# 1959_eng - scribed about these squares with centers P and Q...

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First International Olympiad, 1959 1959/1. Prove that the fraction 21 n +4 14 n +3 is irreducible for every natural number n. 1959/2. For what real values of x is q ( x + 2 x - 1) + q ( x - 2 x - 1) = A, given (a) A = 2 , (b) A = 1, (c) A = 2 , where only non-negative real numbers are admitted for square roots? 1959/3. 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 2 x , whose roots are the same as those of the original equation. Compare the equations in cos x and cos 2 x for a = 4 ,b = 2 ,c = - 1 . 1959/4. 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. 1959/5. 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 circum-

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Unformatted text preview: scribed 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 indepen-dent of the choice of M. (c) Find the locus of the midpoints of the segments PQ as M varies between A and B. 1959/6. 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....
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1959_eng - scribed about these squares with centers P and Q...

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