| 1. At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men M and M' and women W and W' such that M dances with W, M' dances with W', but M does not dance with W', and M' does not dance with W. | |
| 2. P is a point inside the triangle ABC. The line through P parallel to AB meets AC at AC at A0 and BC at B0. Similarly, the line through P parallel to CA meets AB at A1 and BC at C1, and the line through P parallel to BC meets AB at B2 and AC at C2. Find the point P such that A0B0 = A1B1 = A2C2. | |
| 3. Given k > 0, the sequence a1, a2, a3, ... is defined by its first two members and an+2 = an+1 + (k/n)an. For which k can we write an as a polynomial in n? For which k can we write an+1/an = p(n)/q(n)? | |
| 4. Show that there is a number of the form 199...91 (with n 9s) with n > 2 which is divisible by 1991. |
|
| 5. P0 = (1,0), P1 = (1,1), P2 = (0,1), P3 = (0,0). Pn+4 is the midpoint of PnPn+1. Qn is the quadrilateral PnPn+1Pn+2Pn+3. An is the interior of Qn. Find ∩n≥0An. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Brasil home
© John Scholes
jscholes@kalva.demon.co.uk
12 Oct 2003
Last corrected/updated 12 Oct 03