| A1. Show that there is a real m x m matrix A such that A3 = A + I, and show that it must satisfy det A > 0. | |
| A2. Is there a bijection f on the positive integers such that ∑ f(n)/n2 converges? | |
| A3. f: R → R satisfies |∑1n 3k (f(x+ky) - f(x-ky))| ≤ 1 for all x, y and all positive integers n. Show that it must be constant. | |
| A4. Find all strictly monotonic functions f: (0,∞) → (0,∞) such that f(x2/f(x)) = x for all x. | |
| A5. 2n points in an n x n array are colored red. Show that one can select an even number of red points ai such that a2i, a2i+1 are in the same column (for i ≤ n, taking a2n+1 to mean a1) and a2i-1, a2i are in the same row. | |
| A6. Let S be the set of functions f: [-1,1] → R which have continuous derivatives such that f(1) > f(-1) and |f '(x)| ≤ 1 for all x. Show that for 1 < p < ∞ we can find a constant cp, such that given f ∈ S we can find x0 such that |f '(x0)| > 0 and |f(x) - f(x0)| ≤ cp f '(x0)1/p |x-x0| for all x. Does c1 exist? | |
| B1. R is a ring. For every a ∈ R, a2 = 0.Show that abc + abc = 0 for any a, b, c ∈ R. | |
| B2. A fair die is thrown 10 times. What is the probability that the total is divisble by 5? | |
| B3. x1, x2, ... , xn are reals ≥ -1 such that ∑ xi3 = 0. Show that ∑ xi ≤ n/3. | |
| B4. Show that no function f: (0,∞) → (0,∞) satisfies f(x) f(x) ≥ f(x+y)(f(x) + y) for all x,y. | |
| B5. S is the set of all words made from the letters x,y,z. The equivalence relation ~ on S satisfies (1) uu ~ u, (2) if u ~ v, then uw ~ uw and wu ~ wv. Show that every word is equivalent to a word of length ≤ 8. | |
| B6. A is a subset of Zn (the integers mod n) with at most (ln n)/100 elements. Define f(r) = ∑s∈A e2πisr/n. Show that for some r ≠ 0 we have |f(r)| ≥ |A|/2. |
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
1 Dec 2003