Problems

(a) In a regular 10-gon we draw all possible diagonals. How many line segments are drawn? How many diagonals?

(b) Same questions for a regular 100-gon.

(c) Same questions for an arbitrary convex 100-gon.

A hedge fund is intending to buy 50 computers and connect each of them with eight other computers with a cable. Please do not ask why they need to do that, that is a top secret never to be made public! A friend of mine said that it’s related to some cryptocurrency research, but you should immediately forget all I just told you; it would be unwise to spread rumours! Let’s go back to the mathematical part of this story and stop the unrelated talk. The question is, how many cables do they need?

At a party there are people dressed in either blue or green. Every person dressed in blue had a chance to dance with exactly $7$ people in green, only once with each one. On the other hand, every person in green danced exactly with $9$ people in blue, also only once with each. Were there more people dressed in blue or in green at the party?

Draw $6$ points on a plane and join some of them with edges so that every point is joined with exactly $4$ other points.

There are $15$ cities in Wonderland, a foreigner was told that every city is connected with at least seven others by a road. Is this enough information to guarantee that he can travel from any city to any other city by going down one or maybe two roads?

Replace letters with digits to maximize the expression $$NO+MORE+MATH.$$ (In this, and all similar problems in the set, same letters stand for identical digits and different letters stand for different digits.)

Jane wrote a number on the board. Then, she looked at it and she noticed it lacks her favourite digit: $5$. So she wrote $5$ at the end of it. She then realized the new number is larger than the original one by exactly $1661$. What is the number written on the board?

Find the biggest 6-digit number such that each of its digits, except for the last two, is equal to the sum of its two right neighbours.

Replace the letters with digits in a way that makes the following sum as big as possible: $$SEND+MORE+MONEY.$$

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit $5$. Jane then decided to include it, but the number was written too close to the edge, so she decided to fit the $5$ in between the two digits. She noticed that the resulting number is $11$ times larger than the original. What is the sum of digits of the new number?