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Difference between revisions of "2004 JBMO Problems/Problem 1"

(Created page with "==Problem== Prove that the inequality <cmath> \frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } } </cmath> holds for all real numbers <math>x</math> and <math...")
 
(Added an even easier solution)
 
(5 intermediate revisions by 2 users not shown)
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Now squaring both sides of the inequality, we get:
 
Now squaring both sides of the inequality, we get:
 
<cmath> \frac{m}{(m-3)^2 } \leq \frac{8}{m-2} </cmath>
 
<cmath> \frac{m}{(m-3)^2 } \leq \frac{8}{m-2} </cmath>
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after cross multiplication and some simplification we get:
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after cross multiplication and simplification we get:
 
<math>7m^2 -46m + 72 \geq 0</math>
 
<math>7m^2 -46m + 72 \geq 0</math>
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or, <math>7(m-4)^2 +10(m-4) \geq 0</math>
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or, <math>7(m-4)^2 +10(m-4) \geq 0</math> which is always true since <math>m \geq 4</math>.
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The above is always true since <math>m \geq 4</math>.
 
  
  
 
<math>Kris17</math>
 
<math>Kris17</math>
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==Solution 2==
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Again, since the inequality is homogenous, we can assume WLOG that <math>x^2+y^2=8</math>.
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By AM-GM we gave <math>xy\leq4=\frac{x^2+y^2}{2}</math> and by QM-AM  we have that <math>x+y\leq4=2\sqrt{\frac{x^2+y^2}{2}}</math>.
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 +
Substituting we have
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<cmath>\frac{x+y}{x^2-xy+y^2}\leq\frac{4}{4}=\frac{2\sqrt{2}}{\sqrt{8}}=\frac{2\sqrt{2}}{\sqrt{x^2+y^2}}</cmath>
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<math>DVDTSB</math>
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==Solution 3==
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By Trivial Inequality,
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<cmath> (x - y)^2 \geq 0 \iff 2(x^2 - xy + y^2) \geq x^2 + y^2 \iff \frac{2}{x^2 + y^2} \geq \frac{1}{x^2 - xy + y^2}.</cmath>
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 +
Then by multiplying by <math>x + y</math> on both sides, we use the Trivial Inequality again to obtain <math>2(x^2 + y^2) \geq (x + y)^2</math> which means <cmath>\frac{x+y}{x^2 - xy + y^2} \leq \frac{2(x + y)}{x^2 + y^2} \leq \frac{2\sqrt{2(x^2+y^2)}}{x^2 + y^2}</cmath> which after simplifying, proves the problem.

Latest revision as of 08:38, 14 March 2023

Problem

Prove that the inequality \[\frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } }\] holds for all real numbers $x$ and $y$, not both equal to 0.


Solution

Since the inequality is homogeneous, we can assume WLOG that xy = 1.


Now, substituting $m = (x+y)^2$, we have:

$m = x^2 + y^2 + 2xy = x^2 + y^2 + 2$ $\geq 2\sqrt {xy} + 2 = 4$, thus we have $m \geq 4$


Now squaring both sides of the inequality, we get: \[\frac{m}{(m-3)^2 } \leq \frac{8}{m-2}\] after cross multiplication and simplification we get: $7m^2 -46m + 72 \geq 0$ or, $7(m-4)^2 +10(m-4) \geq 0$ which is always true since $m \geq 4$.


$Kris17$

Solution 2

Again, since the inequality is homogenous, we can assume WLOG that $x^2+y^2=8$.

By AM-GM we gave $xy\leq4=\frac{x^2+y^2}{2}$ and by QM-AM we have that $x+y\leq4=2\sqrt{\frac{x^2+y^2}{2}}$.

Substituting we have \[\frac{x+y}{x^2-xy+y^2}\leq\frac{4}{4}=\frac{2\sqrt{2}}{\sqrt{8}}=\frac{2\sqrt{2}}{\sqrt{x^2+y^2}}\]

$DVDTSB$

Solution 3

By Trivial Inequality, \[(x - y)^2 \geq 0 \iff 2(x^2 - xy + y^2) \geq x^2 + y^2 \iff \frac{2}{x^2 + y^2} \geq \frac{1}{x^2 - xy + y^2}.\]

Then by multiplying by $x + y$ on both sides, we use the Trivial Inequality again to obtain $2(x^2 + y^2) \geq (x + y)^2$ which means \[\frac{x+y}{x^2 - xy + y^2} \leq \frac{2(x + y)}{x^2 + y^2} \leq \frac{2\sqrt{2(x^2+y^2)}}{x^2 + y^2}\] which after simplifying, proves the problem.

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