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Problem 703

AMC 10/12, early questions

Algebra Difficulty 3.9 Find the answer China Mathematical Competition · China

Suppose $A = \{1, 2, 3\}$ , $B = \{4x - y \mid x, y \in A\}$ , $C = \{4x + y \mid x, y \in A\}$ . Then the sum of all the elements of $B \cap C$ is ______.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

When $x$ and $y$ take all the elements of $A$ , respectively, $4x - y$ gets exactly the values $1, 2, 3, 5, 6, 7, 9, 10, 11$ , and $4x + y$ gets exactly the values $5, 6, 7, 9, 10, 11, 13, 14, 15$ .

Hence, $B \cap C = \{5, 6, 7, 9, 10, 11\}$ . Then the sum of all the elements of $B \cap C$ is $48$ .

$\square$

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.