Problem 1-3 Coin Rotation

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Problem 1-3 (Geometry) [Difficulty 1] [PDF]

QUESTION
(Original) Three identical cylindrical barrels with radius $\sqrt{3}$ are placed tangent to each other (represented by circles $c_{1}$ , $c_{2}$ , $c_{3}$ .) A metal sheet $AB$ is placed just touching $c_{1}$ and $c_{3}$. A coin $c$ with radius $2-\sqrt{3}$  is placed on the floor tangent to $c_{2}$ and $c_{3}$  (see the diagram), and rolled without slipping about the barrels (so that the coin is rotating clockwise), going through plank $AB$, until it returns to its starting point. How many radians has the coin rotated? (For example, half a turn is $\pi$ .)

Overview
Separate the coin's rotation into two 'components'.

Problem 1-2 Ellipse Tracing

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Problem 1-2 (Geometry) [Difficulty 3] [PDF]

QUESTION
(Original) Define points $A\left(2,5+\sqrt{21}\right)$ , $B\left(2,5-\sqrt{21}\right)$ . Now let a point $P$ be initially at $P_{0}\left(4,5\right)$ . Let $Q$ be the point on ray $\overrightarrow{AP}$  such that $PB=PQ$ . As $P$ moves along the lower half of the ellipse $25x^{2}-100x+4y^{2}-40y+100=0$ (to eventually stop at $\left(0,5\right)$), the point $Q$  traces a path. Find the length of this path.

Overview/Laconic Solution Sketch
Apply geometric ellipse definition, and from that calculate arc length.

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.