SE-SKOLORNAS 1972

Find the largest real number $a$ such that $$\left\{ \begin{array}{l} x - 4y = 1 ax + 3y = 1 \end{array} \right.$$ has an integer solution.

A rectangular grid of streets has $m$ north-south streets and $n$ east-west streets. For which $m, n > 1$ is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?

A steak temperature $5^\circ$ is put into an oven. After $15$ minutes, it has temperature $45^\circ$ . After another $15$ minutes it has temperature $77^\circ$ . The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.

Put $x = \log_{10} 2$ , $y = \log_{10} 3$ . Then $15 < 16$ implies $1 - x + y < 4x$ , so $1 + y < 5x$ . Derive similar inequalities from $80 < 81$ and $243 < 250$ . Hence show that $$0.47 < \log_{10} 3 < 0.482.$$

Show that $$\int\limits_0^1 \frac{1}{(1+x)^n} dx > 1-\frac{1}{n}$$ for all positive integers $n$ .

$a_1,a_2,a_3,\dots$ and $b_1,b_2,b_3,\dots$ are sequences of positive integers. Show that we can find $m < n$ such that $a_m \leq a_n$ and $b_m \leq b_n$ .