Problem 1

Let

\mathbb{Z}^+

be the set of positive integers. Determine all functions

f:\mathbb{Z}^+\to\mathbb{Z}^+

such that

a^2+f(a)f(b)

is divisible by

f(a)+b

for all positive integers

a

and

b

Step 1 of 6: Substitute a=b=1

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f(1)+1 \mid f(1)^2+1 \ \Longrightarrow\ f(1)=1

Detailed analysis

Setting $a=b=1$ in the divisibility condition gives $f(1)+1\mid1+f(1)^2$ . Since $f(1)^2+1=(f(1)-1)(f(1)+1)+2$ , this reduces to $f(1)+1\mid2$ , and as $f(1)$ is a positive integer, $f(1)+1=2$ , so $f(1)=1$ .