20th Swedish 1980

1.  Show that log102 is irrational.
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.
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.
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
29 September 2003
Last corrected/updated 29 Sep 03