Problem 3

Let

\mathbb{N}

denote the set of positive integers. A function

f:\mathbb{N}\to\mathbb{N}

is said to be bonza if

f(a)

divides

b^a-f(b)^{f(a)}

for all positive integers

a

and

b

. Determine the smallest real constant

c

such that

f(n)\le cn

for all bonza functions

f

and all positive integers

n

Step 7 of 7: Conclusion

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c=4

Detailed analysis

Steps 1–5 show $f(n)\le4n$ for every bonza function and every $n$ , while step 6 exhibits a bonza function with $f(4)=4\cdot4$ , so no smaller constant works. Hence the smallest valid constant is $c=4$ .