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Take \(x=-\frac{1}{2}\). It solves \[2x+1=0.\] Add \(x^2\) to both sides of the above equation \[x^2 + 2x +1 = x^2.\] Completing the square, we rewrite the equality as \[(x+1)^2 = x^2.\] Now, taking the square root from the both sides of the equality we get \[x+1=x.\] Subtracting \(x\) from both sides we get \[1=0\] Can you explain what went wrong in our reasoning? Why?

Now! – said the Hatter. – you might just as well say that “I see what I eat” is the same thing as “I don’t eat what I don’t see”!

– You might just as well say, – added the March Hare, –that “I like what I get” is the same thing as “I don’t get what I don’t like”!

– You might just as well say,– added the Dormouse, which seemed to be talking in its sleep, – that “I breathe when I sleep” is the same thing as “I don’t sleep when I don’t breathe”!

– It is the same thing indeed,– said the Hatter, and here the conversation dropped.

Do you agree with them? Why?

We already know that \(x=y\) does not follow from \(x^2=y^2\).

But is it true that if \(x^2 \neq y^2\) then \(x \neq y\)?

Tweedledum and Tweedledee were standing under a tree, each with an arm round the other’s neck. Last time Alice met them she knew immediately which was which, because one of them had “DUM” embroidered on his collar, and the other “DEE”. ‘No embroidery this time,’ she said to herself. ‘How do I distinguish them?’. ‘O, yes!’, she suddenly remembered that one of them always tells the truth, while the other always lies. ‘I have to ask one of them just one question, he will answer ‘yes’ or ‘no’, and I will know which is which’, she thought. What question was Alice going to ask?

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?