# 2008 Indonesia MO Problems/Problem 8

## Solution 1

Since , we know that .

Let , be , , respectively. Then, .

Let , be , , respectively. Then,

Let , be , , respectively. Then,

Let , be , , respectively. Then,

From the last 2 equations, we get that

Since , substituting, we get

\begin{align*} \frac{1}{2}(f(2)^2+1)&=(f(2)(f(1)-1)+1)(f(1)-1)+1\\ \frac{1}{2}(f(2)^2+1)&=f(2)f(1)^2-f(2)f(1)+f(1)-f(2)f(1)-f(2)-1+1\\ f(2)^2+1&=2f(2)f(1)^2-2f(2)f(1)+2f(1)-2f(2)f(1)-2f(2) \end{align*}

If we take modulo of f(2) on both sides, we get

Because , we also know that . If , then .

Suppose :

since , we have . Or that . Thus, Thus, or .

case 1:

Let , and be an arbitrary integer . Then, Thus, .

case 2:

Let , and be an arbitrary integer . Then, This forms a linear line where Thus,

Upon verification for , we get

Upon verification for , we get

Thus, both equations, and are valid