Question bank IMO 1959

(POL) For every integer $n$ prove that the fraction $\frac{21n+4}{14n+3}$ cannot be reduced any further.

(ROM) For which real numbers $x$ do the following equations hold:

(a) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x+\sqrt{2x-1}}=\sqrt2$,

(b) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x+\sqrt{2x-1}}=1$,

(c) $\sqrt{x+\sqrt{2x-1}}+\sqrt{x+\sqrt{2x-1}}=2$?

(HUN) Let $x$ be an angle and let the real numbers $a,b,c,\cos x$ satisfy the following equation:
$$a\cos^2x+b\cos x+c=0.$$
Write the analogous quadratic equation for $a,b,c,\cos2x$. Compare the given and the obtained equality for $a=4,b=2,c=-1$.

(HUN) Construct a right-angled triangle whose hypotenuse $c$ is given if it is known that the median from the right angle equals the geometric mean of the remaining two sides of the triangle.

(ROM) A segment $AB$ is given and on it a point $M$. On the same side of $AB$ squares $AMCD$ and $BMFE$ are constructed. The circumcircles of the two squares, whose centers are $P$ and $Q$, intersect in $M$ and another point $N$.

(a) Prove that lines $FA$ and $BC$ intersect at $N$.

(b) Prove that all such constructed lines $MN$ pass through the same point $S$, regardless of the selection of $M$.

(c) Find the locus of the midpoints of all segments $PQ$, as $M$ varies along the segment $AB$.

(CZS) Let $\alpha$ and $\beta$ be two planes intersecting at a line $p$. In $\alpha$ a point $A$ is given and in $\beta$ a point $C$ is given, neither of which lies on $p$. Construct $B$ in $\alpha$ and $D$ in $\beta$ such that $ABCD$ is an equilateral trapezoid, $AB\parallel CD$, in which a circle can be inscribed.