Solutions: 1
Problem 4A semicircular arc $\gamma$ has diameter $AB$ . Let $C$ be a point of the arc other than $A,B$ , and let $D$ be the foot of the perpendicular from $C$ to $AB$ . Three circles $\gamma_1,\gamma_2,\gamma_3$ are tangent to line $AB$ ; $\gamma_1$ is inscribed in triangle $ABC$ , while $\gamma_2$ and $\gamma_3$ are tangent to $CD$ and to $\gamma$ , on opposite sides of $CD$ . Prove that the three circles have a second common tangent.Solutions: 1
Problem 5Given $n>4$ points in the plane, no three collinear, prove that at least $\binom{n-3}{2}$ convex quadrilaterals have their vertices among the given points.Solutions: 1
Problem 6For 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 $\frac{8}{(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le\frac1{x_1y_1-z_1^2}+\frac1{x_2y_2-z_2^2}$ . Give necessary and sufficient conditions for equality.Solutions: 1