Having had experience with some faulty proofs above, can you now answer the following questions
(a) From the equality \((a-b)^2=(m-n)^2\) may one draw the conclusion that \(a-b=m-n\)?
(b) For what value of \(x\) do the following expressions lose their meaning?
(1) \(\frac{x^3-1}{x-1}\), (2) \(\frac{1}{x^2-1}\), (3) \(\frac{x+1}{1-2^x}\).
(c) If \(a>b\), can one conclude that \(ac>bc\) for any number \(c\)?
We prove by mathematical induction that all horses in the world are of the same colour.
Base case: There is a single horse. It has some coat colour. Because there are no other horses, all the horses have the same coat colour.
Induction step: We have \(n\) horses. We assume all of them have the same coat colour. Now we add an additional \((n+1)\)st horse. We don’t know what colour it has, but if we for now get rid of one horse from the group we had before, we suddenly have a group of \(n\) horses which includes the new one. Since we have our claim proven for \(n\), all of these horses have the same coat colour and therefore the new horse has the same coat colour as all the other ones. So every group of \(n+1\) horses has the same colour.
The third step: due to mathematical induction rule, all the horses in the world have the same coat colour. THUS WE HAVE PROVED THAT ALL HORSES IN THE WORLD ARE OF THE SAME COLOUR!
There are six natural numbers, all different, which sum up to 22. Can you find those numbers? Are they unique, or is there another bunch of such numbers?
In how many ways can you rearrange the numbers 1, 2, ..., 100 so the neighbouring numbers differ by not more than 1?
Alice finally decided to do some arithmetic. She took four different integer numbers, calculated their pairwise sums and products, and the results ( the pairwise sums and products) wrote down in her wonderful book. What could be the smallest number of different numbers Alice wrote in her book?
Alice wants to write down the numbers from 1 to 16 in such a way that the sum of two neighbouring numbers will be a square number. The Hatter tells Alice that he can write down the numbers with this property in a line, but he believes that it is absolutely impossible to write the numbers with this property in a circle. Show that he is right.
Show that \(\frac{x}{y} + {\frac{y}{z}} + {\frac{z}{x}} = 1\) is not solvable in natural numbers.
Two different numbers \(x\) and \(y\) (not necessarily integers) are such that \(x^2-2000x=y^2-2000y\). Find the sum of \(x\) and \(y\).
The board has the form of a cross, which is obtained if corner boxes of a square board of \(4 \times 4\) are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?
You are given 10 different positive numbers. In which order should they be named \(a_1, a_2, \dots , a_{10}\) such that the sum \(a_1 +2a_2 +3a_3 +\dots +10a_{10}\) is at its maximum?