| 1. Show that there are 3k-1 positive integers n such that n has k digits, all odd, n is divisible by 5 and n/5 has k odd digits. [For example, for k = 2, the possible numbers are 55, 75 and 95.] |
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| 2. ABCDEF is a convex hexagon is such that opposite sides are parallel and the perpendicular distance between each pair of opposite sides is equal. The angles at A and D are 90o. Show that the diagonals BE and CF are at 45o. | |
| 3. The polynomials pn(x) are defined by p0(x) = 0, p1(x) = x, pn+2(x) = x pn+1(x) + (1 - x) pn(x). Find the real roots of each pn(x). |
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| 4. The real numbers w, x, y, z have zero sum and sum of squares 1. Show that the sum wx + xy + yz + zw lies between -1 and 0. |
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| 5. P is a convex polyhedron. S is a sphere which meets each edge of P at the two points which divide the edge into three equal parts. Show that there is a sphere which touches every edge of P. |
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| 6. k, n are positive integers such that n > k > 1. Find all real solutions x1, x2, ... , xn to xi3(xi2 + xi+12 + ... + xi+k-12) = xi-12 for i = 1, 2, ... , n. [Note that we take x0 to mean xn, and xn+j to mean xj.] |
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| 7. Show that there are no non-negative integers m, n satisfying m! + 48 = 48 (m + 1)n. |
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| 8. Show that there is no real polynomial of degree 998 such that p(x)2 - 1 = p(x2 + 1) for all x. |
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| 9. A block is a rectangular parallelepiped with integer sides a, b, c which is not a cube. N blocks are used to form a 10 x 10 x 10 cube. The blocks may be different sizes. Show that if N ≥ 100, then at least two of the blocks must have the same dimensions and be placed with corresponding edges parallel. Prove the same for some number smaller than 100. |
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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
29 July 2002