2. a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7} and b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7} are two permutations of 1, 2, 3, 4, 5, 6, 7. Show that |a_{1} - b_{1}|, |a_{2} - b_{2}|, |a_{3} - b_{3}|, |a_{4} - b_{4}|, |a_{5} - b_{5}|, |a_{6} - b_{6}|, |a_{7} - b_{7}| 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 ∫ _{0}^{1} f(x) g(x) dx ≤ ∫ _{0}^{1} 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.