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 \(Q(t) = t\).
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.
Suppose that we have symbols \(a,b,c,d,e\) and an operation \(\clubsuit\) on the symbols satisfying the following rules:
\(x\;\clubsuit\;e = x\), where \(x\) can be any of \(a,b,c,d,e\).
\(a\;\clubsuit\;c = c\;\clubsuit\;a = b\;\clubsuit\;d = d\;\clubsuit\;b = e\).
any bracketing of the same string of symbols are the same; for example, \(((a\;\clubsuit\;c)\;\clubsuit\;d)\;\clubsuit\;(a\;\clubsuit\;d) = (a\;\clubsuit\;(c\;\clubsuit\;(d\;\clubsuit\;(a\;\clubsuit\;d))))\).
\((a\;\clubsuit\;b)\clubsuit\;c = d\).
We use the power notation. If \(n\geq 1\) is a natural number, we write \(a^n\) for \((\dots(a\;\clubsuit\;a)\;\clubsuit\dots)\;\clubsuit\; a\), where \(a\) appears \(n\) times. Similarly for other symbols. Let \(p,q,r,s\geq 1\) be natural numbers. Express \(a^p\;\clubsuit\;b^q\;\clubsuit\;a^r\;\clubsuit\;b^s\) using the symbols \(a,b,c,d\) no more than once (power notation allowed).