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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>
 +
Then we have one hand
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<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
 +
=\sum_{k=1}^5k(i^2j^2-(i^2+j^2)k^2+k^4)x_k </cmath>
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Therefore \\(1)
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<cmath>
 +
\Sigma(i,j)=\sum_{k=1}^5k(k^2-i^2)(k^2-j^2)x_k
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</cmath>
 +
and on the other hand \\
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(2)
 +
<cmath>
 +
\Sigma(i,j)=i^2j^2a-(i^2+j^2)a^2+a^3=a(a-i^2)(a-j^2)
 +
</cmath>
 +
Then from (1) we have<cmath>
 +
\Sigma(0,5)=\sum_{k=1}^5k^3(k^2-5^2)x_k\leq 0
 +
</cmath>
 +
and from (2)
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<cmath>
 +
\Sigma(0,5)=a^2(a-25)
 +
</cmath>
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so <math>a\in [0,25]</math>
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Besides we also have from (1)
 +
<cmath>
 +
\Sigma(0,1)=\sum_{k=1}^5k^3(k^2-1)x_k\geq 0
 +
</cmath>
 +
and from (2)
 +
<cmath>
 +
\Sigma(0,1)=a^2(a-1)\geq 0 \implies a\notin (0,1)
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</cmath>
 +
and for <math>n=1,2,3,4</math>
 +
<cmath>
 +
\Sigma(n,n+1)=\sum_{k=1}^5k(k^2-n^2)(k^2-(n+1)^2)x_k
 +
</cmath>
 +
where in  the right hand we have that  <cmath>k<n \implies (k^2-n^2)<0, (k^2-(n+1)^2)<0</cmath>, so
 +
<cmath>(k^2-n^2)(k^2-(n+1)^2)>0</cmath>,  <cmath>k=n,n+1 , \implies (k^2-n^2)(k^2-(n+1)^2)=0</cmath> and <cmath>k>n \implies (k^2-n^2)(k^2-(n+1)^2)>0</cmath>, so
 +
<cmath>
 +
\Sigma(n,n+1)\geq 0
 +
</cmath>
 +
for <math>n=1,2,3,4</math>
 +
From the later and (2) we also have
 +
<cmath>
 +
\Sigma(n,n+1)=a(a-n^2)(a-(n+1)^2))\geq 0\implies a\notin (n^2,(n+1)^2)
 +
</cmath>
 +
So we have that <cmath>a\in [0,25]-\bigcup_{n=0}^4(n^2,(n+1^2))=\{0,1,4,9,16,25\}</cmath>
 +
 +
If <math>a=k^2</math>, <math>k=0,1,2,3,4,5</math> take <math>x_k=k</math>, <math> x_j=0 </math> for <math>j\neq k</math>. Then <math>\Sigma_1=k^2=a</math>, <math>\Sigma_2=k^3k=k^4=a^2</math>, and <math>\Sigma_3=k^5k=k^6=a^3</math>

Revision as of 20:17, 27 November 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

Let $\Sigma_1= \sum_{k=1}^{5} kx_{k}$, $\Sigma_2=\sum_{k=1}^{5} k^{3}x_{k}$ and $\Sigma_3=\sum_{k=1}^{5} k^{5}x_{k}$. For all pairs $i,j\in \mathbb{Z}$, let \[\Sigma(i,j)=i^2j^2\Sigma_1-(i^2+j^2)\Sigma_2+\Sigma_3\] Then we have one hand \[\Sigma(i,j)=i^2j^2\Sigma_1-(i^2+j^2)\Sigma_2+\Sigma_3=\sum_{k=1}^5(i^2j^2k-(i^2+j^2)k^3+k^5)x_k =\sum_{k=1}^5k(i^2j^2-(i^2+j^2)k^2+k^4)x_k\] Therefore \\(1) \[\Sigma(i,j)=\sum_{k=1}^5k(k^2-i^2)(k^2-j^2)x_k\] and on the other hand \\ (2) \[\Sigma(i,j)=i^2j^2a-(i^2+j^2)a^2+a^3=a(a-i^2)(a-j^2)\] Then from (1) we have\[\Sigma(0,5)=\sum_{k=1}^5k^3(k^2-5^2)x_k\leq 0\] and from (2) \[\Sigma(0,5)=a^2(a-25)\] so $a\in [0,25]$ Besides we also have from (1) \[\Sigma(0,1)=\sum_{k=1}^5k^3(k^2-1)x_k\geq 0\] and from (2) \[\Sigma(0,1)=a^2(a-1)\geq 0 \implies a\notin (0,1)\] and for $n=1,2,3,4$ \[\Sigma(n,n+1)=\sum_{k=1}^5k(k^2-n^2)(k^2-(n+1)^2)x_k\] where in the right hand we have that \[k<n \implies (k^2-n^2)<0, (k^2-(n+1)^2)<0\], so \[(k^2-n^2)(k^2-(n+1)^2)>0\], \[k=n,n+1 , \implies (k^2-n^2)(k^2-(n+1)^2)=0\] and \[k>n \implies (k^2-n^2)(k^2-(n+1)^2)>0\], so \[\Sigma(n,n+1)\geq 0\] for $n=1,2,3,4$ From the later and (2) we also have \[\Sigma(n,n+1)=a(a-n^2)(a-(n+1)^2))\geq 0\implies a\notin (n^2,(n+1)^2)\] So we have that \[a\in [0,25]-\bigcup_{n=0}^4(n^2,(n+1^2))=\{0,1,4,9,16,25\}\]

If $a=k^2$, $k=0,1,2,3,4,5$ take $x_k=k$, $x_j=0$ for $j\neq k$. Then $\Sigma_1=k^2=a$, $\Sigma_2=k^3k=k^4=a^2$, and $\Sigma_3=k^5k=k^6=a^3$

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