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?
\[\begin{align} \color{red}S_{\Delta APD}&=27\\ \color{purple}S_{CPDQ}&=37\\ \color{blue}S_{\Delta BPC}&=12\\ \color{green}S_{\Delta APB}&=\ ? \end{align}\]
Bonus: The four points need not to be concyclic; solve it without this information.
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