Using mathematical induction prove that \(2^n \geq n + 1\) 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) \[1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{1}{6} n (n+1)(2n+1)\]
(b) \[1^2 + 3^2 + 5^2 + \dots + (2n-1)^2 = \frac{1}{3} n (2n-1)(2n+1)\].
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 \(2 \neq (-2)!\) Therefore, from \((a-b)^2 = (b-a)^2\) we cannot conclude \(a-b = b-a\).
Was the Dormouse right? Is “I breathe when I sleep” the same thing as “I sleep when I breathe”?
Mary Ann yawns every time it rains. In fact Mary Ann is yawning right now. Is it raining at the moment?