| 1. Find g(1/1996) + g(2/1996) + g(3/1996) + ... + g(1995/1996) where g(x) = 9x/(3 + 9x). |
|
| 2. Show that xxyyzz >= (xyz)(x+y+z)/3 for positive reals x, y, z. |
|
| 3. A convex n-gon is divided into m quadrilaterals. Show that at most m - n/2 + 1 of the quadrilaterals have an angle exceeding 180o. |
|
| 4. Show that for any n > 0 and k ≥ 0 we can find infinitely many solutions in positive integers to x13 + x23 + ... + xn3 = y3k+2. |
|
| 5. 0 < k < 1 is a real number. Define f: [0, 1] → [0, 1] by f(x) = 0 for x ≤ k, 1 - (√(kx) + √( (1-k)(1-x) ) )2 for x > k. Show that the sequence 1, f(1), f( f(1) ), f( f( f(1) ) ), ... eventually becomes zero. |
|
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Canada home
© John Scholes
jscholes@kalva.demon.co.uk
15 June 2002
Last corrected/updated 28 Nov 03