There are some coins laying flat on the table, each with a head side and a tail side. Fifteen of them are heads up, the others are tails up. You can’t feel, see or in any other way find out which side is up, but you can turn them upside down. Split the coins into two piles such that there is the same number of heads in each pile.
For an experiment a researcher puts a dot of invisible ink on a piece of paper and also draws a square with regular ink on the paper. In the experiment, a subject will draw a visible straight line on the page and the researcher, who has on special eyeglasses for spotting the dot, will tell the subject which side of the line the dot of invisible ink is on. If the dot is on the line, the researcher will tell the subject it is on the line. What is the smallest number of straight lines the subject needs to draw to figure out for sure whether the invisible dot lies in the square?
The product of 22 integers is equal to 1. Show that their sum cannot be zero.
A bus, a truck, and a motorcycle move without acceleration and pass a stationary observer at equal time intervals in the order mentioned above. They pass another observer farther down the road at the same equal time intervals but in different order. This time the order is the bus, the motorcycle, the truck. Find the speed of the bus, if the speed of the truck is 30 mph, and the speed of the motorcycle is 60 mph.
The area of a rectangle is 1 cm\(^2\). Can its perimeter be greater than 1 km?
Two pirates Bob and John were boasting that they could make the strongest coctail. Bob mixed together rum and gin, and John mixed vodka and port. It is known that rum is stronger than vodka, and gin is stronger than port. Can it be that John’s drink was stronger than Bob’s?
Is it possible that the sum and the product of some given natural numbers is equal to 99?
Selena wrote down some positive numbers. She added up those numbers, and the resulting sum was greater than 10. Then she decided to add up the squares of those numbers. Could it be possible that the sum of the squares of the numbers was less than 0.1?
There were two retired couples Robinsons and Morrises who lived next to each other in a quiet street. They loved animals, especially cats and dogs, but did not consider themselves fit enough to have the actual animals in the house. Instead, they were collecting stamps depicting cats and dogs. Mr Robinson had some stamps with cats and dogs, Mrs Robinson had her own stamps with cats and dogs, and so did Mr and Mrs Morris. It was known that Mrs Robinson had bigger proportion of stamps with cats (the number of stamps with cats to the number of all stamps she owned, i.e. stamps with cats and dogs) than Mrs Morris, and Mr Robinson had bigger proportion of stamps with cats than Mr Morris. Does it mean that the proportion of stamps with cats Mr & Mrs Robinson owned together was larger than proportion of stamps with cats owned by Mr & Mrs Morris?
Anna, Sasha, and India were running races on a sports day. Could it be that Anna was faster than Sasha in more than half of the races, Sasha was faster than India in more than half of the races, and India was faster than Anna in more than half of the races?