2023 SSMO Team Round Problems/Problem 1

Problem

Let $(a, b, c, d)$ be a permutation of $(2, 0, 2, 3)$. Find the largest possible value of $a^b + b^c + c^d + d^a$

Solution

We can assume because of the symmetry that $a=0$. Then, the problem is reduced to $1+b^c+c^d$. Since there are only $3$ possible permutations for $b$, $c$, and $d$, we can try them and find that the maximum possible value is obtained when $b=2$, $c=3$, and $d=2$. Therefore, the answer is $1+2^3+3^2=\boxed{18}$.

~alexanderruan