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. Among any group of ten children, there are always at least three who go to the same school. Prove that there must be at least fifteen children from one school among all sixty who came to the maths circle.
The people of Wonderland are having an election. Each voter writes the names of 10 candidates on their ballot paper (they cannot write the same name twice on their ballot paper).
There are 11 ballot boxes in total and each ballot box has at least one ballot paper inside. The March Hare, who is counting the votes, notices something:
If he takes one ballot paper from each ballot box (so 11 ball together), there will always be at least one candidate whose name appears on all 11 of those papers.
Prove that there is at least one ballot box and a candidate’s name such that every ballot paper on 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?