# Problems of the Week

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# 2018-10-15 Intermediate

True or False?

The sum $$a+b+c$$ varies depending on the location of $$P.$$

Clarification: Point $$P$$ lies inside of this equilateral triangle, at perpendicular distances $$a$$, $$b$$, and $$c$$ from the sides of the triangle.

True or False?

If $$a,b,c,$$ and $$n$$ are positive integers such that $n=a^2+b^2+c^2,$ then there exist positive integers $$d,e,$$ and $$f$$ such that $n^2=d^2+e^2+f^2.$

In other words, if $$n$$ is the sum of three squares of positive integers, then $$n^2$$ is also the sum of three squares of positive integers.

Three $$60^{\circ}$$-sectors of a unit circle pack neatly inside an equilateral triangle.

The side length of this triangle can be written as $$A+\frac{B}{\sqrt{C}},$$ where $$A, B, C$$ are integers with $$C$$ square-free.

What is the value of $$A+B+C?$$

Twenty people at a convention are of different ages and have different names.

The following is true of four of the people:

• Allen is older than Bob.
• Bob is older than Carrie.
• Bob is older than Dave.

If Edward is part of this convention, the probability that Dave is older than Edward is $$\frac{a}{b},$$ where $$a$$ and $$b$$ are coprime positive integers.

What is $$a+b?$$

Assume that for any particular age, each person has the same chance to be that age.

In the diagram, $$A, B, C,$$ and $$D$$ are concyclic points.

What is the area of the green triangle $$APB,$$ given the areas of the three other colored regions?

\begin{align} \color{red}S_{\Delta APD}&=27\\ \color{purple}S_{CPDQ}&=37\\ \color{blue}S_{\Delta BPC}&=12\\ \color{green}S_{\Delta APB}&=\ ? \end{align}

Bonus: The four points need not to be concyclic; solve it without this information.

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