Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 3.9 AMC 10/12 Find the answer 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.

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 $48$ .

$\square$

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