Question 29
a, b, c are three positive real numbers such that $a+b+c=1$. Which of the following conditions is equivalent to requiring $a, b, c$ to be the measures of the sides of a nondegenerate triangle?
(A) $0\lt |b-a|\lt \frac{1}{2},\ 0\lt |c-b|\lt \frac{1}{2},\ 0\lt |c-a|\lt \frac{1}{2}$
(B) $a\lt \frac{1}{2},\ b\lt \frac{1}{2},\ c\lt \frac{1}{2}$
(C) $a+b\lt \frac{1}{2},\ \quad b+c\lt \frac{1}{2},\ \quad c+a\lt \frac{1}{2}$
(D) $a \leq \frac{1}{3},\ b \leq \frac{1}{3},\ c \leq \frac{1}{3}$
(E) none of the above.
(A) $0\lt |b-a|\lt \frac{1}{2},\ 0\lt |c-b|\lt \frac{1}{2},\ 0\lt |c-a|\lt \frac{1}{2}$
(B) $a\lt \frac{1}{2},\ b\lt \frac{1}{2},\ c\lt \frac{1}{2}$
(C) $a+b\lt \frac{1}{2},\ \quad b+c\lt \frac{1}{2},\ \quad c+a\lt \frac{1}{2}$
(D) $a \leq \frac{1}{3},\ b \leq \frac{1}{3},\ c \leq \frac{1}{3}$
(E) none of the above.