Difference between revisions of "1967 IMO Problems"
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[[1967 IMO Problems/Problem 6|Solution]] | [[1967 IMO Problems/Problem 6|Solution]] | ||
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+ | * [[1967 IMO]] | ||
+ | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1967 IMO 1967 Problems on the Resources page] | ||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] | ||
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+ | {{IMO box|year=1967|before=[[1966 IMO]]|after=[[1968 IMO]]}} |
Latest revision as of 11:40, 29 January 2021
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?
- 1967 IMO
- IMO 1967 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1967 IMO (Problems) • Resources | ||
Preceded by 1966 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1968 IMO |
All IMO Problems and Solutions |