Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.6 AIME Find the answer

19. Given positive real numbers $a$ , $b$ , $c$ satisfy the system of equations
$\left\{\begin{array}{l} a+b^{2}+2 a c=29, \\ b+c^{2}+2 a b=18, \\ c+a^{2}+2 b c=25 . \end{array}\right.$

Then the value of $a+b+c$ is

Pick one

Solution

19.E.

Let $x=a+b+c$ . By adding the three equations, we get $x+x^{2}=72$ , which simplifies to $(x-8)(x+9)=0$ . Since $a, b, c$ are all positive numbers, then $x>0$ , hence $a+b+c=8$ .
Note that $a=4, b=1, c=3$ satisfies this system of equations.

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