| A1. Show that the equation x2 + y2 + z2 = 3xyz has infinitely many solutions in positive integers. |
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| A2. Does there exist a set of n > 2 points in the plane such that no three are collinear and the circumcenter of any three points of the set is also in the set? | |
| A3. Let f(n) be the smallest number of 1s needed to represent the positive integer n using only 1s, + signs, x signs and brackets. For example, you could represent 80 with 13 1s as follows: (1+1+1+1+1)x(1+1+1+1)x(1+1+1+1). Show that 3 log3n ≤ f(n) ≤ 5 log3n for n > 1. |
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| B1. ABC is acute-angled. D s a variable point on the side BC. O1 is the circumcenter of ABD, O2 is the circumcenter of ACD, and O is the circumcenter of AO1O2. Find the locus of O. | |
| B2. There are n boys B1, B2, ... , Bn and n girls G1, G2, ... , Gn. Each boy ranks the girls in order of preference, and each girl ranks the boys in order of preference. Show that we can arrange the boys and girls into n pairs so that we cannot find a boy and a girl who prefer each other to their partners. For example if (B1, G3) and (B4, G7) are two of the pairs, then it must not be the case that B4 prefers G3 to G7 and G3 prefers B4 to B1. |
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| B3. Let p(x) be the polynomial x3 + 14x2 - 2x + 1. Let pn(x) denote p(pn-1(x)). Show that there is an integer N such that pN(x) - x is divisible by 101 for all integers x. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
24 January 2004
Last corrected/updated 24 Jan 04