A rectangle \(5 \times 9\) is cut into 10 small rectangles with sides of integer lengths. Show that there are two identical rectangles among them.
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 the maths circle, coming from several different schools, and each school sent at least two children. Show that if among any group of ten children, there are always at least three who attend the same school, then among all sixty children there must be at least fifteen who come from one particular school.
The people of Wonderland are having an election. Each voter writes the names of 10 candidates on their ballot. No name can be written twice on the same ballot.
There are 11 ballot boxes in total and each box has at least one ballot inside. The March Hare, who is counting the votes, notices something:
If he takes one ballot from each box (so 11 altogether), there is always at least one candidate whose name appears on all 11 of those papers.
Prove that there is at least one ballot box and one candidate such that every ballot in that box contains the name of that 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?