| A1. The positive integers x1, x2, ... , x7 satisfy x6 = 144, xn+3 = xn+2(xn+1+xn) for n = 1, 2, 3, 4. Find x7. | |
| A2. Find all real solutions to 3(x2 + y2 + z2) = 1, x2y2 + y2z2 + z2x2 = xyz(x + y + z)3. | |
| A3. ABCD is a tetrahedron. DE, DF, DG are medians of triangles DBC, DCA, DAB. The angles between DE and BC, between DF and CA, and between DG and AB are equal. Show that area DBC ≤ area DCA + area DAB. | |
| B1. The sequence a1, a2, a3, ... is defined by a1 = 0, an = a[n/2] + (-1)n(n+1)/2. Show that for any positive integer k we can find n in the range 2k ≤ n < 2k+1 such that an = 0. | |
| B2. ABCDE is a convex pentagon such that DC = DE and ∠C = ∠E = 90o. F is a point on the side AB such that AF/BF = AE/BC. Show that ∠FCE = ∠FDE and ∠FEC = ∠BDC. | |
| B3. Given any n points on a unit circle show that at most n2/3 of the segments joining two points have length > √2. |
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
1 March 2004
Last corrected/updated 1 Mar 04