Difference between revisions of "1983 IMO Problems"
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== See Also == | == See Also == | ||
{{IMO box|year=1983|before=[[1982 IMO]]|after=[[1984 IMO]]}} | {{IMO box|year=1983|before=[[1982 IMO]]|after=[[1984 IMO]]}} | ||
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Latest revision as of 13:47, 14 October 2024
Problem 1
Find all functions defined on the set of positive reals which take positive real values and satisfy:
for all
; and
as
.
Problem 2
Let be one of the two distinct points of intersection of two unequal coplanar circles
and
with centers
and
respectively. One of the common tangents to the circles touches
at
and
at
, while the other touches
at
and
at
. Let
be the midpoint of
and
the midpoint of
. Prove that
.
Problem 3
Let and
be positive integers, no two of which have a common divisor greater than
. Show that
is the largest integer which cannot be expressed in the form
, where
are non-negative integers.
Problem 4
Let be an equilateral triangle and
the set of all points contained in the three segments
,
, and
(including
,
, and
). Determine whether, for every partition of
into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.
Problem 5
Is it possible to choose distinct positive integers, all less than or equal to
, no three of which are consecutive terms of an arithmetic progression?
Problem 6
Let ,
and
be the lengths of the sides of a triangle. Prove that
Determine when equality occurs.
See Also
1983 IMO (Problems) • Resources | ||
Preceded by 1982 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1984 IMO |
All IMO Problems and Solutions |