Gerard says: the day before yesterday I was 10 years old, and next year I will turn 13. Can this be?
We call a natural number “amazing” if it has the form \(a^b + b^a\) (where \(a\) and \(b\) are natural numbers). For example, the number 57 is amazing, since \(57 = 2^5 + 5^2\). Is the number 2006 amazing?
Homework. Cut a hole in an exercise book of a size so that you yourself can climb through it.
Henry did not manage to get into the elevator on the first floor of the building and decided to go up the stairs. It takes 2 minutes to rise to the third floor. How long does it take to rise to the ninth floor?
A message is encrypted by replacing the letters of the source text with pairs of digits according to some table (known only to the sender and receiver) in which different letters of the alphabet correspond to different pairs of digits. The cryptographer was given the task to restore the encrypted text. In which case will it be easier for him to perform the task: if it is known that the first word of the second line is a “thermometer” or that the first word of the third line is “smother”? Justify your answer. (It is assumed that the cryptographic table is not known).
Find the coefficient of \(x\) for the polynomial \((x - a) (x - b) (x - c) \dots (x - z)\).