IMO Problems and Solutions 1959

Problem 1: Prove that the fraction  is irreducible for every natural number n:

Problem 2:

For what real values of x is

given (a) A = √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:

Using the number, a, b, c forms a quadratic equation in cos2x, whose roots are the same as those of the original equation. Compare the equations 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 squares AMCD and MBEF are constructed on the same side of AB; with the 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 the points 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 to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively.