Among some number of mathematicians, every seventh is a philosopher, and among some number of philosophers every ninth is a mathematician. Who are there more of: philosophers or mathematicians?
Which five-digit numbers are there more of: ones that are not divisible by 5 or those with neither the first nor the second digit on the left being a five?
In the garden of Sandra and Lewis 2006 rose bushes were growing. Lewis watered half of all the bushes, and Sandra watered half of all the bushes. At the same time, it turned out that exactly three bushes, the most beautiful, were watered by both Sandra and Lewis. How many rose bushes have not been watered?
In a class there are 50 children. Some of the children know all the letters except “h” and they miss this letter out when writing. The rest know all the letters except “c” which they also miss out. One day the teacher asked 10 of the pupils to write the word “cat”, 18 other pupils to write “hat” and the rest to write the word “chat”. The words “cat” and “hat” each ended up being written 15 times. How many of the pupils wrote their word correctly?
11 scouts are working on 5 different badges. Prove that there will be two scouts \(A\) and \(B\), such that every badge that \(A\) is working towards is also being worked towards by \(B\).
Arrange in a row the numbers from 1 to 100 so that any two neighbouring ones differ by at least 50.
A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?
In a corridor of length 100 m, 20 sections of red carpet are laid out. The combined length of the sections is 1000 m. What is the largest number there can be of distinct stretches of the corridor that are not covered by carpet, given that the sections of carpet are all the same width as the corridor?
Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?
In the gymnasium, all students know at least one of the ancient languages – Greek or Latin, some – both languages. 85% of all children know the Greek language and 75% know Latin. How many students know both languages?