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Algebra Difficulty 4.8 AIME Prove it United States

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

Solution

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

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