| A1. Show that x1 + 2x2 + 3x3 + ... + nxn ≤ ½n(n-1) + x1 + x22 + x33 + ... + xnn for all non-negative reals xi. | |
| A2. P is a point inside a regular tetrahedron with edge 1. Show that the sum of the distances from P to the vertices is at most 3. | |
| A3. The sequence x1, x2, x3, ... is defined by x1 = a, x2 = b, xn+2 = xn+1 + xn, where a and b are reals. A number c is a repeated value if it occurs in the sequence more than once. Show that we can choose a, b so that the sequence has more than 2000 repeated values, but not so that it has infinitely many repeated values. | |
| B1. a and b are integers such that 2na + b is a square for all non-negative integers n. Show that a = 0. | |
| B2. ABCD is a parallelogram. K is a point on the side BC and L is a point on the side CD such that BK·AD = DL·AB. DK and BL meet at P. Show that ∠DAP = ∠BAC. | |
| B3. Given a set of 2000 distinct positive integers under 10100, show that one can find two non-empty disjoint subsets which have the same number of elements, the same sum and the same sum of squares. |
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