Is it possible that odd integers \(a\), \(b\), \(c\), \(d\) satisfy \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1\)?
The Cheshire Cat wrote one of the numbers \(1, 2,\dots, 15\) into each box of a \(15\times15\) square table in such a way, that boxes which are symmetric to the main diagonal contain equal numbers. Every row and column consists of 15 different numbers. Show that no two numbers along the main diagonal are the same.
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.
Is it possible to divide the numbers 1, 2, 3, ..., 100 into pairs of one odd and one even number, such that in every pair except one the even number is greater than the odd number
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?