1967 IMO Problems
Problems of the 9th IMO 1967 in Yugoslavia.
Contents
[hide]Day I
Problem 1
Let be a parallelogram with side lengths
,
, and with
. If
is acute, prove that the four circles of radius
with centers
,
,
,
cover the parallelogram if and only if
Problem 2
Prove that if one and only one edge of a tetrahedron is greater than , then its volume is
.
Problem 3
Let ,
,
be natural numbers such that
is a prime greater than
. Let
. Prove that the product
is divisible by the product
.
Day II
Problem 4
Let and
be any two acute-angled triangles. Consider all triangles
that are similar to
(so that vertices
,
,
correspond to vertices
,
,
, respectively) and circumscribed about triangle
(where
lies on
,
on
, and
on
). Of all such possible triangles, determine the one with maximum area, and construct it.
Problem 5
Consider the sequence , where
in which
,
,
,
are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence
are equal to zero. Find all natural numbers
for which
.
Problem 6
In a sports contest, there were medals awarded on
successive days (
). On the first day, one medal and
of the remaining
medals were awarded. On the second day, two medals and
of the now remaining medals were awarded; and so on. On the
-th and last day, the remaining
medals were awarded. How many days did the contest last, and how many medals were awarded altogether?