1959 IMO Problems
Problems of the 1st IMO 1959 in Romania.
Prove that is irreducible for every natural number .
For what real values of is
given (a) , (b) , (c) , where only non-negative real numbers are admitted for square roots?
Let be real numbers. Consider the quadratic equation in :
Using the numbers , form a quadratic equation in , whose roots are the same as those of the original equation. Compare the equations in and for .
Construct a right triangle with a given hypotenuse such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.
An arbitrary point is selected in the interior of the segment . The squares and are constructed on the same side of , with the segments and as their respective bases. The circles about these squares, with respective centers and , intersect at and also at another point . Let denote the point of intersection of the straight lines and .
(a) Prove that the points and coincide.
(b) Prove that the straight lines pass through a fixed point independent of the choice of .
(c) Find the locus of the midpoints of the segments as varies between and .
Two planes, and , intersect along the line . The point is in the plane , and the point is in the plane ; neither of these points lies on the straight line . Construct an isosceles trapezoid (with parallel to ) in which a circle can be constructed, and with vertices and lying in the planes and , respectively.
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