IMO 1959

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

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?

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 $a=4$ , $b=2$ , $c=-1$ .

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.

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 $B$ .

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.