Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$. [i]B. Berceanu[/i]

Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.

Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that \[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \; \left|P(x)+\frac{1}{x-4}\right| \leqslant 0.01.\] Are there linear polynomials with this property? [i]Octavian Stănășilă[/i]

Let $ p$ be a prime and let $ d \in \left\{0,\ 1,\ \ldots,\ p\right\}$. Prove that \[ \sum_{k \equal{} 0}^{p \minus{} 1}{\binom{2k}{k \plus{} d}}\equiv r \pmod{p}, \]where $ r \equiv p\minus{}d \pmod 3$, $ r\in\{\minus{}1,0,1\}$.

Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? $\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$

The equation \[ x^{10}+(13x-1)^{10}=0 \] has 10 complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},$ where the bar denotes complex conjugation. Find the value of \[ \frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}. \]

Are there positive real numbers $x$, $y$ and $z$ such that $ x^4 + y^4 + z^4 = 13\text{,} $ $ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $ $ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $ [i]Proposed by Matko Ljulj.[/i]

The polynomials $P_n(x,y), n=1,2,... $ are defined by \[P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)\] Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $.

A quadratic polynomial $f(x) = Ax^2 + Bx + C$ is [I]small[/I] if $A, B, C$ are single-digit positive integers. It is [I]full[/I] if there are only finitely many positive integers that cannot be expressed as $f(x) + 3y$ for some positive integers $x$ and $y.$ Find the number of quadratic polynomials that are both small and full.

[b]p1.[/b] Fleming has a list of 8 mutually distinct integers between $90$ to $99$, inclusive. Suppose that the list has median $94$, and that it contains an even number of odd integers. If Fleming reads the numbers in the list from smallest to largest, then determine the sixth number he reads. [b]p2.[/b] Find the number of ordered pairs $(x,y)$ of three digit base-$10$ positive integers such that $x-y$ is a positive integer, and there are no borrows in the subtraction $x-y$. For example, the subtraction on the left has a borrow at the tens digit but not at the units digit, whereas the subtraction on the right has no borrows. $$\begin{tabular}{ccccc} & 4 & 7 & 2 \\ - & 1 & 9 & 1\\ \hline & 2 & 8 & 1 \\ \end{tabular}\,\,\, \,\,\, \begin{tabular}{ccccc} & 3 & 7 & 9 \\ - & 2 & 6 & 3\\ \hline & 1 & 1 & 6 \\ \end{tabular}$$ [b]p3.[/b] Evaluate $$1 \cdot 2 \cdot 3-2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5- 4 \cdot 5 \cdot 6+ ... +2017 \cdot 2018 \cdot 2019 -2018 \cdot 2019 \cdot 2020+1010 \cdot 2019 \cdot 2021$$ [b]p4.[/b] Find the number of ordered pairs of integers $(a,b)$ such that $$\frac{ab+a+b}{a^2+b^2+1}$$ is an integer. [b]p5.[/b] Lin Lin has a $4\times 4$ chessboard in which every square is initially empty. Every minute, she chooses a random square $C$ on the chessboard, and places a pawn in $C$ if it is empty. Then, regardless of whether $C$ was previously empty or not, she then immediately places pawns in all empty squares a king’s move away from $C$. The expected number of minutes before the entire chessboard is occupied with pawns equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Find $m+n$. A king’s move, in chess, is one square in any direction on the chessboard: horizontally, vertically, or diagonally. [b]p6.[/b] Let $P(x) = x^5-3x^4+2x^3-6x^2+7x+3$ and $a_1,...,a_5$ be the roots of$ P(x)$. Compute $$\sum^5_{k=1}(a^3_k -4a^2_k +a_k +6).$$ [b]p7.[/b] Rectangle $AXCY$ with a longer length of $11$ and square $ABCD$ share the same diagonal $\overline{AC}$. Assume $B$,$X$ lie on the same side of $\overline{AC}$ such that triangle$ BXC$ and square $ABCD$ are non-overlapping. The maximum area of $BXC$ across all such configurations equals $\frac{m}{n}$ for relatively prime positive integers $m$,$n$. Compute $m+n$. [b]p8.[/b] Earl the electron is currently at $(0,0)$ on the Cartesian plane and trying to reach his house at point $(4,4)$. Each second, he can do one of three actions: move one unit to the right, move one unit up, or teleport to the point that is the reflection of its current position across the line $y=x$. Earl cannot teleport in two consecutive seconds, and he stops taking actions once he reaches his house. Earl visits a chronologically ordered sequence of distinct points $(0,0)$, $...$, $(4,4)$ due to his choice of actions. This is called an [i]Earl-path[/i]. How many possible such [i]Earl-paths[/i] are there? [b]p9.[/b] Let $P(x)$ be a degree-$2022$ polynomial with leading coefficient $1$ and roots $\cos \left( \frac{2\pi k}{2023} \right)$ for $k = 1$ , $...$,$2022$ (note $P(x)$ may have repeated roots). If $P(1) =\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, then find the remainder when $m+n$ is divided by $100$. [b]p10.[/b] A randomly shuffled standard deck of cards has $52$ cards, $13$ of each of the four suits. There are $4$ Aces and $4$ Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the first King and before the first Ace is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p11.[/b] The following picture shows a beam of light (dashed line) reflecting off a mirror (solid line). The [i]angle of incidence[/i] is marked by the shaded angle; the[i] angle of reflection[/i] is marked by the unshaded angle. [img]https://cdn.artofproblemsolving.com/attachments/9/d/d58086e5cdef12fbc27d0053532bea76cc50fd.png[/img] The sides of a unit square $ABCD$ are magically distorted mirrors such that whenever a light beam hits any of the mirrors, the measure of the angle of incidence between the light beam and the mirror is a positive real constant $q$ degrees greater than the measure of the angle of reflection between the light beam and the mirror. A light beam emanating from $A$ strikes $\overline{CD}$ at $W_1$ such that $2DW_1 =CW_1$, reflects off of $\overline{CD}$ and then strikes $\overline{BC}$ at $W_2$ such that $2CW_2 = BW_2$, reflects off of $\overline{BC}$, etc. To this end, denote $W_i$ the $i$-th point at which the light beam strikes $ABCD$. As $i$ grows large, the area of $W_iW_{i+1}W_{i+2}W_{i+3}$ approaches $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] For any positive integer $m$, define $\phi (m)$ the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime. Find the smallest positive integer $N$ such that $\sqrt{ \phi (n) }\ge 22$ for any integer $n \ge N$. [b]p13.[/b] Let $n$ be a fixed positive integer, and let $\{a_k\}$ and $\{b_k\}$ be sequences defined recursively by $$a_1 = b_1 = n^{-1}$$ $$a_j = j(n- j+1)a_{j-1}\,\,\, , \,\,\, j > 1$$ $$b_j = nj^2b_{j-1}+a_j\,\,\, , \,\,\, j > 1$$ When $n = 2021$, then $a_{2021} +b_{2021} = m \cdot 2017^2$ for some positive integer $m$. Find the remainder when $m$ is divided by $2017$. [b]p14.[/b] Consider the quadratic polynomial $g(x) = x^2 +x+1020100$. A positive odd integer $n$ is called $g$-[i]friendly[/i] if and only if there exists an integer $m$ such that $n$ divides $2 \cdot g(m)+2021$. Find the number of $g$-[i]friendly[/i] positive odd integers less than $100$. [b]p15.[/b] Let $ABC$ be a triangle with $AB < AC$, inscribed in a circle with radius $1$ and center $O$. Let $H$ be the intersection of the altitudes of $ABC$. Let lines $\overline{OH}$, $\overline{BC}$ intersect at $T$. Suppose there is a circle passing through $B$, $H$, $O$, $C$. Given $\cos (\angle ABC-\angle BCA) = \frac{11}{32}$ , then $TO = \frac{m\sqrt{p}}{n}$ for relatively prime positive integers $m$,$n$ and squarefree positive integer $p$. Find $m+n+ p$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities $$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$ Prove that the degrees of the three polynomials are all even.

Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.

Let $p(z)$ be a polynomial of degree $n>0$ with complex coefficients. Prove that there are at least $n+1$ complex numbers $z$ for which $p(z)\in \{0,1\}$.