| A1. Let S be the set of all points in the plane with integer coordinates. Let T be the set of all segments AB, where A, B ∈ S and AB has integer length. Prove that we cannot find two elements of T making an angle 45o. Is the same true in three dimensions? | |
A2. a, b are distinct elements of {0,1,-1}. A is the matrix:
a+b a+b2 a+b3 ... a+bm a2+b a2+b2 a2+b3 ... a2+bm a3+b a3+b2 a3+b3 ... a3+bm ... an+b an+b2 an+b3 ... an+bmFind the smallest possible number of columns of A such that any other column is a linear combination of these columns with integer coefficients. |
|
| A3. What condition must be satisfied by the coefficients u, v, w if the roots of the polynomial x3 - ux2 + vx - w can be the sides of a triangle? | |
| B1. The incircle of ABC touches BC, CA, AB at A', B', C' respectively. The line A'C' meets the angle bisector of A at D. Find ∠ADC. | |
| B2. Let s(n) be the sum of the binary digits of n. Find s(1) + s(2) + s(3) + ... + s(2k) for each positive integer k. | |
| B3. Find the integral part of 1/√1 + 1/√2 + 1/√3 + ... + 1/√1000. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Spain home
© John Scholes
jscholes@kalva.demon.co.uk
25 March 2004
Last corrected/updated 25 Mar 04