| 1. How many positive integers n have a non-negative power which is a sum of 2001 non-negative powers of n? | |
| 2. Take n > 2. Solve x1 + x2 = x32, x2 + x3 = x42, ... , xn-2 + xn-1 = xn2, xn-1 + xn = x12, xn + x1 = x22 for the real numbers x1, x2, ... , xn. | |
| 3. Show that 2 < (a + b)/c + (b + c)/a + (c + a)/b - (a3 + b3 + c3)/(abc) ≤ 3, where a, b, c are the sides of a triangle. |
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| 4. Show that the area of a quadrilateral is at most (ac + bd)/2, where the side lengths are a, b, c, d (with a opposite c). When does equality hold? | |
| 5. Label the squares of a chessboard according to a knights' tour. So for i = 1, 2, ... , 63, the square labeled i+1 is one away from i in the direction parallel to one side of the board and two away in the perpendicular direction. Take any positive numbers x1, x2, ... , x64 and let yi = 1 + xi2 - (xi-12xi+1 )1/3 for i a white square and 1 + xi2 - (xi-1xi+12 )1/3 for i a black square (x0 means x64 and x65 means x1). Show that y1 + y2 + ... + y64 ≥ 48. | |
| 6. Define a0 = 1, an+1 = an + [an1/k], where k is a positive integer. Find Sk = {n | n = am for some m}. | |
| 7. Show that there are infinitely many positive integers n which do not contain the digit 0, whose digit sum divides n and in which each digit that does occur occurs the same number of times. Show that there is a positive integer n which does not contain the digit 0, whose digit sum divides n, and which has k digits. | |
| 8. The top and bottom faces of a prism are regular octagons and the sides are squares. Every edge has length 1. The midpoints of the faces are Mi. A point P inside the prism is such that each ray MiP meets the prism again in a different face at Ni. Show that PM1/M1N1 + PM2/M2N2 + ... + PM10/M10M10 = 5. | |
| 9. Find the largest possible number of subsets of {1, 2, ... , 2n} each with n elements such that the intersection of any three distinct subsets has at most one element. | |
| 10. A sequence of real numbers is such that the product of each pair of consecutive terms lies between -1 and 1 and the sum of every twenty consecutive terms is non-negative. What is the largest possible value for the sum of the first 2010 terms? |
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
5 July 2002