1. Suppose $a, b$ and $c$ are nonzero real numbers such that
$$
\frac{a^{2}}{b^{2}+c^{2}}<\frac{b^{2}}{c^{2}+a^{2}}<\frac{c^{2}}{a^{2}+b^{2}} \text {. }
$$

Which of the following statements is always true?
(A) $a<b<c$
(B) $|a|<|b|<|c|$
(C) $c<b<a$
(D) $|b|<|c|<|a|$
(E) $|c|<|b|<|a|$
1. Answer: (B)
Since $a, b$ and $c$ are nonzero real numbers and $\frac{a^{2}}{b^{2}+c^{2}}\frac{c^{2}+a^{2}}{b^{2}}>\frac{a^{2}+b^{2}}{c^{2}} .
$$

Adding 1 throughout, we obtain
$$
\frac{a^{2}+b^{2}+c^{2}}{a^{2}}>\frac{a^{2}+b^{2}+c^{2}}{b^{2}}>\frac{a^{2}+b^{2}+c^{2}}{c^{2}} .
$$

Thus $\frac{1}{a^{2}}>\frac{1}{b^{2}}>\frac{1}{c^{2}}$, which implies that $a^{2}<b^{2}<c^{2}$. So we have $|a|<|b|<|c|$.