2023 WSMO Speed Round Problems/Problem 7: Difference between revisions
Created page with "==Problem== Let <math>e, a, j</math> be real numbers such that <math>e + a + j = 1</math> and <math>e\geq -\frac{1}{3}</math>, <math>a\geq 1</math> and <math>j\geq-\frac{5}{3..." |
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==Solution== | ==Solution== | ||
From the Cauchy-Schwarz inequality, we have | |||
<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)\\ | |||
&\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}. | |||
\end{align*}</cmath> | |||
~pinkpig | |||
Latest revision as of 10:33, 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
