Let \(n!= n\times (n-1) \times(n-2)\times \dots \times 2\times 1\). Prove that if \(n!+1\) is divisible by \(n+1\), then \(n+1\) is a prime number.
Denote by \(\overline{ab} = 10a +b\) the two-digit number whose first and second digits are \(a\) and \(b\) respectively. Do there exist two \(2\)-digit numbers \(\overline{ab}\) and \(\overline{cd}\) such that \(\overline{ab} \times \overline{cd} = \overline{abcd}\)? (Here \(\overline{abcd}\) is a four-digit number with digits \(a\), \(b\), \(c\) and \(d\), i.e. \(\overline{abcd} = 1000a + 100b +10c +d\).)
Sixty children came to a maths circle at UCL. Among any ten children who came to the circle there are three from the same school. Show that there are 15 children from the same school among all the children who came to the maths circle.
The people in Wonderland are having an election. Every voter writes 10 candidate names on a bulletin and puts it in a ballot box.
There are 11 ballot boxes all together. The March Hare, who is counting the votes, is very surprised to discover that there is at least one bulletin in each ballot box. Moreover, he learned that if he takes one bulletin from each ballot box (11 bulletins all together), then there is always a candidate whose name is written in each of the 11 chosen bulletins. Prove that there is a ballot box, in which all the bulletins contain the name of the same candidate.
A train was moving in one direction for 5.5 hours. During any one hour period during the journey the train covered exactly 100 km.
(a) Was the train moving always with the same speed during the trip?
(b) Is it true that the average speed of the train was equal to 100 km per hour?
Is it true that among any six natural numbers one can always choose either three mutually prime numbers or three numbers with a common divisor?
The White Hare was very good at keeping his accounts. Every month he wrote his income and expenses in a big book.
Alice looked into his book and discovered that during any five consecutive months his income was less than his expenses, but over the past year his total income was larger than his total expenses. How could it be?
(a) Is it true that among any six natural numbers one can always choose either three mutually prime numbers, or three numbers, such that each two have a common divisor?
(b) Is it true that among any six people one can always choose either three strangers, or three people who know each other pairwise?
(a) Can you represent 203 as a sum of natural numbers in such a way that both the product and the sum of those numbers are equal to 203?
(b) Which numbers you cannot represent as a sum of natural numbers in such a way that both the product and the sum of those numbers are equal to the original number?
(a) The sum of some numbers is equal to one. Can it be that the sum of the cubes of these numbers is greater than one?
(b) The sum of some numbers is equal to one. Moreover, it is known that each of the numbers is less than one. Can it be that the sum of the cubes of these numbers is greater than one?