Difference between revisions of "1979 IMO Problems/Problem 5"
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==Problem== | ==Problem== | ||
− | Determine all real numbers a for which there exists | + | Determine all real numbers a for which there exists non-negative reals <math>x_{1}, \ldots, x_{5}</math> which satisfy the relations <math> \sum_{k=1}^{5} kx_{k}=a,</math> <math> \sum_{k=1}^{5} k^{3}x_{k}=a^{2},</math> <math> \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.</math> |
==Solution== | ==Solution== |
Revision as of 12:55, 30 November 2021
Problem
Determine all real numbers a for which there exists non-negative reals which satisfy the relations
Solution
Discussion thread can be found here: [1]
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See Also
1979 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |
Let , and . For all pairs , let Then we have on one hand Therefore \(1) and on the other hand \ (2) Then from (1) we have and from (2) so Besides we also have from (1) and from (2) and for where in the right hand we have that , so , and , so for From the latter and (2) we also have So we have that
If , take , for . Then , , and