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China Mathematical Competition

Algebra Difficulty 3.8 AMC 10/12 Find the answer China

Suppose that $A = \{1, 2, 3\}$ , $B = \{2x + y \mid x, y \in A, x < y\}$ , $C = \{2x + y \mid x, y \in A, x > y\}$ . 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.

Solution

By enumeration, we get $B = \{4, 5, 7\}$ , $C = \{5, 7, 8\}$ . Thus, $B \cap C = \{5, 7\}$ . Therefore, the sum of all the elements of $B \cap C$ is $5 + 7 = 12$ . $\square$

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