For any positive integer $n$, we define the integer $P(n)$ by : $P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$. Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$.

A polynomial $ f(x)\equal{}ax^4\plus{}bx^3\plus{}cx^2\plus{}dx$ with $ a,b,c,d>0$ is such that $ f(x)$ is an integer for $ x \in \{ \minus{}2,\minus{}1,0,1,2 \}$ and $ f(1)\equal{}1$ and $ f(5)\equal{}70$. $ (a)$ Show that $ a\equal{}\frac{1}{24}, b\equal{}\frac{1}{4},c\equal{}\frac{11}{24},d\equal{}\frac{1}{4}$. $ (b)$ Prove that $ f(x)$ is an integer for all $ x \in \mathbb{Z}$.

Let $ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients. Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?

Find the number of ordered pairs $(P(x),Q(x))$ of polynomials with integer coefficients such that \[ P(x)^2+Q(x)^2=\left(x^{4096}-1\right)^2. \] [i]Proposed by Michael Kural[/i]

The roots of $ x(x^2\plus{}8x\plus{}16)(4\minus{}x)\equal{}0$ are: $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 0,4 \qquad\textbf{(C)}\ 0,4,\minus{}4 \qquad\textbf{(D)}\ 0,4,\minus{}4,\minus{}4 \qquad\textbf{(E)}\ \text{none of these}$

[b]p1.[/b] A teenager coining home after midnight heard the hall clock striking the hour. At some moment between $15$ and $20$ minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then? [b]p2.[/b] The ratio of two positive integers $a$ and $b$ is $2/7$, and their sum is a four digit number which is a perfect cube. Find all such integer pairs. [b]p3.[/b] Given the integers $1, 2 , ..., n$ , how many distinct numbers are of the form $\sum_{k=1}^n( \pm k) $ , where the sign ($\pm$) may be chosen as desired? Express answer as a function of $n$. For example, if $n = 5$ , then we may form numbers: $ 1 + 2 + 3- 4 + 5 = 7$ $-1 + 2 - 3- 4 + 5 = -1$ $1 + 2 + 3 + 4 + 5 = 15$ , etc. [b]p4.[/b] $\overline{DE}$ is a common external tangent to two intersecting circles with centers at $O$ and $O'$. Prove that the lines $AD$ and $BE$ are perpendicular. [img]https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png[/img] [b]p5.[/b] Find all polynomials $f(x)$ such that $(x-2) f(x+1) - (x+1) f(x) = 0$ for all $x$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose $f(x)$ is a polynomial with integer coefficients such that $f(0) = 11$ and $f(x_1) = f(x_2) = ... = f(x_n) = 2002$ for some distinct integers $x_1, x_2, . . . , x_n$. Find the largest possible value of $n$.

Let $r, s \geq 1$ be integers and $a_{0}, a_{1}, . . . , a_{r-1}, b_{0}, b_{1}, . . . , b_{s-1} $ be real non-negative numbers such that $(a_0+a_1x+a_2x^2+. . .+a_{r-1}x^{r-1}+x^r)(b_0+b_1x+b_2x^2+. . .+b_{s-1}x^{s-1}+x^s) =1 +x+x^2+. . .+x^{r+s-1}+x^{r+s}$. Prove that each $a_i$ and each $b_j$ equals either $0$ or $1$.

Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$. [K. Gyory]

Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\hspace{1pt}\overline{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ has four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let $Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$ be the polynomial whose roots are $f(z_1)$, $f(z_2)$, $f(z_3)$, and $f(z_4)$, where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. What is $B + D?$ $(\textbf{A})\: {-}304\qquad(\textbf{B}) \: {-}208\qquad(\textbf{C}) \: 12i\qquad(\textbf{D}) \: 208\qquad(\textbf{E}) \: 304$

One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers $a_{-1},\ldots,a_{-k}$ on a blackboard, so he decided to do the following process just for fun: if there are real numbers $a_{-k},\ldots,a_{n-1}$ on the blackboard, then he computes the polynomial $$P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t).$$ He then writes a real number $a_n$, where $$a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}.$$ If $a_n$ is undefined (that is, $P_n(i)+P_n(-i)=0$), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists $m>n\ge0$ such that $am=an$), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)

$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$. Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$ [i]A. Golovanov[/i]

Let $P(x)$ be a polynomial with integer coefficients that has at least one rational root. Let $n$ be a positive integer. Alan and Allan are playing a game. First, Alan writes down $n$ integers at $n$ different locations on a board. Then Allan may make moves of the following kind: choose a position that has integer $a$ written, then choose a different position that has integer $b$ written, then at the first position erase $a$ and in its place write $a+P(b)$. After any nonnegative number of moves, Allan may choose to end the game. Once Allan ends the game, his score is the number of times the mode (most common element) of the integers on the board appears. Find, in terms of $P(x)$ and $n$, the maximum score Allan can guarantee. [i]Henrick Rabinovitz[/i]

Let $P(x)$ be a quadratic polynomial with real coefficients such that $P(3) = 7$ and \[P(x) = P(0) + P(1)x + P(2)x^2\] for all real $x$. What is $P(-1)$?