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 found 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 |