1959 IMO Problems
Problems of the 1st IMO 1959 in Romania.
Contents
[hide]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 |