Problems

In any group of 10 children, out of a total of 60 pupils, there will be three who are in the same class. Will it always be the case that amongst the 60 pupils there will be: 1) 15 classmates? 2) 16 classmates?

Find the smallest four-digit number $CEEM$ for which there exists a solution to the rebus $MN+PORG=CEEM$. (The same letters correspond to the same numbers, different – different.)

A square napkin was folded in half, the resulting rectangle was then folded in half again (see the figure). The resulting square was then cut with scissors (in a straight line). Could the napkin have been broken up a) into 2 parts? b) into 3 parts? c) into 4 parts? d) into 5 parts? If yes – illustrate such a cut, if not – write the word “no”.

Peter has some coins in his pocket. If Peter pulls $3$ coins from his pocket, without looking, there will always be a £1 coin among them. If Peter pulls $4$ coins from his pocket, without looking, there will always be a £2 coin among them. Peter pulls $5$ coins from his pocket. Identify these coins.

One day, Claudia, Sofia and Freia noticed that they brought the same toy cars to kindergarten. Claudia has a car with a trailer, a small car and a green car without a trailer. Sofia has a car without a trailer and a small green one with a trailer, and Freia has a big car and a small blue car with a trailer. What kind of car (in terms of colour, size and availability of a trailer) did all of the girls bring to the kindergarten? Explain the answer.

The $KUB$ is a cube. Prove that the ball, $CIR$, is not a cube. ($KUB$ and $CIR$ are three-digit numbers, where different letters denote different numbers).

How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,

a) if each number can occur only once?

b) if each number can occur several times?

The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from $n$ books of different heights, if: a) $n=4$; b) $n=5$?

Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.

Specify any solution of the puzzle: $2014+YES=BEAR$.