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Problem 576

AMC 10/12, early questions

Algebra Difficulty 3.7 Find the answer China Mathematical Competition · China

Suppose $A = \{x \mid 5x - a \le 0\}$ , $B = \{x \mid 6x - b > 0\}$ , $a, b \in \mathbb{N}$ , and $A \cap B \cap \mathbb{N} = \{2, 3, 4\}$ . The number of such pairs $(a, b)$ is ( ).

This was a multiple-choice question, but the options didn't survive into the source we have, so there is nothing here to pick from. Work it on paper and mark yourself against the solution below.

Official solution

Since $5x - a \le 0 \Rightarrow x \le \frac{a}{5}$ , $6x - b > 0 \Rightarrow x > \frac{b}{6}$ . In order to satisfy $A \cap B \cap \mathbb{N} = \{2, 3, 4\}$ , we have
$\begin{cases} 1 \le \frac{b}{6} < 2, \\ 4 \le \frac{a}{5} < 5, \end{cases}$
or
$\begin{cases} 6 \le b < 12, \\ 20 \le a < 25. \end{cases}$
So the number of pairs $(a, b)$ is $\binom{6}{1} \cdot \binom{5}{1} = 30$ . Answer: C.

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