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Difference between revisions of "1979 IMO Problems/Problem 5"

(Created page with "==Problem== Determine all real numbers a for which there exists positive reals <math>x_{1}, \ldots, x_{5}</math> which satisfy the relations <math> \sum_{k=1}^{5} kx_{k}=a,</m...")
 
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==Solution==
 
==Solution==
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Let <math>A</math> be the first sum, <math>B</math> the second and <math>C</math> the third.
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Discussion thread can be found here: [https://aops.com/community/p367346]
  
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From Cauchy, we have <math>AC \ge B^2</math>. but, <math>AC = B^2</math>.
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{{solution}}
  
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The equality of C-S occurs if the terms are proportional.
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== See Also == {{IMO box|year=1979|num-b=4|num-a=6}}
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so, <math>jx_j/j^5x_j = ix_i/i^5x_i</math> for all <math>i</math>,<math>j</math> <math>\Longrightarrow i = j</math>, thus no such <math>a</math> exists.
 

Revision as of 22:04, 29 January 2021

Problem

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $\sum_{k=1}^{5} kx_{k}=a,$ $\sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $\sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

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
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