| 1. Show that log102 is irrational. |
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| 2. a1, a2, a3, a4, a5, a6, a7 and b1, b2, b3, b4, b5, b6, b7 are two permutations of 1, 2, 3, 4, 5, 6, 7. Show that |a1 - b1|, |a2 - b2|, |a3 - b3|, |a4 - b4|, |a5 - b5|, |a6 - b6|, |a7 - b7| are not all different. |
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| 3. Let T(n) be the number of dissimilar (non-degenerate) triangles with all side lengths integral and ≤ n. Find T(n+1) - T(n). | |
| 4. The functions f and g are positive and continuous. f is increasing and g is decreasing. Show that ∫ 01 f(x) g(x) dx ≤ ∫ 01 f(x) g(1-x) dx. |
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| 5. A word is a string of the symbols a, b which can be formed by repeated application of the following: (1) ab is a word; (2) if X and Y are words, then so is XY; (3) if X is a word, then so is aXb. How many words have 12 letters? | |
| 6. Find the smallest constant c such that for every 4 points in a unit square there are two a distance ≤ c apart. |
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 September 2003
Last corrected/updated 29 Sep 03