Prove that if \(x_0^4 + a_1x_0^3 + a_2x_0^2 + a_3x_0 + a_4\) and \(4x_0^3 + 3a_1x_0^2 + 2a_2x_0 + a_3 = 0\) then \(x^4 + a_1x^3 + a_2x^2 + a_3x + a_4\) is divisible by \((x - x_0)^2\).
The expression \(ax^2+bx+c\) is an exact fourth power for all integers \(x\). Prove that \(a=b=0\).
Solving the problem: “What is the solution of the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^3 + 2000x^2 + 2000x + 3000\) (\(x\) is a real number) if \(x^2 + x + 1 = 0\)?”, Vasya got the answer of 3000. Is Vasya right?
Let \(n\) numbers are given together with their product \(p\). The difference between \(p\) and each of these numbers is an odd number.
Prove that all \(n\) numbers are irrational.
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 \(Q(t) = t\).