Difference between revisions of "1989 APMO Problems"
(New page: == Problem 1 == Let <math>x_1, x_2, x_3, \dots , x_n</math> be positive real numbers, and let <cmath>S=x_1+x_2+x_3+\cdots +x_n</cmath>. Prove that <cmath>(1+x_1)(1+x_2)(1+x_3)\cdots (1+...) |
(→See also) |
||
Line 45: | Line 45: | ||
== See also == | == See also == | ||
− | * [[Asian Pacific | + | * [[Asian Pacific Mathematics Olympiad]] |
* [[APMO Problems and Solutions]] | * [[APMO Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] |
Revision as of 13:19, 14 April 2008
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
.)