International Mathematical Olympiad · 1959

Problems

Problem 1Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$ .Solutions: 1
Problem 2For what real values of $x$ is $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A$ , given (a) $A=\sqrt2$ , (b) $A=1$ , (c) $A=2$ , where square roots are non-negative?Solutions: 1
Problem 3Let $a,b,c$ be real numbers. Consider $a\cos^2x+b\cos x+c=0$ . Using $a,b,c$ , form a quadratic equation in $\cos2x$ whose roots are the same values of $x$ . Compare the equations for $a=4$ , $b=2$ , $c=-1$ .Solutions: 1
Problem 4Construct a right triangle with a given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs.Solutions: 1
Problem 5An arbitrary point $M$ is selected in the interior of segment $AB$ . Squares $AMCD$ and $MBEF$ are constructed on the same side of $AB$ , with circumcenters $P$ and $Q$ . Their circumcircles meet again at $N$ . Let $N'$ be the intersection of $AF$ and $BC$ . (a) Prove $N=N'$ . (b) Prove that $MN$ passes through a fixed point independent of $M$ . (c) Find the locus of the midpoint of $PQ$ as $M$ varies.Solutions: 1
Problem 6Two planes $P$ and $Q$ intersect along line $p$ . Points $A\in P$ and $C\in Q$ are not on $p$ . Construct an isosceles trapezoid $ABCD$ with $AB\parallel DC$ , an incircle, and vertices $B\in P$ , $D\in Q$ .Solutions: 1