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A1. Prove that we can find a real polynomial p(y) such that p(x - 1/x) = xn - 1/xn (where n is a positive integer) iff n is odd.
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A2. Let ω3 = 1, ω ≠ 1. Show that z1, z2, -ωz1 - ω2z2 are the vertices of an equilateral triangle.
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A3. Let C be the complex numbers. f : C → C satisfies f(z) + z f(1 - z) = 1 + z for all z. Find f.
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A4. R is the reals. f, g : [0, 1] → R are arbitary functions. Show that we can find x, y such that |xy - f(x) - g(y)| ≥ 1/4.
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A5. At a particular moment, A, T and B are in a vertical line, with A 50 feet above T, and T 100 feet above B. T flies in a horizontal line at a fixed speed. A flies at a fixed speed directly towards B, B flies at twice T's speed, also directly towards T. A and B reach T simultaneously. Find the distance traveled by each of A, B and T, and A's speed.
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A6. Given any real numbers α1, α2, ... , αm, β, show that for m, n > 1 we can find m real n x n matrices A1, ... , Am such that det Ai = αi, and det(∑ Ai) = β.
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A7. Let R be the reals. Let f : [a, b] → R have a continuous derivative, and suppose that if f(x) = 0, then f '(x) ≠ 0. Show that we can find g : [a, b] → R with a continuous derivative, such that f(x)g'(x) > f '(x)g(x) for all x ∈ [a, b].
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B1. Join each of m points on the positive x-axis to each of n points on the positive y-axis. Assume that no three of the resulting segments are concurrent (except at an endpoint). How many points of intersection are there (excluding endpoints)?
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B2. Show that any positive real can be expressed in infinitely many ways as a sum ∑ 1/(10 an), where a1 < a2 < a3 < ... are positive integers.
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B3. Find a continuous function f : [0, 1] → [0, 1] such that given any β ∈ [0, 1], we can find infinitely many α such that f(α) = β.
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B4. A is the 5 x 5 array:
11 17 25 19 16
24 10 13 15 3
12 5 14 2 18
23 4 1 8 22
6 20 7 21 9
Pick 5 elements, one from each row and column, whose minimum is as large as possible (and prove it so).
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B5. L1 is the line { (t + 1, 2t - 4, -3t + 5) : t real } and L2 is the line { (4t - 12, -t + 8, t + 17) : t real }. Find the smallest sphere touching L1 and L2.
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B6. α and β are positive irrational numbers satisfying 1/α + 1/β = 1. Let an = [n α] and bn = [n β], for n = 1, 2, 3, ... . Show that the sequences an and bn are disjoint and that every positive integer belongs to one or the other.
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B7. Given any finite ordered tuple of real numbers X, define a real number [X], so that for all xi, α:
(1) [X] is unchanged if we permute the order of the numbers in the tuple X;
(2) [(x1 + α, x2 + α, ... , xn + α)] = [(x1, x2, ... , xn)] + α;
(3) [(-x1, -x2, ... , -xn)] = - [(x1, x2, ... , xn)];
(4) for y1 = y2 = ... = yn = [(x1, x2, ... , xn)], we have [(y1, y2, ... , yn, xn+1)] = [(x1, x2, ... , xn+1)].
Show that [(x1, x2, ... , xn)] = (x1 + x2 + ... + xn)/n.
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