Prove that $1280000401$ is composite.

Find all polynomials $P(x)$ of odd degree $d$ and with integer coefficients satisfying the following property: for each positive integer $n$, there exists $n$ positive integers $x_1, x_2, \ldots, x_n$ such that $\frac12 < \frac{P(x_i)}{P(x_j)} < 2$ and $\frac{P(x_i)}{P(x_j)}$ is the $d$-th power of a rational number for every pair of indices $i$ and $j$ with $1 \leq i, j \leq n$.

Let $K$ be a finite field with $q$ elements and let $n \ge q$ be an integer. Find the probability that by choosing an $n$-th degree polynomial with coefficients in $K,$ it doesn't have any root in $K.$

Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$, \[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.

Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying (i) $A$ and $B$ are disjoint; (ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$. Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)

An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions $ f$ and $ g$ to be the arithmetic function $ f * g$, where \[ (f * g)(n) \equal{} \sum_{ij\equal{}n} f(i) \cdot g(j),\] and $ f^{*k} \equal{} f * f * \ldots * f$ ($ k$ times) We say that two arithmetic functions $ f$ and $ g$ are dependent if there exists a nontrivial polynomial of two variables $ P(x, y) \equal{} \sum_{i,j} a_{ij} x^i y^j$ with real coefficients such that \[ P(f,g) \equal{} \sum_{i,j} a_{ij} f^{*i} * g^{*j} \equal{} 0,\] and say that they are independent if they are not dependent. Let $ p$ and $ q$ be two distinct primes and set \[ f_1(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} p, \\ 0 & \text{ otherwise}. \end{cases}\] \[ f_2(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} q, \\ 0 & \text{ otherwise}. \end{cases}\] Prove that $ f_1$ and $ f_2$ are independent.

Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$. [i]Proposed by Milan Haiman and Carl Schildkraut[/i]

Let $z_1$ and $z_2$ be the zeros of the polynomial $f(x) = x^2 + 6x + 11$. Compute $(1 + z^2_1z_2)(1 + z_1z_2^2)$.

Consider a polynomial $f(x) = x^4 +ax^3 +bx^2 +cx+d$ with integer coefficients. Prove that if $f(x)$ has exactly one real root, then it can be factored into nonconstant polynomials with rational coefficients

Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

\[ \begin{array}{c} {x^{2} \plus{} y^{2} \plus{} z^{2} \equal{} 21} \\ {x \plus{} y \plus{} z \plus{} xyz \equal{} \minus{} 3} \\ {x^{2} yz \plus{} y^{2} xz \plus{} z^{2} xy \equal{} \minus{} 40} \end{array} \] The number of real triples $ \left(x,y,z\right)$ satisfying above system is $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None}$

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

Given 2 positive reals $a$ and $b$. There exists 2 polynomials $F(x)=x^2+ax+b$ and $G(x)=x^2+bx+a$ such that all roots of polynomials $F(G(x))$ and $G(F(x))$ are real. Show that $a$ and $b$ are more than $6$. [i]Raja Oktovin, Pekanbaru[/i]

[b]p1.[/b] Consider the following function $f(x) = \left(\frac12 \right)^x - \left(\frac12 \right)^{x+1}$. Evaluate the infinite sum $f(1) + f(2) + f(3) + f(4) +...$ [b]p2.[/b] Find the area of the shape bounded by the following relations $$y \le |x| -2$$ $$y \ge |x| - 4$$ $$y \le 0$$ where |x| denotes the absolute value of $x$. [b]p3.[/b] An equilateral triangle with side length $6$ is inscribed inside a circle. What is the diameter of the largest circle that can fit in the circle but outside of the triangle? [b]p4.[/b] Given $\sin x - \tan x = \sin x \tan x$, solve for $x$ in the interval $(0, 2\pi)$, exclusive. [b]p5.[/b] How many rectangles are there in a $6$ by $6$ square grid? [b]p6.[/b] Find the lateral surface area of a cone with radius $3$ and height $4$. [b]p7.[/b] From $9$ positive integer scores on a $10$-point quiz, the mean is $ 8$, the median is $ 8$, and the mode is $7$. Determine the maximum number of perfect scores possible on this test. [b]p8.[/b] If $i =\sqrt{-1}$, evaluate the following continued fraction: $$2i +\frac{1}{2i +\frac{1}{2i+ \frac{1}{2i+...}}}$$ [b]p9.[/b] The cubic polynomial $x^3-px^2+px-6$ has roots $p, q$, and $r$. What is $(1-p)(1-q)(1-r)$? [b]p10.[/b] (Variant on a Classic.) Gilnor is a merchant from Cutlass, a town where $10\%$ of the merchants are thieves. The police utilize a lie detector that is $90\%$ accurate to see if Gilnor is one of the thieves. According to the device, Gilnor is a thief. What is the probability that he really is one? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].