 # Doubt from algebra and geometry

Q1)Let C_{1} , C_{2} , . . . , C_{2008} be complex numbers such that
|C_{1}| = |C_{2}| = |C_{3}| = · · · = |C_{2008}| = 2011, and let S(2008, t) be the sum of all products of these 2008 complex numbers taken t at a time. Let Q be the
maximum possible value of
|S(2008,1492)| \over |S(2008, 516)|
Find the remainder when Q is divided by 1000.

Q2)Given a convex, n-sided polygon P, form a 2n-sided polygon clip(P) by cutting off each corner of P at the
edges’ trisection points. In other words, clip(P) is the polygon whose vertices are the 2n edge trisection
points of P, connected in order around the boundary of P. Let P_{1} be an isosceles trapezoid with side lengths 13, 13, 13, and 3, and for each i ≥ 2, let P_{i} = clip( P_{i} −1). This iterative clipping process approaches a limiting
shape P∞ = lim_{i→∞} P_{i}
. If the difference of the areas of P_{10} and P∞ is written as a fraction x \over y in lowest terms, calculate the number of positive integer factors of x · y.

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