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Berkeley Math Circle: Monthly Contest 2

Algebra Difficulty 4.8 AIME Prove it United States

Problem:
Given that
$\begin{aligned} a+b & =23 \\ b+c & =25 \\ c+a & =30 \end{aligned}$
determine (with proof) the value of $a b c$ .

Solution

Solution:
We add all three equations to obtain
$2(a+b+c)=78$
so $a+b+c=39$ . Therefore,
$\begin{aligned} & a=39-25=14 \\ & b=39-30=9 \\ & c=39-23=16 . \end{aligned}$
Thus $a b c=14 \cdot 9 \cdot 16=2016$ .

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