International Mathematical Olympiad  – IMO 1959  Problems

Problem 1

Prove that the fraction $ \dfrac{21n + 4}{14n + 3}$ is irreducible for every natural number $ n$.

Problem 2

For what real values of $x$ is\[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \]given

a) $A=\sqrt{2}$;

b) $A=1$;

c) $A=2$,

where only non-negative real numbers are admitted for square roots?

Problem 3

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$\[ a \cos^2{x}+b \cos{x}+c=0. \]Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$..

Problem 4

Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

Problem 5

An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N’$ denote the point of intersection of the straight lines $AF$ and $BC$.

a) Prove that $N$ and $N’$ coincide;

b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$;

c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$

Problem 6

Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.