AoPS Wiki talk:Problem of the Day/July 24, 2011

Solution to Problem 1: If the circumference is $18\pi$ , then we can find the radius. Because of our cirumference formula, we know that $r = \frac{18\pi}{2\pi} = 9$ . So, the radius is $9$ units. We can insert this into the circle area formula to find the area: $(9^2)\pi = 81\pi$ . So, the area is $81\pi$ .

Solution to Problem 2: Rearrange the terms on the left side:

$a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \geq 0$

This is just

$(a + b + c)^2 \geq 0$

This is trivial, so the inequality must be true. (Idea for wording of the end of the solution: solutions for the Pre-Olympiad Inequalities marathon by other users)

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AoPS Wiki talk:Problem of the Day/July 24, 2011