Difference between revisions of "2008 Indonesia MO Problems/Problem 2"
Rockmanex3 (talk | contribs) (Solution to Problem 2 -- oh baby a triple!) |
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<cmath> \frac {1}{(1 + \sqrt {x})^{2}} + \frac {1}{(1 + \sqrt {y})^{2}} \ge \frac {2}{x + y + 2}.</cmath> | <cmath> \frac {1}{(1 + \sqrt {x})^{2}} + \frac {1}{(1 + \sqrt {y})^{2}} \ge \frac {2}{x + y + 2}.</cmath> | ||
− | ==Solution== | + | ==Solution 1== |
By the [[Cauchy-Schwarz Inequality]], <math>(1+1)(1+x) \ge (1 + \sqrt{x})^2</math> and <math>(1+1)(1+y) \ge (1 + \sqrt{y})^2</math>, with equality happening in the earlier inequality when <math>x = 1</math> and equality happening in the latter inequality when <math>y = 1</math>. Because <math>x,y > 0</math>, | By the [[Cauchy-Schwarz Inequality]], <math>(1+1)(1+x) \ge (1 + \sqrt{x})^2</math> and <math>(1+1)(1+y) \ge (1 + \sqrt{y})^2</math>, with equality happening in the earlier inequality when <math>x = 1</math> and equality happening in the latter inequality when <math>y = 1</math>. Because <math>x,y > 0</math>, | ||
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\end{align*}</cmath> | \end{align*}</cmath> | ||
Therefore, since <math>\frac{1}{(1 + \sqrt{x})^2} + \frac{1}{(1 + \sqrt{y})^2} \ge \frac12 \cdot (\frac{1}{1+x} + \frac{1}{1+y})</math> and <math>\frac12 \cdot (\frac{1}{1+x} + \frac{1}{1+y}) \ge \sqrt{\frac{1}{(1+x)(1+y)}}</math> and <math>\sqrt{\frac{1}{(x+1)(y+1)}} \ge \frac{2}{x+y+2}</math>, we must have <math>\frac{1}{(1 + \sqrt{x})^2} + \frac{1}{(1 + \sqrt{y})^2} \ge \frac{2}{x+y+2}</math>, with equality happening when <math>x = y = 1</math>. | Therefore, since <math>\frac{1}{(1 + \sqrt{x})^2} + \frac{1}{(1 + \sqrt{y})^2} \ge \frac12 \cdot (\frac{1}{1+x} + \frac{1}{1+y})</math> and <math>\frac12 \cdot (\frac{1}{1+x} + \frac{1}{1+y}) \ge \sqrt{\frac{1}{(1+x)(1+y)}}</math> and <math>\sqrt{\frac{1}{(x+1)(y+1)}} \ge \frac{2}{x+y+2}</math>, we must have <math>\frac{1}{(1 + \sqrt{x})^2} + \frac{1}{(1 + \sqrt{y})^2} \ge \frac{2}{x+y+2}</math>, with equality happening when <math>x = y = 1</math>. | ||
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+ | ==Solution 2== | ||
+ | Let <math>f(j)=\frac{1}{(1+\sqrt{j})^2}</math> | ||
+ | Since this function is concave up, according to Jensen's inequality, we can get <math>\frac{f(x)+f(y)}{2}\geq f(\frac{x+y}{2})</math> which means <math>f(x)+f(y)\geq 2f(\frac{x+y}{2})</math>. | ||
+ | In this problem, it turns into <math>f(x)+f(y)\geq \frac{2}{(1+\sqrt\frac{x+y}{2})^2}</math>.The conclusion we try to find is that <math>f(x)+f(y) \geq \frac{2}{x+y+2}</math> | ||
+ | So we can see that <math>\frac{2}{x+y+2} \leq \frac{2}{(1+\sqrt\frac{x+y}{2})^2}</math>. | ||
+ | Take reciprocal for both sides we can get <math>(x+y+2)\geq (1+\sqrt\frac{x+y}{2})^2</math>. | ||
+ | Take RHS, <math>(1+\sqrt\frac{x+y}{2})^2=1+\frac{x+y}{2}+\sqrt{2}\sqrt{x+y}</math>. | ||
+ | Now we have to prove that <math>(x+y+2)\geq (1+\frac{x+y}{2}+\sqrt{2}\sqrt{x+y})</math>. | ||
+ | which turns to <math>(\frac{x+y}{2}+1)\geq \sqrt{2}\sqrt{x+y}</math>. It is always correct according to <math>AM-GM</math> inequality, it happens when <math>x=y=1</math>. | ||
+ | <math>Q.E.D</math> ~bluesoul | ||
==See Also== | ==See Also== |
Latest revision as of 23:09, 3 December 2021
Contents
[hide]Problem
Prove that for every positive reals and ,
Solution 1
By the Cauchy-Schwarz Inequality, and , with equality happening in the earlier inequality when and equality happening in the latter inequality when . Because , By the AM-GM Inequality, we know that . For the equality case, , so . Additionally, by the AM-GM Inequality, . For the equality case, , so . Because , Therefore, since and and , we must have , with equality happening when .
Solution 2
Let Since this function is concave up, according to Jensen's inequality, we can get which means . In this problem, it turns into .The conclusion we try to find is that So we can see that . Take reciprocal for both sides we can get . Take RHS, . Now we have to prove that . which turns to . It is always correct according to inequality, it happens when . ~bluesoul
See Also
2008 Indonesia MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 3 |
All Indonesia MO Problems and Solutions |