Each whole number is painted either blue or yellow, with the following rules:
The sum of two numbers painted in different colours is painted yellow.
The product of two numbers painted in different colours is painted blue.
There is at least one number of each colour. What possible colours can the product of two blue numbers have?
We make a long list of numbers in the following way. We start with \(1\) and \(1\). After that, each new number is the last digit of the sum of the two numbers right before it. For example, the beginning of the list is \[1,\,1,\,2,\,3,\,5,\,8,\,3,\,1,\,4,\ldots\]
Show that, if we keep making numbers like this forever, the list must eventually start repeating in a loop.
In Problemtown there are \(n\) farms and also \(n\) wells which we think of as points on a plane. We know that no three points lie on a straight line. The mayor wants to build straight roads so that each farm is connected to exactly one well, and each well is connected to exactly one farm. The mayor insists that no two roads are allowed to cross each other. Prove that this is always possible.
Let \(P(x)\) be a polynomial with integral coefficients. Suppose there exist four distinct integers \(a,b,c,d\) with \(P(a) = P(b) = P(c) = P(d) = 5\). Prove that there is no integer \(k\) with \(P(k) = 8\).
For which natural number \(n\) is the polynomial \(1+x^2+x^4+\dots+x^{2n-2}\) divisible by the polynomial \(1 +x+x^2+\dots+x^{n-1}\)?
Let \(P(x)\) be a polynomial with integer coefficients. Set \(P^1(x) = P(x)\) and \(P^{i+1}(x) = P(P^i(x))\). Show that if \(t\) is an integer such that \(P^k(t)=t\) for some natural number \(k\), then in fact we have \(P^2(t) = t\).
(IMO 2006) Let \(P(x)\) be a polynomial of degree \(n > 1\) with integer coefficients and let \(k\) be a positive integer. Consider the polynomial \(Q(x) = P^k(x)\). Prove that there are at most \(n\) integers \(t\) such that \(Q(t) = t\).
Look back at Problem 3. You will now prove that actually we can do better! Show that given \(5\) distinct whole numbers - not necessarily consecutive - we can also find three of them such that their sum is divisible by \(3\).
Calculate the value of: \[1\cdot \left(1+\frac{1}{2025} \right)^1 + 2\cdot \left(1+\frac{1}{2025} \right)^2 +\dots + 2025\cdot \left(1+\frac{1}{2025} \right)^{2025},\] and provide proof that your calculation is correct.
On the questioner’s planet, every alien is either a Crick or a Goop. A Crick can only ask questions whose answer is “yes,” while a Goop can only ask questions whose answer is “no.”
An alien stands on each cell of a \(4\times 4\) chessboard. Every alien asks the same question:
“Do I have an equal number of Cricks and Goops among my neighbours?”
(Here, a neighbour means any alien on a horizontally or vertically adjacent cell.)
How many Cricks and how many Goops could be on the chessboard?