On the first day of school, in all three of the first year classes (1A, 1B, 1C), there were three lessons: Maths, French and Biology. Two classes cannot have the same lesson at the same time. 1B’s first lesson was Maths. The Biology teacher praised the students in 1B: “You have even better marks than 1A”. 1A’s second lesson was not French. Which class’s last lesson was Biology?
A bag contains balls of two different colours – black and white. What is the minimum number of balls you need to remove, without looking, to guarantee that within the removed balls at least two are the same colour.
A forest contains a million fir trees. It is known that any given tree has at most 600,000 needles. Prove that there will be two trees with the same number of needles.
You are given 12 different whole numbers. Prove that it is possible to choose two of these whose difference is divisible by 11.
A supermarket received a delivery of 25 crates of apples of 3 different types; each crate contains only one type of apple. Prove that there are at least 9 crates of apples of the same sort in the delivery.
In Scotland there are \(m\) football teams containing 11 players each. All of the players met at the airport in order to travel to England for a match. The plane made 10 journeys from Scotland to England, carrying 10 passengers each time. One player also flew to the location of the match by helicopter. Prove that at least one team made it in its entirety to the other country to play the match.
You are given 8 different natural numbers that are no greater than 15. Prove that there are three pairs of these numbers whose positive difference is the same.
Prove that in any group of 5 people there will be two who know the same number of people in that group.
Several football teams are taking part in a football tournament, where each team plays every other team exactly once. Prove that at any point in the tournament there will be two teams who have played exactly the same number of matches up to that point.
a) What is the maximum number of squares on an \(8\times 8\) grid that can be shaded in with a black pen such that each ‘L’ shaped group of 3 squares has at least one unshaded square.
b) What is the maximum number of squares on an \(8\times 8\) grid that can be shaded in with a black pen, such that each ‘L’ shaped group of 3 squares has at least one shaded square.