You start with the number \(1\) on a piece of paper. You may perform two operations:
double up (multiply the number on your paper by \(2\), and erase the old number), increase by \(1\) (add \(1\) to the number on your paper, and erase the old number).
So for example, you may end up with the numbers \(1,2,3,6,\cdots\). Show that it is possible to obtain \(975\) starting from \(1\) in at most \(16\) operations.
A collection of weights is made from the weights \(1,2,4,8,\dots\) grams (that is, all powers of \(2\)). Some weights may appear several times. The weights are placed on the two pans of a balance scale and the scale is in balance. It is known that all the weights on the left pan are different.
Prove that the number of weights on the right pan is at least as large as the number of weights on the left pan.
Cut a \(7\times 7\) square into \(9\) rectangles, out of which you can construct any rectangle whose sidelengths are less than \(7\). Show how to construct the rectangles.
There are \(2^4\) cities in a kingdom. Show that it is possible to build a system of roads so that one can travel from any city to any other while passing through at most one intermediate city, and so that at most five roads leave each city.