Problem 1-9: Triangle Side Expression

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 [Difficulty 4] [PDF]

QUESTION
(Hong Kong Team Selection Test 2009) Let $a$, $b$, $c$ be the sides of a triangle. Determine all possible values of $$\frac{a^2+b^2+c^2}{ab+bc+ac}$$

Laconic Solution Sketch
Apply Triangle Inequality, or Ravi Transformation.

Problem 1-1 Kookie's Cookies

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Problem 1-1 (Number Theory) [Difficulty 1] [PDF]

QUESTION
(Sipnayan 2012 High School Final Round: Weightlifting 4) Kookie has a kooky way of eating cookies. He lays them out on a circle. After Kookie eats a cookie, he skips the next (clockwise) remaining cookie in the circle and eats the next (clockwise) remaining cookie after that. Kookie places a batch of $2012$ cookies numbered $1, 2, \ldots 2012$ in that clockwise order, and begins to eat them, cookie $2$ first. Let $k$ be the number on the last of the $2012$ cookies that he eats. Kookie, unsatisfied, arranges another batch of $k$ cookies numbered $2, 4, 6, \ldots 2k$ in that clockwise order, and begins again with the cookie numbered $2$. What is the number on the last cookie Kookie eats from this batch?

Overview
This problem comprises two iterations of the Josephus problem.