Problems

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?

Every point in the plane is coloured red or blue. Show that there is a colour such that for any distance \(d\), there is a pair of points of that colour that are exactly distance \(d\) apart.

Downtown MathHattan has a grid pattern, with \(4\) streets going east-west and \(6\) streets south-north. You take a taxi from School (A) to cinema (point B), but you would like to stop by an ice cream shop first. In how many ways can a taxi get you there if you don’t want to take a route that is longer than necessary?

Gabby the Gnome has \(3\) cloaks of different colours: blue, green, and brown. He also has \(5\) different hats: \(3\) yellow and \(2\) red. Finally, he owns \(6\) different pairs of shoes: \(2\) yellow, and \(4\) red. Gabby is selecting an outfit: a cloak, a hat, and a pair of shoes. In how many ways can he do it if he wants the colour of his shoes to match the colour of the hat?

An airplane is flying from Prague to Tokyo, which are cities in the northern hemisphere with different latitudes. Suppose that the airplane must touch the equator. Could you help the pilot find the shortest path that the airplane can take, assuming that the Earth is a perfect sphere?