Peter recorded an example of an addition on a board, after which he replaced some digits with letters, with the same figures being replaced with the same letters, and different figures with different letters. He did it such that he was left with the sum: \(CROSS + 2011 = START\). Prove that Peter made a mistake.
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
Matt, Conrad and Louie ate some sweets. Their surnames are Smith, Jones and Cooper. Smith ate 2 sweets fewer than Matt, Jones – 2 sweets fewer than Conrad, and Conrad ate more than anyone. Which of them has which last name?
Going to school, Michael found everything he needed under the pillow, under the sofa, on the table or under the table. The items he needed to find were a notebook, a cheat sheet, an mp3 player and sneakers. Under the table, he did not find a notebook or an mp3 player. His cheat sheet never lies on the floor. The mp3 player was neither on the table nor under the sofa. What was lying where, if there was only one object in each of the places?
When cleaning her children’s room, a mother found \(9\) socks. In a group of any \(4\) of the socks at least two belonged to the same child. In a group of any \(5\) of the socks no more than \(3\) had the same owner. How many children are there in the room and how many socks belong to each child?
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.
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.