1975 AHSME Problems/Problem 3

Which of the following inequalities are satisfied for all real numbers $a, b, c, x, y, z$ which satisfy the conditions $x < a, y < b$ , and $z < c$ ?

$\text{I}. \ xy + yz + zx < ab + bc + ca \\ \text{II}. \ x^2 + y^2 + z^2 < a^2 + b^2 + c^2 \\ \text{III}. \ xyz < abc$

$\textbf{(A)}\ \text{None are satisfied.} \qquad \textbf{(B)}\ \text{I only} \qquad \textbf{(C)}\ \text{II only} \qquad \textbf{(D)}\ \text{III only} \qquad \textbf{(E)}\ \text{All are satisfied.}$

Solution

Solution by e_power_pi_times_i

Notice if $a$ , $b$ , and $c$ are $0$ , then we can find $x$ , $y$ , and $z$ to disprove $\text{I}$ and $\text{II}$ . For example, if $(a, b, c, x, y, z) = (0, 0, 0, -1, -1, -1)$ , then $\text{I}$ and $\text{II}$ are disproved. If $(a, b, c, x, y, z) = (0, 1, 2, -1, -1, 1)$ , then $\text{III}$ is disproved. Therefore the answer is $\boxed{\textbf{(A) } \text{None are satisfied}}$ .

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1975 AHSME Problems/Problem 3

Solution