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China Mathematical Competition

Algebra Difficulty 3.8 AMC 10/12 Find the answer China

Given set $A = \{1, 2, m\}$ , $m$ is real. Let $B = \{a^2 \mid a \in A\}$ , $C = A \cup B$ . If the sum of all the elements of $C$ is $6$ , then the product of all the elements of $C$ is ______.

Your answer

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

Solution

By the condition, it is known that $1$ , $2$ , $4$ , $m$ , $m^2$ (allowing for repetition) are all the elements of $C$ .
Note that when $m$ is real, $1+2+4+m+m^2 > 6$ , $1+2+4+m^2 > 6$ , so it can only be $C = \{1, 2, 4, m\}$ , and $1+2+4+m = 6$ . Therefore, $m = -1$ , and is tested to be consistent with the question. At this point the product of all the elements of $C$ is $1 \times 2 \times 4 \times (-1) = -8$ .

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