1971 IMO Problems/Problem 1
Problem
Prove that the following assertion is true for and
, and that it is false for every other natural number
If are arbitrary real numbers, then
Solution
Take , and the remaining
. Then
for
even,
so the proposition is false for even
.
Suppose and odd. Take any
, and let
,
,
and
.
Then
.
So the proposition is false for odd
.
Assume .
Then in
the sum of the first two terms is non-negative, because
.
The last term is also non-negative.
Hence
, and the proposition is true for
.
It remains to prove .
Suppose
.
Then the sum of the first two terms in
is
.
The third term is non-negative (the first two factors are non-positive and the last two non-negative).
The sum of the last two terms is:
.
Hence
.
This solution was posted and copyrighted by e.lopes. The original thread can be fond here: [1]
See Also
1971 IMO (Problems) • Resources | ||
Preceded by First Question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |