Difference between revisions of "2023 WSMO Speed Round Problems/Problem 7"
| Line 6: | Line 6: | ||
From the Cauchy-Schwarz inequality, we have | From the Cauchy-Schwarz inequality, we have | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
| − | \left(\sqrt{3e+1}+\sqrt{3a+3}+\sqrt{3j+5}\right)^2&\le\left((3e+1)+(3a+3)+(3j+5)\right)\left(1+1+1\right) | + | \left(\sqrt{3e+1}+\sqrt{3a+3}+\sqrt{3j+5}\right)^2&\le\left((3e+1)+(3a+3)+(3j+5)\right)\left(1+1+1\right)\\ |
&\le\left(3(e+a+j)+9\right)(3) = (3(1)+9)(3)=36\implies\\ | &\le\left(3(e+a+j)+9\right)(3) = (3(1)+9)(3)=36\implies\\ | ||
\sqrt{3e+1}+\sqrt{3a+3}+\sqrt{3j+5}\le\sqrt{36}=\boxed{6}. | \sqrt{3e+1}+\sqrt{3a+3}+\sqrt{3j+5}\le\sqrt{36}=\boxed{6}. | ||
Latest revision as of 11:33, 12 September 2025
Problem
Let 
be real numbers such that 
and 
, 
and 
. Find the maximum value of 
Solution
From the Cauchy-Schwarz inequality, we have
~pinkpig
