Found problems: 3597
Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]
Prove that for each $n\in\mathbb N$ the polynomial $(x^2+x)^{2^n}+1$ is irreducible over the polynomials with integer coefficients.
Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 \minus{} 2x \plus{} 2 \le P(x) \le 2x^2 \minus{} 4x \plus{} 3\] for all real numbers $ x$, and suppose $ P(11) \equal{} 181$. Find $ P(16)$.
Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that
$p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$
For non-negative integers $n$ and $k$, let $P_{n, k}(x)$ denote the rational function \[\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.\] Show that $P_{n, k}(x)$ is actually a polynomial for all $n, k \in \mathbb{N}$.
Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.
Let $p,q$ be integers such that the polynomial $x^2+px+q+1$ has two positive integer roots. Show that $p^2+q^2$ is composite.
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
For real numbers $x_i$, the statement \[ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0\] is always true. (Prove!)
For which $n\geq 4$ integers, the statement \[x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0\] is always true. Justify your answer.
If $p (x) = c_0 + c_1x + c_2x^2 + c_3x^3$ is a polynomial with integer coefficients with $a, b,c$ integers and different from each other, prove that it cannot happen simultaneously that $p (a) = b$, $p (b) = c$ and $p (c) = a$.
Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number $a_1,a_2,...,a_{2017}$ such that $g(a_i)=f(a_{i+1})$ for i=1,2,...,2017 where indices taken modulo 2017.
The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient $1$, meets the $x$-axis at the points $(1,0),\, (2,0),\,(3,0),\dots,\, (2020,0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $a!/b$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a+b$.
Prove that the polynomial $P(X) = (X^2-12X +11)^4+23$ can not be written as the product of three non-constant polynomials with integer coefficients.
(Le Anh Vinh)
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself).
Show that $f(1)=0$.
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$.
Determine if there exists a (three-variable) polynomial $P(x,y,z)$ with integer coefficients satisfying the following property: a positive integer $n$ is [i]not[/i] a perfect square if and only if there is a triple $(x,y,z)$ of positive integers such that $P(x,y,z) = n$.
Solve the system equations
\[\left\{\begin{array}{cc}x^{4}-y^{4}=240\\x^{3}-2y^{3}=3(x^{2}-4y^{2})-4(x-8y)\end{array}\right.\]
Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
Let $f(n)$ and $g(n)$ be polynomials of degree $2014$ such that $f(n)+(-1)^ng(n)=2^n$ for $n=1,2,\ldots,4030$. Find the coefficient of $x^{2014}$ in $g(x)$.
Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$. Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$.
Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$.
[i](Walther Janous)[/i]
Let be a finite field $ K. $ Say that two polynoms $ f,g $ from $ K[X] $ are [i]neighbours,[/i] if the have the same degree and they differ by exactly one coefficient.
[b]a)[/b] Show that all the neighbours of $ 1+X^2 $ from $ \mathbb{Z}_3[X] $ are reducible in $ \mathbb{Z}_3[X] . $
[b]b)[/b] If $ |K|\ge 4, $ show that any polynomial of degree $ |K|-1 $ from $ K[X] $ has a neighbour from $ K[X] $ that is reducible in $ K[X] , $ and also has a neighbour that doesn´t have any root in $ K. $
Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.
Denote $f(x) = x^4 + 2x^3 - 2x^2 - 4x+4$. Prove that there are infinitely many primes $p$ that satisfies the following.
For all positive integers $m$, $f(m)$ is not a multiple of $p$.