2. Let $a, b$, and $c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $a x^{2}+b x+c, b x^{2}+c x+a$, and $c x^{2}+a x+b$.
Answer: 4 If all the polynomials had real roots, their discriminants would all be nonnegative: $a^{2} \geq$ $4 b c, b^{2} \geq 4 c a$, and $c^{2} \geq 4 a b$. Multiplying these inequalities gives $(a b c)^{2} \geq 64(a b c)^{2}$, a contradiction. Hence one of the quadratics has no real roots. The maximum of 4 real roots is attainable: for example, the values $(a, b, c)=(1,5,6)$ give $-2,-3$ as roots to $x^{2}+5 x+6$ and $-1,-\frac{1}{5}$ as roots to $5 x^{2}+6 x+1$.