3. If a, b, c are distinct non-negative real numbers, what is the smallest possible number of distinct numbers among the numbers $a+b, b+c, c+a, a^{2}+b^{2}, b^{2}+c^{2}, c^{2}+a^{2}$? (Patrik Bak)
SOLUTION. The problem is symmetric in the variables $a, b, c$: changing their order only changes the order of the six numbers being examined. In the first part of the solution, we will assume that $a < b < c$ and $a > 1$. Then it will hold that

$$
11$ will satisfy the condition $a^{2}=a+1$. A simple calculation reveals that this number is

$$
a=\frac{1+\sqrt{5}}{2}
$$

known as the golden ratio, not only in mathematics but also in art.

Conclusion. The smallest possible number of different numbers in the given sextet is 4.