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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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