Competition problem — Algebra — Maths Olympiad Prep

Library / /5 of 100

Algebra Difficulty 3.7 AMC 10/12 Find the answer 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. The answer given is C, and the solution below works it through.

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.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.