

A1. The rectangle with vertices (0, 0), (0, 3), (2, 0) and (2, 3) is rotated clockwise through a right angle about the point (2, 0), then about (5, 0), then about (7, 0), and finally about (10, 0). The net effect is to translate it a distance 10 along the xaxis. The point initially at (1, 1) traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the xaxis and the lines parallel to the yaxis through (1, 0) and (11, 0) ).


A2. M and N are real unequal n x n matrices satisfying M^{3} = N^{3} and M^{2}N = N^{2}M. Can we choose M and N so that M^{2} + N^{2} is invertible?


A3. P(x) of is a polynomial of degree n ≥ 2 with real coefficients, such that (1) it has n unequal real roots, (2) for each pair of adjacent roots a, b the derivative P'(x) is zero halfway between the roots (at x = (a + b)/2 ). Find all possible P(x).


A4. Can we find an (infinite) sequence of disks in the Euclidean plane such that: (1) their centers have no (finite) limit point in the plane; (2) the total area of the disks is finite; and (3) every line in the plane intersects at least one of the disks?


A5. Let f(z) = ∫_{0}^{z} √(x^{4} + (z  z^{2})^{2}) dx. Find the maximum value of f(z) in the range 0 ≤ z ≤ 1.


A6. An nsum of type 1 is a finite sequence of positive integers a_{1}, a_{2}, ... , a_{r}, such that: (1) a_{1} + a_{2} + ... a_{r} = n; and (2) a_{1} > a_{2} + a_{3}, a_{2} > a_{3} + a_{4}, ... , a_{r2} > a_{r1} + a_{r}, and a_{r1} > a_{r}. For example, there are five 7sums of type 1, namely: 7; 6, 1; 5, 2; 4, 3; 4, 2, 1. An nsum of type 2 is a finite sequence of positive integers b_{1}, b_{2}, ... , b_{s} such that: (1) b_{1} + b_{2} + ... + b_{s} = n; (2) b_{1} ≥ b_{2} ≥ ... ≥ b_{s}; (3) each b_{i} is in the sequence 1, 2, 4, ... , g_{j}, ... defined by g_{1} = 1, g_{2} = 2, g_{j} = g_{j1} + g_{j2} + 1; and (4) if b_{1} = g_{k}, then 1, 2, 4, ... , g_{k} is a subsequence. For example, there are five 7sums of type 2, namely: 4, 2, 1; 2, 2, 2, 1; 2, 2, 1, 1, 1; 2, 1, 1, 1, 1, 1; 1, 1, 1, 1, 1, 1, 1. Prove that for n ≥ 1 the number of type 1 and type 2 nsums is the same.


B1. For positive integers n define d(n) = n  m^{2}, where m is the greatest integer with m^{2} ≤ n. Given a positive integer b_{0}, define a sequence b_{i} by taking b_{k+1} = b_{k} + d(b_{k}). For what b_{0} do we have b_{i} constant for sufficiently large i?


B2. R is the real line. f, g: R > R are nonconstant, differentiable functions satisfying: (1) f(x + y) = f(x)f(y)  g(x)g(y) for all x, y; (2) g(x + y) = f(x)g(y) + g(x)f(y) for all x, y; and (3) f '(0) = 0. Prove that f(x)^{2} + g(x)^{2} = 1 for all x.


B3. Can we find N such that all m x n rectangles with m, n > N can be tiled with 4 x 6 and 5 x 7 rectangles?


B4. p is a prime > 2. Prove that ∑_{0≤n≤p} pCn (p+n)Cn = 2^{p} + 1 (mod p^{2}). [aCb is the binomial coefficient a!/(b! (ab)!).]


B5. p a prime > 2. How many residues mod p are both squares and squares plus one?


B6. Let a and b be positive numbers. Find the largest number c, in terms of a and b, such that for all x with 0 < x ≤ c and for all α with 0 < α < 1, we have:
a^{α} b^{1α} ≤ a sinh αx/sinh x + b sinh x(1  α)/sinh x.

