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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=a2+b2+c2, then there exist positive integers d,e, and f such that n2=d2+e2+f2.
In other words, if n is the sum of three squares of positive integers, then n2 is also the sum of three squares of positive integers.
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Three 60∘-sectors of a unit circle pack neatly inside an equilateral triangle.
The side length of this triangle can be written as A+CB, 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 ba, 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?
SΔAPDSCPDQSΔBPCSΔAPB=27=37=12= ?
Bonus: The four points need not to be concyclic; solve it without this information.
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