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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.
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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.
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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 $A+B+C?$
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Twenty people at a convention are of different ages and have different names.
The following is true of four of the people:
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.
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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?
$\begin{aligned} \color{#D61F06}S_{\Delta APD}&=27\\ \color{#69047E}S_{CPDQ}&=37\\ \color{#3D99F6}S_{\Delta BPC}&=12\\ \color{#20A900}S_{\Delta APB}&=\ ? \end{aligned}$
Bonus: The four points need not to be concyclic; solve it without this information.
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