SE-SKOLORNAS 1980

$a_1$ , $a_2$ , $a_3$ , $a_4$ , $a_5$ , $a_6$ , $a_7$ 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.

Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$ . Find $T(n+1)-T(n)$ .

The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that $$\int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx$$

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?

Find the smallest constant $c$ such that for every $4$ points in a unit square there are two a distance $\leq c$ apart.