Find all polynomials of the form $P(x) = (-1)^nx^n + a_1x^{n-1} + a_2x^{n-2} + ...+ a_{n-1}x + a_n$ with the following two properties: (i) $\{a_1, a_2, . . . , a_n-1, a_n\} =\{0, 1\}$, and (ii) all roots of $P(x)$ are distinct real numbers

[b]p1.[/b] A light beam shines from the origin into the unit square at an angle of $\theta$ to one of the sides such that $\tan \theta = \frac{13}{17}$ . The light beam is reflected by the sides of the square. How many times does the light beam hit a side of the square before hitting a vertex of the square? [img]https://cdn.artofproblemsolving.com/attachments/5/7/1db0aad33ed9bf82bee3303c7fbbe0b7c2574f.png[/img] [b]p2.[/b] Alex is given points $A_1,A_2,...,A_{150}$ in the plane such that no three are collinear and $A_1$, $A_2$, $...$, $A_{100}$ are the vertices of a convex polygon $P$ containing $A_{101}$, $A_{102}$, $ ...$, $A_{150}$ in its interior. He proceeds to draw edges $A_iA_j$ such that no two edges intersect (except possibly at their endpoints), eventually dividing $P$ up into triangles. How many triangles are there? [img]https://cdn.artofproblemsolving.com/attachments/d/5/12c757077e87809837d16128b018895a8bcc94.png[/img] [b]p3. [/b]The polynomial P(x) has the property that $P(1)$, $P(2)$, $P(3)$, $P(4)$, and $P(5)$ are equal to $1$, $2$, $3$, $4$,$5$ in some order. How many possibilities are there for the polynomial $P$, given that the degree of $P$ is strictly less than $4$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square. (proposed by G. Batzaya, folklore)

If $A$ and $B$ are numbers such that the polynomial $x^{2017} + Ax + B$ is divisible by $(x + 1)^2$, what is the value of $B$?

Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that $$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$

Let $\mathcal{P}$ be the set of all polynomials $p(x)=x^4+2x^2+mx+n$, where $m$ and $n$ range over the positive reals. There exists a unique $p(x) \in \mathcal{P}$ such that $p(x)$ has a real root, $m$ is minimized, and $p(1)=99$. Find $n$. [i]Proposed by [b]AOPS12142015[/b][/i]

Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$

Find the number of nonzero terms of the polynomial $P(x)$ if $$x^{2018}+x^{2017}+x^{2016}+x^{999}+1=(x^4+x^3+x^2+x+1)P(x).$$

Define a positive integer $n$ to be a fake square if either $n = 1$ or $n$ can be written as a product of an even number of not necessarily distinct primes. Prove that for any even integer $k \geqslant 2$, there exist distinct positive integers $a_1$, $a_2, \cdots, a_k$ such that the polynomial $(x+a_1)(x+a_2) \cdots (x+a_k)$ takes ‘fake square’ values for all $x = 1,2,\cdots,2023$. [i]Proposed by Prof. Aditya Karnataki[/i]

Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$ and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$.

We say that a number is [i]angelic[/i] if it is greater than $10^{100}$ and all of its digits are elements of $\{1,3,5,7,8\}$. Suppose $P$ is a polynomial with nonnegative integer coefficients such that over all positive integers $n$, if $n$ is angelic, then the decimal representation of $P(s(n))$ contains the decimal representation of $s(P(n))$ as a contiguous substring, where $s(n)$ denotes the sum of digits of $n$. Prove that $P$ is linear and its leading coefficient is $1$ or a power of $10$. [i]Proposed by Grant Yu[/i]

Find all polynomials $P(x)$ with real coefficents, such that for all $x,y,z$ satisfying $x+y+z=0$, the equation below is true: \[P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))\]

Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$

Depending on the real number \( a \), find all polynomials \( P(x) \) with real coefficients such that \[ (x^3 - ax^2 + 1)P(x) = (x^3 + ax^2 + 1)P(x-1) \] for every real number \( x \).

Find the free coefficient of the polynomial $P(x)$ with integer coefficients, knowing that it is less than $1000$ in absolute value and that $P(19) = P(94) = 1994$.