Problems

Determine the number of \(5\)-digit numbers that have only one odd digit and all other digits are even and distinct.

A group of \(4\) adults and \(5\) children is on a mountain hiking trip. At one point, the path becomes really narrow, and the hikers have to move in a line. They agreed that the line has to both start and end with an adult, for safety reasons. In how many ways can they arrange themselves?

A restaurant offers \(5\) choices of starter, \(10\) choices of the main course, and \(4\) choices of dessert. A customer can choose to eat just one course, or two different courses, or all three courses. Assuming that all food choices are available, how many different possible meals does the restaurant offer?

Find how many \(5\)-digit numbers contain only one \(0\) and one \(1\) in their decimal representation

Given a pile of cards, is it true that reversing the order of the pile by counting the cards out one by one leaves no card in its original position?

How many permutations of \(13\) cards swap the third card and the fourth card?

Consider lifting the top \(2\) cards from a pile of \(6\) cards and placing them at the bottom of the pile. Show how the same process can be done by switching adjacent pairs of cards.

Lennart is a skilled magician - he can shuffle a deck of card in the most arbitrary ways and replicate the shuffle. Once he performed a terribly complicated shuffle and repeated it right away. It turned out all the cards returned to their original position. Show that this complicated shuffle consists of only switching distinct pairs of cards (or keeping some cards where they are).

Take Ace to King of any one suit to make a pile. The cards are in numerical order, so that Ace is at the top and King is at the bottom. Count out four cards from the top one by one and put the rest of the pile on top of the four cards.

If you perform this action a total of \(7\) times, what is now the top card?

Let \(ABC\) be a triangle with midpoints \(D\) on the side \(BC\), \(E\) on the side \(AC\), and \(F\) on the side \(AB\). Prove that the perpendicular bisectors to the sides \(AB\), \(BC\), \(AC\) intersect at one point.