$23$ frat brothers are sitting in a circle. One, call him Alex, starts with a gallon of water. On the first turn, Alex gives each person in the circle some rational fraction of his water. On each subsequent turn, every person with water uses the same scheme as Alex did to distribute his water, but in relation to themselves. For instance, suppose Alex gave $\frac{1}{2}$ and $\frac{1}{6}$ of his water to his left and right neighbors respectively on the first turn and kept $\frac{1}{3}$ for himself. On each subsequent turn everyone gives $\frac{1}{2}$ and $\frac{1}{6}$ of the water they started the turn with to their left and right neighbors, respectively, and keep the final third for themselves. After $23$ turns, Alex again has a gallon of water. What possibilities are there for the scheme he used in the first turn? (Note: you may find it useful to know that $1+x+x^2+\cdot +x^{23}$ has no polynomial factors with rational coefficients)

Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of $a$.

Given a positive integer $k > 1$, find all positive integers $n$ such that the polynomial $$P(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n$$ has a complex root $w$ such that $|w| = 1$.

Is there an infinite sequence $ a_0, a_1, a_2, \cdots $ of nonzero real numbers such that for $ n = 1, 2, 3, \cdots $ the polynomial \[ p_n(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \] has exactly $n$ distinct real roots?

Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win? [i](Anant Mudgal)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

Prove that there exists a uniqe $P(x)$ polynomial with real coefficients such that\\ $xy-x-y|(x+y)^{1000}-P(x)-P(y)$ for all real $x,y$.

We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$. A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial. Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that $$P(n)|f(n)$$ for all positive integers $n$.

[b]p17.[/b] Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ? [b]p18.[/b] Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? [b]p19.[/b] Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime. [b]p20.[/b] In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find, with proof, all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f$($1$) = $8$ and $f$($2$) = $2012$.

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

Find the first degree polynomial function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equation $$ f(x-1)=-3x-5-f(2), $$ for all real numbers $ x. $

Let \( P(x) \) be a polynomial with natural coefficients. We denote by \( d(n) \) the number of positive divisors of the natural number \( n \), and by \( \sigma(n) \), the sum of these divisors. The sequence \( a_n \) is defined as follows: \[ a_{n+1} \in \left\{ \begin{array}{ll} \sigma(P(d(a_n))) \\ d(P(\sigma(a_n))) \end{array} \right. \] That is, \( a_{n+1} \) is one of the two terms above. Show that there exists a constant \( C \), depending on \( a_1 \) and \( P(x) \), such that for all \( i \), \( a_i < C \); in other words, show that the sequence \( a_n \) is bounded.

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.

Let $ r_1 $, $ r_2 $, $ r_3 $ be the (possibly complex) roots of the polynomial $ x^3 + ax^2 + bx + \dfrac{4}{3} $. How many pairs of integers $ a $, $ b $ exist such that $ r_1^3 + r_2^3 + r_3^3 = 0 $?