IMO 1959

1st IMO 1959


A1. Prove that (21n+4)/(14n+3) is irreducible for every natural number n.
A2. For what real values of x is √(x + √(2x-1)) + √(x - √(2x-1)) = A given (a) A = √2, (b) A = 1, (c) A = 2, where only non-negative real numbers are allowed in square roots and the root always denotes the non-negative root?
A3. Let a, b, c be real numbers. Given the equation for cos x: a cos²x + b cos x + c = 0, form a quadratic equation in cos 2x whose roots are the same values of x. Compare the equations in cos x and cos 2x for a=4, b=2, c=-1.
B1. Given the length \|AC\|, construct a triangle ABC with ∠ABC = 90^o, and the median BM satisfying BM² = AB·BC.
B2. An arbitrary point M is taken in the interior of the segment AB. Squares AMCD and MBEF are constructed on the same side of AB. The circles circumscribed about these squares, with centers P and Q, intersect at M and N. (a) prove that AF and BC intersect at N; (b) prove that the lines MN pass through a fixed point S (independent of M); (c) find the locus of the midpoints of the segments PQ as M varies.
B3. The planes P and Q are not parallel. The point A lies in P but not Q, and the point C lies in Q but not P. Construct points B in P and D in Q such that the quadrilateral ABCD satisfies the following conditions: (1) it lies in a plane, (2) the vertices are in the order A, B, C, D, (3) it is an isosceles trapezoid with AB parallel to CD (meaning that AD = BC, but AD is not parallel to BC unless it is a square), and (4) a circle can be inscribed in ABCD touching the sides.