Find the maximum value

Sep 2013
38
0
New Delhi
Let \(\displaystyle a, b, c\) be positive real numbers such that \(\displaystyle a + b + c = 3\). Determine, with certainty, the largest possible value of the expression:
\(\displaystyle \frac {a}{a^3+b^2+c}+\frac {b}{b^3+c^2+a}+\frac {c}{c^3+a^2+b}\)
 

romsek

Math Team
Sep 2015
2,961
1,674
USA
well by symmetry of the expression you know that $a=b=c$ and since they sum to 3 you thus know that $a=b=c=1$

so the max of the sum will be 1