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A1. C is the complex numbers. f : C → R is defined by f(z) = |z3 - z + 2|. What is the maximum value of f on the unit circle |z| = 1?
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A2. K is a cone. s is a sphere radius r, and S is a sphere radius R. s is inside K touches it along all points of a circle. S is also inside K and touches it along all points of a circle. s and S also touch each other. What is the volume of the finite region between the two spheres and inside K?
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A3. an is a sequence of positive reals decreasing monotonically to zero. bn is defined by bn = an - 2an+1 + an+2 and all bn are non-negative. Prove that b1 + 2b2 + 3b3 + ... = a1.
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A4. Let D be a disk radius r. Given (x, y) ∈ D, and R > 0, let a(x, y, R) be the length of the arc of the circle center (x, y), radius R, which is outside D. Evaluate limR→0 R-2 ∫D a(x, y, R) dx dy.
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A5. Let α1, α2, ... , αn be the nth roots of unity. Find ∏i<j (αi - αj)2.
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A6. Do either (1) or (2):
(1) On each element ds of a closed plane curve there is a force 1/R ds, where R is the radius of curvature. The force is towards the center of curvature at each point. Show that the curve is in equilibrium.
(2) Prove that x + 2/3 x3 + 2.4/3.5 x5 + ... + 2.4. ... .2n/(3.5. ... .2n+1) x2n+1 + ... = (1 - x2)-1/2 sin-1x
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B1. p(x) is a cubic polynomial with roots α, β, γ and p'(x) divides p(2x). Find the ratios α : β : γ.
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B2. A circle radius r is tangent to the three coordinate planes (x =0, y =0, z = 0) in space. Find the locus of its center.
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B3. Show that [√n + √(n + 1)] = [√(4n + 2)] for positive integers n.
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B4. R is the reals. For what λ can we find a continuous function f : (0, 1) → R, not identically zero, such that ∫01 min(x, y) f(y) dy = λ f(x) for all x ∈ (0, 1)?
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B5. Find the area of the region { (x, y) : |x + yt + t2| ≤ 1 for all t ∈ [0, 1] }.
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B6. Do either (1) or (2):
(1) Take the origin O of the complex plane to be the vertex of a cube, so that OA, OB, OC are edges of the cube. Let the feet of the perpendiculars from A, B, C to the complex plane be the complex numbers u, v, w. Show that u2 + v2 + w2 = 0.
(2) Let (aij) be an n x n matrix. Suppose that for each i, 2 |aii| > ∑1n |aij|. By considering the corresponding system of linear equations or otherwise, show that det aij ≠ 0.
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