Let $n$ be an even positive integer, and let $p(x)$ be an $n$-degree polynomial such that $p(-k) = p(k)$ for $k = 1, 2, \dots , n$. Prove that there is a polynomial $q(x)$ such that $p(x) = q(x^2)$.
Let $a, b, c$ be distinct integers. Can the polynomial $(x - a)(x - b)(x - c) - 1$ be factored into the product of two polynomials with integer coefficients?
Let $p_1, p_2, \cdots, p_n$ be distinct integers and let $f(x)$ be the polynomial of degree $n$ given by $$f(x) = (x - p_1)(x - p_2)\cdots (x -p_n)$$ Prove that the polynomial $g(x) = (f(x))^2 + 1$ cannot be expressed as the product of two non-constant polynomials with integral coefficients.
Find the remainder when you divide $(x^{81} + x^{49} + x^{25} + x^9 + x)$ by $(x^3 - x)$.
Does there exist a polynomial $f(x)$ for which $xf(x - 1) = (x + 1)f(x)$
Is it possible to write the polynomial $f(x) = x^{105}-9$ as the product of two polynomials of degree less than 105 with integer coefficients?
Find all prime numbers $p$ that can be written $p = x^4 + 4y^4$, where $x, y$ are positive integers.
Is $4^{545} + 545^{4}$ a prime?
Prove that if $n>1$, then $(n^4 + 4^n)$ is a composite number.
Compute $$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
Find the largest prime divisor of $25^2+72^2$
Calculate the value of $$\dfrac{2014^4+4 \times 2013^4}{2013^2+4027^2}-\dfrac{2012^4+4 \times 2013^4}{2013^2+4025^2}$$
Let $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ be a polynomial with integral coefficients. Suppose that 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 $k$ satisfying $P(k) = 8$.
Show that $(1 + x + \cdots + x^n)^2 - x^n$ is the product of two polynomials.
Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x - 5$ in four distinct points $P_i = (x_i, y_i), i = 1, 2, 3, 4$. Prove that
$$\frac{x_1 + x_2 + x_3 + x_44}{4}$$
is independent of the line, and compute its value.
Find the value of $(2 + \sqrt{5})^{1/3} - (-2 + \sqrt{5})^{1/3}$.
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$
Which of these describes the graph of $x^2(x+y+1)=y^2(x+y+1)$ ?
There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is the value of $b?$
All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
There exist some integers, $a$, such that the equation $(a+1)x^2 -(a^2+1)x+2a^2-6=0$ is solvable in integers. Find the sum of all such $a$.
Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$, and $(P(2))^2 = P(3)$. Then $P\big(\frac72\big)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Given $7x + 13 = 328$, what is the value of $14x + 13$?
If $\frac{x + 5}{x-2} = \frac{2}{3}$, what is the value of $x$?
Ross and Max have a combined weight of 184 pounds. Ross and Seth have a combined weight of 197 pounds. Max and Seth have a combined weight of 189 pounds. How many pounds does Ross weigh?