Difference between revisions of "1989 APMO Problems"
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[[1989 APMO Problems/Problem 5|Solution]] | [[1989 APMO Problems/Problem 5|Solution]] | ||
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* [[Asian Pacific Mathematics Olympiad]] | * [[Asian Pacific Mathematics Olympiad]] | ||
* [[APMO Problems and Solutions]] | * [[APMO Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] |
Revision as of 08:42, 30 May 2012
Problem 1
Let be positive real numbers, and let
.
Prove that
.
Problem 2
Prove that the equation
has no solutions in integers except .
Problem 3
Let be three points in the plane, and for convenience, let
,
. For
and
, suppose that
is the midpoint of
, and suppose that
is the midpoint of
. Suppose that
and
meet at
, and that
and
meet at
. Calculate the ratio of the area of triangle
to the area of triangle
.
Problem 4
Let be a set consisting of
pairs
of positive integers with the property that
. Show that there are at least
triples such that
,
, and
belong to
.
Problem 5
Determine all functions from the reals to the reals for which
is strictly increasing,
for all real
,
where is the composition inverse function to
. (Note:
and
are said to be composition inverses if
and
for all real
.)