Problems

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 $a+b-1$?

Prove that

(a) $$1^2+2^2+3^2+\cdots +n^2=\frac{1}{6}n(n+1)(2n+1)$$

(b) $$1^2+3^2+5^2+\cdots +(2n-1{)}^2=\frac{1}{3}n(2n-1)(2n+1)$$.

Is “I see what I eat” the same thing as “I eat what I see”?

To make it not so confusing let’s change the wording to make it more “mathematical”

“I see what I eat”=“If I eat it then I see it”

“I eat what I see”= “If I see it then I eat it”

Was the March Hare right? Is “I like what I get” the same thing as “I get what I like”?

Do you remember the example from the previous maths circle?

“Take any two non-equal numbers $a$ and $b$, then we can write; $a^2-2ab+b^2=b^2-2ab+a^2$.

Using the formula $(x-y{)}^2=x^2-2xy+y^2$, we complete the squares and rewrite the equality as $(a-b{)}^2=(b-a{)}^2$.

As we take a square root from the both sides of the equality, we get $a-b=b-a$. Finally, adding to both sides $a+b$ we get $a-b+(a+b)=b-a+(a+b)$. It simplifies to $2a=2b$, or $a=b$. Therefore, All NON-EQUAL NUMBERS ARE EQUAL! (This is gibberish, isn’t it?)”

Do you remember what the mistake was? In fact we have mixed up two things. It is indeed true “if $x=y$, then $x^2=y^2$”. But is not always true “if $x^2=y^2$, then $x=y$.” For example, consider $2^2=(-2{)}^2$, but $2\ne (-2)!$ Therefore, from $(a-b{)}^2=(b-a{)}^2$ we cannot conclude $a-b=b-a$.