1959 IMO Problems
Problems of the 1st IMO 1959 in Romania.
Contents
Day I
Problem 1
Prove that is irreducible for every natural number .
Problem 2
For what real values of is
given (a) , (b) , (c) , where only non-negative real numbers are admitted for square roots?
Problem 3
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 .
Day II
Problem 4
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.
Problem 5
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 .
Problem 6
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.
See Also
1959 IMO (Problems) • Resources) | ||
Preceded by First IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1960 IMO |
All IMO Problems and Solutions |