Found problems: 117
Let $f(x)$ be a polynomial with rational coefficients, and let $\alpha$ be a real number. If
\[\alpha^3-2019\alpha=(f(\alpha))^3-2019f(\alpha)=2021,\]
prove that $(f^n(\alpha))^3-2019f^n(\alpha)=2021$ for any positive integer $n$.
(Here, we define $f^n(x)=\underbrace{f(f(f\cdots f}_{n\text{ times}}(x)\cdots ))$.)
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$.
i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$.
ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$.
Show that $f(x)$ has two roots that are in between $-1$ and $1$.
The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
$$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$
Prove that $P(z)=Q(z).$
How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?
Find real roots $x_1$, $x_2$ of equation $x^5-55x+21=0$, if we know $x_1\cdot x_2=1$
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$ holds for all real numbers $x$.
[i]Proposed by Walther Janous, Austria[/i]
Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
Let $a,b,c,d$ be distinct integers such that
$$(x-a)(x-b)(x-c)(x-d) -4=0$$
has an integer root $r.$ Show that $4r=a+b+c+d.$
Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions
Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer
Let $a$ and $b$ be the roots of the equation $x^2 + x - 3 = 0$. Find the value of the expression $4 b^2 -a^3$.
Prove that the equation $x^n - a_1x^{n-1} - a_2x^{n-2} - ... -a_{n-1}x - a_n = 0$, where $a_1 \ge 0, a_2 \ge 0, . . . , a_n \ge 0$, cannot have two positive roots.
Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively, then there is a root $x_3$ of the equation $$\frac{a}{2}x^2 + bx + c = 0$$ such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$.
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
Prove that the number $\left(\frac{76}{\frac{1}{\sqrt[3]{77}-\sqrt[3]{75}}-\sqrt[3]{5775}}+\frac{1}{\frac{76}{\sqrt[3]{77}+\sqrt[3]{75}}+\sqrt[3]{5775}}\right)^3$ is an integer.