| A1. The numbers from 1 to n2 are entered in an n x n array, starting at the top left, moving along the top row, then left to right along the second row and so on. What are the possible values for the sum of n entries, one from each row and column? | |
| A2. G is an abelian group. a and b are elements of G such that am = bn = (ab)k where m, n, k, are positive integers no two of which have a common factor. Show that a = b = 1. Is this necessarily true in a non-abelian group? | |
| A3. Find lim (1 - t) ( t/(1 + t) + t2/(1 + t2) + t3/(1 + t3) + ... ). Where the limit is taken as t approaches 1 from below. | |
| A4. p(x) is a polynomial of degree n with every coefficient 0 or ±1, and p(x) is divisible by (x - 1)k for some integer k > 0. q is a prime such that q/ln q < k/ln(n+1). Show that the complex qth roots of unity must be roots of p(x). | |
| A5. A is an n x n matrix, which is not a (complex) multiple of the identity matrix. Show that there are matrices B, C such that A = B C B-1, where C has at most one non-zero diagonal entry. | |
| A6. f, g, a, b are real-valued differentiable functions on the reals such that: (1) f '(x)/g'(x) + a(x) f(x)/g(x) = b(x); (2) f(x) and g(x) → ∞ as x → ∞ (3) a(x) → A > 0, and b(x) → B > 0 as x → ∞. Show that f(x)/g(x) → B/(A + 1) as x → ∞. | |
| B1. ai and bj are non-negative reals such that (a0 + a1x + a2x2 + ... + an-1xn-1 + xn)(b0 + b1x + b2x2 + ... + bm-1xm-1 + xm) = 1 + x + x2 + ... + xm+n. Show that all ai and bj are 0 or 1. | |
| B2. The sequences a0, a1, a2, ... and b0, b1, b2, ... are defined by a0 = √2, b0 = 2, an+1 = √(2 - √(4 - an2)), bn+1 = 2 bn/(2 + √(4 + bn2)). Show that both sequences are decreasing and converge to 0. Show that 2nan increases to a limit, and that 2nbn decreases to the same limit. Show that 0 < bn - an < c/8n for some constant c. | |
| B3. Find the largest number of points on a sphere radius 1 in Rn such that the distance between any two exceeds √2. | |
| B4. A is a complex n x n matrix such that the determinant of the matrix formed by any m rows and m columns of A is zero (for m = 1, 2, 3, ... , n). Show that An = 0 and that we can apply a permutation to the rows and the same permutation to the columns so that the resulting matrix has all the elements on or below the diagonal zero. | |
| B5. Show that there is no real-valued function f(x) on the reals such that (1) f(x + y) ≥ f(x) + y f(f(x)) for all x, y and (2) f(0) > 0. | |
| B6. Show that |sin x sin 2x sin 4x ... sin 2nx | ≤ 2/√3 (sin k sin 2k sin 4k ... sin 2nk) for all x, where k = π/3. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
28 Jul 2002