\section*{Problem 3 - 051033}

Determine all real numbers \(a, b\) and \(c\) for which the following holds: \(a+b c=(a+b)(a+c)\).
}

The given equation is equivalent to each of the following equations

\[
a+b c=a^{2}+a c+a b+b c \quad ; \quad a(a+b+c-1)=0
\]

Since the product of two numbers is zero if and only if at least one of its factors is zero, it follows that

\[
a=0 \quad \text { or } \quad a+b+c-1=0
\]

must hold, and conversely, in each of these cases, equation (1) and thus the given equation is satisfied. The given equation is therefore satisfied for all number triples \((a, b, c)\) with
1. \(a=0\) and real numbers \(b\) and \(c\) and

2. real numbers \(a, b\) and \(c\) for which \(a+b+c=1\) holds.

In all other cases, it is not satisfied.

Taken from \([5]\)