
3. Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct nonzero real numbers $a, b, c$ we have

$$
f(a)=b c ; f(b)=c a ; f(c)=a b
$$

Determine $f(a+b+c)$ in terms of $a, b, c$.


## Solution

Observe that

$$
a f(a)=b f(b)=c f(c)=a b c
$$

. Let

$$
P(x)=x f(x)-a b c
$$

Then $P(x)$ is a polynomial of degree 3 with three distinct roots $a, b, c$. Therefore

$$
P(x)=k(x-a)(x-b)(x-c)
$$

where $k$ is a real constant. Thus

$$
x f(x)=k(x-a)(x-b)(x-c)+a b c .
$$

Putting $x=0$ gives $k=1$. Hence

$$
f(x)=x^{2}-(a+b+c) x+a b+b c+c a
$$

and

$$
f(a+b+c)=a b+b c+c a
$$
