Difference between revisions of "1979 IMO Problems/Problem 5"
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== See Also == {{IMO box|year=1979|num-b=4|num-a=6}} | == See Also == {{IMO box|year=1979|num-b=4|num-a=6}} | ||
Let <math>\Sigma_1= \sum_{k=1}^{5} kx_{k}</math>, <math>\Sigma_2=\sum_{k=1}^{5} k^{3}x_{k}</math> and <math>\Sigma_3=\sum_{k=1}^{5} k^{5}x_{k}</math>. For all pairs <math>i,j\in \mathbb{Z}</math>, let <cmath>\Sigma(i,j)=i^2j^2\Sigma_1-(i^2+j^2)\Sigma_2+\Sigma_3</cmath> | Let <math>\Sigma_1= \sum_{k=1}^{5} kx_{k}</math>, <math>\Sigma_2=\sum_{k=1}^{5} k^{3}x_{k}</math> and <math>\Sigma_3=\sum_{k=1}^{5} k^{5}x_{k}</math>. For all pairs <math>i,j\in \mathbb{Z}</math>, let <cmath>\Sigma(i,j)=i^2j^2\Sigma_1-(i^2+j^2)\Sigma_2+\Sigma_3</cmath> | ||
− | Then we have one hand | + | Then we have on one hand |
<cmath> | <cmath> | ||
\Sigma(i,j)=i^2j^2\Sigma_1-(i^2+j^2)\Sigma_2+\Sigma_3=\sum_{k=1}^5(i^2j^2k-(i^2+j^2)k^3+k^5)x_k | \Sigma(i,j)=i^2j^2\Sigma_1-(i^2+j^2)\Sigma_2+\Sigma_3=\sum_{k=1}^5(i^2j^2k-(i^2+j^2)k^3+k^5)x_k | ||
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</cmath> | </cmath> | ||
for <math>n=1,2,3,4</math> | for <math>n=1,2,3,4</math> | ||
− | From the | + | From the latter and (2) we also have |
<cmath> | <cmath> | ||
\Sigma(n,n+1)=a(a-n^2)(a-(n+1)^2))\geq 0\implies a\notin (n^2,(n+1)^2) | \Sigma(n,n+1)=a(a-n^2)(a-(n+1)^2))\geq 0\implies a\notin (n^2,(n+1)^2) |
Revision as of 20:46, 27 November 2021
Problem
Determine all real numbers a for which there exists positive reals which satisfy the relations
Solution
Discussion thread can be found here: [1]
This problem needs a solution. If you have a solution for it, please help us out by adding it.
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