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Difference between revisions of "2003 JBMO Problems/Problem 4"

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==Problem==
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Let <math>x, y, z > -1</math>. Prove that
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<math>\frac {1+x^2}{1+y+z^2}+\frac {1+y^2}{1+z+x^2}+\frac {1+z^2}{1+x+y^2} \geq 2</math>
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==Solution==
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Since <math>x, y, z > -1</math> and <math>x^2, y^2, z^2 \geq 0</math>, we have that <math>1+x^2, 1+y^2, 1+z^2</math> and <math>1+y+z^2, 1+z+x^2, 1+x+y^2</math> are always positive.  
 
Since <math>x, y, z > -1</math> and <math>x^2, y^2, z^2 \geq 0</math>, we have that <math>1+x^2, 1+y^2, 1+z^2</math> and <math>1+y+z^2, 1+z+x^2, 1+x+y^2</math> are always positive.  
  
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We have
 
We have
 
<math>\Sigma \frac {p}{q+2r} = \frac {\Sigma \frac {p}{q+2r}*\Sigma p(q+2r)}{\Sigma p(q+2r)}</math>
 
<math>\Sigma \frac {p}{q+2r} = \frac {\Sigma \frac {p}{q+2r}*\Sigma p(q+2r)}{\Sigma p(q+2r)}</math>
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By Cauchy-Schwarz,
 
By Cauchy-Schwarz,
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<math>\Sigma \frac {\frac {p}{q+2r}*\Sigma p*(q+2r)}{\Sigma p(q+2r)} \geq \frac {(\Sigma p)^2}{\Sigma p(q+2r)} = \frac {(\Sigma p)^2}{3\Sigma pq}</math>
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<math>\Sigma \frac {\frac {p}{q+2r}*\Sigma p(q+2r)}{\Sigma p(q+2r)} \geq \frac {(\Sigma p)^2}{\Sigma p(q+2r)} = \frac {(\Sigma p)^2}{3\Sigma pq}</math>
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Since <math>\Sigma p^2 \geq \Sigma pq</math>, we have <math>(\Sigma p)^2 \geq 3\Sigma pq</math>.
 
Since <math>\Sigma p^2 \geq \Sigma pq</math>, we have <math>(\Sigma p)^2 \geq 3\Sigma pq</math>.
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Thus,
 
Thus,
 
<math>\Sigma\frac {p}{q+2r} \geq \frac{(\Sigma p)^2}{3\Sigma pq} \geq \frac {3\Sigma pq}{3\Sigma pq} = 1</math>
 
<math>\Sigma\frac {p}{q+2r} \geq \frac{(\Sigma p)^2}{3\Sigma pq} \geq \frac {3\Sigma pq}{3\Sigma pq} = 1</math>
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So,
 
So,
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<math>\Sigma \frac {1+x^2}{1+y+z^2} \geq \Sigma\frac {2(1+x^2)}{(y^2+1)+2(z^2+1)} = 2\Sigma \frac {1+x^2}{y^2+1+2(z^2+1)} = 2\Sigma \frac {p}{q+2r} \geq 2*1 = 2</math>
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<math>\Sigma \frac {1+x^2}{1+y+z^2} \geq \Sigma\frac {2(1+x^2)}{(y^2+1)+2(z^2+1)} = 2\Sigma \frac {1+x^2}{y^2+1+2(z^2+1)} = 2\Sigma \frac {p}{q+2r} \geq 2*1 = 2</math>, as desired.

Latest revision as of 17:17, 25 April 2021

Problem

Let $x, y, z > -1$. Prove that

$\frac {1+x^2}{1+y+z^2}+\frac {1+y^2}{1+z+x^2}+\frac {1+z^2}{1+x+y^2} \geq 2$


Solution

Since $x, y, z > -1$ and $x^2, y^2, z^2 \geq 0$, we have that $1+x^2, 1+y^2, 1+z^2$ and $1+y+z^2, 1+z+x^2, 1+x+y^2$ are always positive.

Hence, $\frac {1+x^2}{1+y+z^2}, \frac {1+y^2}{1+z+x^2}$ and $\frac {1+z^2}{1+x+y^2}$ must also be positive.

From the inequality $(x-1)^2 \geq 0$, we obtain that $x \leq \frac {x^2+1}{2}$ and, analogously, $\frac {1}{x} \geq \frac{2}{x^2+1}$. Similarly, $\frac {1}{y} \geq \frac {2}{y^2+1}$ and $\frac {1}{z} \geq \frac {2}{z^2+1}$.

Now, 

$\Sigma \frac {1+x^2}{1+y+z^2} \geq \Sigma \frac {1+x^2}{1+\frac {y^2+1}{2}+z^2} = \Sigma \frac {2(1+x^2)}{y^2+2z^2+3} = \Sigma \frac {2(1+x^2)}{(y^2+1)+2(z^2+1)}$

Substituting $p=x^2+1, q=y^2+1$ and $r=z^2+1$, we now need to prove $\Sigma \frac{p}{q+2r} \geq 1$.

We have $\Sigma \frac {p}{q+2r} = \frac {\Sigma \frac {p}{q+2r}*\Sigma p(q+2r)}{\Sigma p(q+2r)}$


By Cauchy-Schwarz, $\Sigma \frac {\frac {p}{q+2r}*\Sigma p(q+2r)}{\Sigma p(q+2r)} \geq \frac {(\Sigma p)^2}{\Sigma p(q+2r)} = \frac {(\Sigma p)^2}{3\Sigma pq}$


Since $\Sigma p^2 \geq \Sigma pq$, we have $(\Sigma p)^2 \geq 3\Sigma pq$.


Thus, $\Sigma\frac {p}{q+2r} \geq \frac{(\Sigma p)^2}{3\Sigma pq} \geq \frac {3\Sigma pq}{3\Sigma pq} = 1$


So, $\Sigma \frac {1+x^2}{1+y+z^2} \geq \Sigma\frac {2(1+x^2)}{(y^2+1)+2(z^2+1)} = 2\Sigma \frac {1+x^2}{y^2+1+2(z^2+1)} = 2\Sigma \frac {p}{q+2r} \geq 2*1 = 2$, as desired.

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