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 \(Q(t) = t\).

A goofy robot named Zippity only speaks using \(0\)s and \(1\)s. Every message Zippity sends is made of \(10\) digits. How many different \(10\)-digit messages can Zippity send if each message must include exactly one run of five zeros in a row? For example, \(0011000001\) would count as a valid message, but not \(1001010001\).

Four points \(A,B,C,D\) are chosen on the sides of a square of side length \(1\). The quadrilateral with vertices \(A,B,C,D\) has side lengths \(a,b,c,d\) as in the picture below. Show that \(2\leq a^2+b^2+c^2+d^2\leq 4\).

Let \(a, b, c\) be numbers such that \(a^2 + b^2 + c^2 = 1\). Show that \[-\frac12 \leq ab + bc + ac \leq 1.\]

An ordered triple of numbers is given. It is permitted to perform the following operation on the triple: to change two of them, say \(a\) and \(b\), to \(\frac{a+b}{\sqrt{2}}\) and \(\frac{a-b}{\sqrt{2}}\). Is it possible to obtain the triple \((1,\sqrt{2},1+\sqrt{2})\) from the triple \((2,\sqrt{2},\frac{1}{\sqrt{2}})\) using this operation?

(USAMO 1997) Let \(p_1, p_2, p_3,\dots\) be the prime numbers listed in increasing order, and let \(0 < x_0 < 1\) be a real number between 0 and 1. For each positive integer \(k\), define \[x_k = \begin{cases} 0 & \text{ if } x_{k-1} = 0 \\ \left\{\frac{p_k}{x_{k-1}} \right\} & \text{ if } x_{k-1} \neq 0 \end{cases}\] where \(\{x\}\) denotes the fractional part of \(x\). For example, \(\{2.53\} = 0.53\) and \(\{3.1415926...\} = 0.1415926...\). Find, with proof, all \(x_0\) satisfying \(0 <x_0 <1\) for which the sequence \(x_0, x_1, x_2,\dots\) eventually becomes 0.

Take the number \(2026^{2026}\). We remove the leading digit and add it to the remaining number. This action is repeated until there are exactly \(10\) digits left. Show that there must be two digits that are the same in the end.