| A1. Find all triples (x,y,z) of positive rationals such that x + y + z, 1/x + 1/y + 1/z and xyz are all integers. | |
| A2. L, L' are parallel lines. C is a circle that does not intersect L. A is a variable point on L. The two tangents to C from A meet L' in two points with midpoint M. Show that the line AM passes through a fixed point (as A varies). | |
| A3. k is a fixed positive integer. Let an be the number of maps f from the subsets of {1, 2, ... , n} to {1, 2, ... , k} such that for all subsets A, B of {1, 2, ... , n} we have f(A ∩ B) = min(f(A), f(B)). Find limn→∞ an1/n. | |
| B1. m, n are relatively prime. We have three jugs which contain m, n and m+n liters. Initially the largest jug is full of water. Show that for any k in {1, 2, ... , m+n} we can get exactly k liters into one of the jugs. | |
| B2. A parallelepiped has vertices A1, A2, ... , A8 and center O. Show that 4 ∑ |OAi|2 ≤ (∑|OAi|)2. | |
| B3. The distinct reals x1, x2, ... , xn (n > 3) satisfy ∑ xi = 0, &sum xi2 = 1. Show that four of the numbers a, b, c, d must satisfy a + b + c + nabc ≤ ∑ xi3 ≤ a + b + d + nabd. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Polish home
© John Scholes
jscholes@kalva.demon.co.uk
1 March 2004
Last corrected/updated 1 Mar 04