1960 IMO Problems
Problems of the 2nd IMO 1960 Romania.
Contents
Day I
Problem 1
Determine all three-digit numbers having the property that
is divisible by 11, and
is equal to the sum of the squares of the digits of
.
Problem 2
For what values of the variable does the following inequality hold:
Problem 3
In a given right triangle , the hypotenuse
, of length
, is divided into
equal parts (
and odd integer). Let
be the acute angle subtending, from
, that segment which contains the midpoint of the hypotenuse. Let
be the length of the altitude to the hypotenuse of the triangle. Prove that:
![$\tan{\alpha}=\frac{4nh}{(n^2-1)a}.$](http://latex.artofproblemsolving.com/d/7/6/d7667ee6a4058553b8290f72ffe77f099ab473bf.png)
Day II
Problem 4
Construct triangle , given
,
(the altitudes from
and
), and
, the median from vertex
.
Problem 5
Consider the cube (with face
directly above face
).
a) Find the locus of the midpoints of the segments , where
is any point of
and
is any point of
;
b) Find the locus of points which lie on the segment
of part a) with
.
Problem 6
Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let be the volume of the cone and
be the volume of the cylinder.
a) Prove that ;
b) Find the smallest number for which
; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.