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 1

WLOG, assume that $a = 0$ . Therefore, we have $(a,b,c,d) = (0,2,2,3),(0,2,3,2),$ or $(0,3,2,2).$ The value of $a^b+b^c+c^d+d^a$ for these three permutations are $13,18,$ and $14,$ respectively, meaning the greatest possible sum is $\boxed{18}.$

~SMO_Team

Solution 2

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

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2023 SSMO Team Round Problems/Problem 1

Problem

Solution 1

Solution 2