Prove that if the irreducible rational fraction \(p/q\) is a root of the polynomial \(P (x)\) with integer coefficients, then \(P (x) = (qx - p) Q (x)\), where the polynomial \(Q (x)\) also has integer coefficients.
There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.
Using mathematical induction prove that \[1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}.\]
There are \(n\) lines on a plane, and all the lines intersect at exactly one point. Prove that the lines divide the plane into \(2n\) parts.
There are \(n\) lines on a plane, no two lines are parallel, and no three lines cross at one point. Show that those lines dived the plane into \(\frac{n(n+1)}{2}+1\) regions.
In a sequence 2, 6, 12, 20, 30, ... find the number
(a) in the 6th place
(b) in the 2016th place.
Using mathematical induction prove that \[1 +3 +5 +\dots + (2n-1) = n^2.\]
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.
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) \[1^2 + 3^2 + 5^2 + \dots + (2n-1)^2 = \frac{1}{3} n (2n-1)(2n+1).\]