Found problems: 3597
The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$.
[i]Proposed by Eugene Chen
[/i]
Let $m$ and $n$ be integers greater than $2$, and let $A$ and $B$ be non-constant polynomials with complex coefficients, at least one of which has a degree greater than $1$. Prove that if the degree of the polynomial $A^m-B^n$ is less than $\min(m,n)$, then $A^m=B^n$.
[i]Proposed by Tobi Moektijono, Indonesia[/i]
Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?
Let $n, \in \mathbb {N^*} , n\ge 3$
a) Prove that the polynomial $f (x)=\frac {X^{2^n-1}-1}{X-1}-X^n $ has a divisor of form $X^p +1$ where $p\in\mathbb {N^*} $
b) Show that for $n=7$ the polynomial $f (X) $ has three divisors with integer coefficients .
$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root.
$(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.
We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.
Find a polynomial $ p\left(x\right)$ with real coefficients such that
$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$
for all real $ x$ and $ p\left(1\right)\equal{}210$.
Let $f$ be a primitive polynomial with integral coefficients (their highest common factor is $1$ ) such that $f$ is irreducible in $\mathbb{Q}[X]$ , and $f(X^2)$ is reducible in $\mathbb{Q}[X]$ . Show that $f= \pm(u^2-Xv^2)$ for some polynomials $u$ and $v$ with integral coefficients.
Let $x>0$ be a real number and $A$ a square $2\times 2$ matrix with real entries such that $\det {(A^2+xI_2 )} = 0$.
Prove that $\det{ (A^2+A+xI_2) } = x$.
Find all functions $f: \mathbb N \to \mathbb N$ for which
\[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\]
holds for all positive integers $n$.
Find all polynomials $P(X)$ with real coefficients such that if real numbers $x,y$ and $z$ satisfy $x+y+z=0,$ then the points $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$ are all colinear.
Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$:
\[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]
Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.
Consider the polynomials
\[P(x) = x^4 + ax^3 + bx^2 + cx + 1 \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.\]
Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$
Determine all polynomials $P(x)$ with real coefficients and which satisfy the following properties:
i) $P(0) = 1$
ii) for any real numbers $x$ and $y,$
\[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\]
Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that
$$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$
is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial.
[i]Proposed by Vlad Matei, Romania[/i]
Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1,\dots,v_{4n-1} \in V,$ there exists a sequence $1\leq i_1<\dots<i_{2n}\leq 4n-1$ of indices such that $v_{i_1}+\dots+v_{i_{2n}}=0.$
Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\]
Holds.
[i] Nikolai Nikolov, Oleg Mushkarov[/i]
Suppose that $x$ and $y$ are complex numbers such that \[\frac{x^{n}-y^{n}}{x-y}\] are integers for some four consecutive positive integers $n$. Prove that it is an integer for all positive integers $n$.
Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$.
Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying
\[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \]
for all $x,y \in S$.
Let $N$ be the product of all possible nonzero values of $f(81)$.
Find the remainder when when $N$ is divided by $p$.
[i]Proposed by Yang Liu and Ryan Alweiss[/i]
Find all integer pairs $(a,b)$ such that $a\cdot b$ divides $a^2+b^2+3$.
Determine all polynomials $P(x)$ with integer coefficients which satisfies $P(n)\mid n!+2$ for all postive integer $n$.
Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$
Determine all real polynomials $P$ of degree at most $22$ for which
$$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$
for all $k \in \{1, 2, 3, . . . , 21, 22\}$.
Prove that every polynomial of fourth degree can be represented in the form $P(Q(x))+R(S(x))$, where $P,Q,R,S$ are quadratic trinomials.
[i]A. Golovanov[/i]
[b]EDIT.[/b] It is confirmed that assuming the coefficients to be [b]real[/b], while solving the problem, earned a maximum score.