Problems

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!

Using mathematical induction show that $2^n>n$ for all natural numbers $n$.

Illustrate with a picture

(a) $(a-b{)}^2=a^2-2ab+b^2$,

(b) $a^2-b^2=(a-b)(a+b)$,

(c) $(a+b+c{)}^2=a^2+b^2+c^2+2ab+2ac+2bc$.

Suppose $a>b$. Explain using the number line why

(a) $a-c>b-c$, (b) $2a>2b$.

Suppose $a>b$ and $c>d$. Prove that $a+c>b+d$.

Using mathematical induction prove that $2^n\ge n+1$ for all natural numbers.

Circles and lines are drawn on the plane. They divide the plane into non-intersecting regions, see the picture below.

Show that it is possible to colour the regions with two colours in such a way that no two regions sharing some length of border are the same colour.

Consider a number consisting of $3^n$ digits, all ones, such as 111 or 111111111 for example. Show that such a number with $3^n$ digits is divisible by $3^n$.

Numbers $1,2,\dots ,n$ are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers $a$ and $b$, and write their sum $a+b$ instead. Louise enjoys erasing the numbers, and continues the procedure until only one number is left on the whiteboard. What number is it? What if instead of $a+b$ she writes $a+b-1$?