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 $x,y$.

Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$ $y-x+3=\frac{11x-12y}{x^2+y^2}$

Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property: For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.

Let $P(x)$ be a polynomial with integer coefficients of degree $n \ge 3$. If $x_1,...,x_m$ ($n\ge m\ge3$) are different integers such that $P(x_1) = P(x_2) = ... = P(x_m) = 1$, prove that $P$ cannot have integer roots$.

Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]

Given that $P(x)$ is the least degree polynomial with rational coefficients such that \[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.

Find all polynomials with real coefficients such that one can find an integer valued series $a_0, a_1, \dots$ satisfying $\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}$ for all $x$ real numbers.

Positive real numbers $a$ and $b$ satisfy the following conditions: the function $f(x)=x^3+ax^2+2bx-1$ has three different real roots, while the function $g(x)=2x^2+2bx+a$ doesn't have real roots. Prove that $a-b>1$. [i](V. Karamzin)[/i]

Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$, $p_{2}(x)=2x(x^{2}-1)$, and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$, for $n\geq 2$, let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$. For each $n$, find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

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 $P(-1)$?

(a) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac{k+2}k$ for all positive integers $k$? (b) Is there a polynomial $P(x)$ with real coefficients such that $P\left(\frac1k\right)=\frac1{2k+1}$ for all positive integers $k$?

find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $ 3$ rows of small congruent equilateral triangles, with $ 5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $ 2003$ small equilateral triangles? [asy]unitsize(15mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60); pair Fp=shift(Gp)*Bp, Ep=shift(Gp)*Cp; pair Hp=shift(Gp)*Gp, Ip=shift(Gp)*Fp; pair Jp=shift(Gp)*Hp; pair[] points={Ap,Bp,Cp,Dp,Ep,Fp,Gp,Hp,Ip,Jp}; draw(Ap--Dp--Jp--cycle); draw(Gp--Bp--Ip--Hp--Cp--Ep--cycle); for(pair p : points) { fill(circle(p, 0.07),white); } pair[] Cn=new pair[5]; Cn[0]=centroid(Ap,Bp,Gp); Cn[1]=centroid(Gp,Bp,Fp); Cn[2]=centroid(Bp,Fp,Cp); Cn[3]=centroid(Cp,Fp,Ep); Cn[4]=centroid(Cp,Ep,Dp); label("$1$",Cn[0]); label("$2$",Cn[1]); label("$3$",Cn[2]); label("$4$",Cn[3]); label("$5$",Cn[4]); for (pair p : Cn) { draw(circle(p,0.1)); }[/asy] $ \textbf{(A)}\ 1,\!004,\!004 \qquad \textbf{(B)}\ 1,\!005,\!006 \qquad \textbf{(C)}\ 1,\!507,\!509 \qquad \textbf{(D)}\ 3,\!015,\!018 \qquad \textbf{(E)}\ 6,\!021,\!018$

Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if \[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\] for every prime number $p$ and every real number $y.$

Let $a$ and $b$ be complex non-zero numbers and $z_1,z_2$ the roots of the polynomials $X^2+aX+b$. Show that $|z_1+z_2|=|z_1|+|z_2|$ if and only if there exists a real number $\lambda\ge 4$ such that $a^2=\lambda b$.

Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that $\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$ Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$

Prove the following statement: If a polynomial $p(x) = x^3 + Ax^2 + Bx +C$ has three real positve roots at least two of which are distinct, then $A^2 +B^2 +18C > 0$.

Let $u, v, w$ be the roots of $x^3 -5x^2 + 4x-3 = 0$. Find a cubic polynomial having $u^3, v^3, w^3$ as roots.

Find all polynomials $P(x)$ with integer coefficients such that for all positive number $n$ and prime $p$ satisfying $p\nmid nP(n)$, we have $ord_p(n)\ge ord_p(P(n))$.

A polynomial is called positivist if it can be written as a product of two non-constant polynomials with non-negative real coefficients. Let $f(x)$ be a polynomial of degree greater than one such that $f(x^n)$ is positivist for some positive integer $n$. Show that $f(x)$ is positivist.

Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\]

Find with proof all non–zero real numbers $a$ and $b$ such that the three different polynomials $x^2 + ax + b, x^2 + x + ab$ and $ax^2 + x + b$ have exactly one common root.

Find the polynomials $ f(x)$ having the following properties: (i) $ f(0) \equal{} 1$, $ f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0$ (ii) $ f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0$