Let $P (x)$ be a polynomial of degree $2n$, such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$. Determine $P (2n + 1)$.

Let $ f(x)$and $ g(y)$ be two monic polynomials of degree=$ n$ having complex coefficients. We know that there exist complex numbers $ a_i,b_i,c_i \forall 1\le i \le n$, such that $ f(x)\minus{}g(y)\equal{}\prod_{i\equal{}1}^n{(a_ix\plus{}b_iy\plus{}c_i)}$. Prove that there exists $ a,b,c\in\mathbb{C}$ such that $ f(x)\equal{}(x\plus{}a)^n\plus{}c\text{ and }g(y)\equal{}(y\plus{}b)^n\plus{}c$.

Let $p = 2027$ be the smallest prime greater than $2018$, and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$. Let $\mathrm{GF}(p)$ be the integers modulo $p$, and let $\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\mathrm{GF}(p)$ (so that all coefficients are taken modulo $p$). That is, $\mathrm{GF}(p)(X)$ is the set of fractions $\frac{P(X)}{Q(X)}$ of polynomials with coefficients in $\mathrm{GF}(p)$, where $Q(X)$ is not the zero polynomial. Let $D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X)$ be a function satisfying \[ D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2} \]for any $f,g\in \mathrm{GF}(p)(X)$ with $g\neq 0$, and such that for any nonconstant polynomial $f$, $D(f)$ is a polynomial with degree less than that of $f$. If the number of possible values of $D(P(X))$ can be written as $a^b$, where $a$, $b$ are positive integers with $a$ minimized, compute $ab$. [i]Proposed by Brandon Wang[/i]

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Let $n \ge 2$ be an integer. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{\frac{n}{3}}$, then $k^2 +k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

Let \[f(x)=\int_0^1 |t-x|t \, dt\] for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

Let $ P,Q,R $ be polynomials of degree $ 3 $ with real coefficients such that $ P(x)\le Q(x)\le R(x) , $ for every real $ x. $ Suppose $ P-R $ admits a root. Show that $ Q=kP+(1-k)R, $ for some real number $ k\in [0,1] . $ What happens if $ P,Q,R $ are of degree $ 4, $ under the same circumstances?

Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals. Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$.

The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$. Determine the value of $P(1)/P(-1)$.

Let $P(X)$ be a polynomial with integer coefficients of degree $d>0$. $(a)$ If $\alpha$ and $\beta$ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$ , then prove that $|\beta - \alpha|$ divides $2$. $(b)$ Prove that the number of distinct integer roots of $P^2(x)-1$ is atmost $d+2$.

Let $P(x)$ be a polynomial with rational coefficients such that $P(n)$ is integer for all integers $n$. Moreover: $gcd(P(1), \ldots , P(k), \ldots) = 1$. Prove that every integer $k$ can be represented in infinitely many ways of the form $\pm P(1) \pm P(2) \pm \ldots \pm P(m)$, for some positive integer $m$ and certain choices of $\pm$.

The product of uncommon real roots of the two polynomials $ x^4 \plus{} 2x^3 \minus{} 8x^2 \minus{} 6x \plus{} 15$ and $ x^3 \plus{} 4x^2 \minus{} x \minus{} 10$ is ? $\textbf{(A)}\ \minus{} 4 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ \minus{} 6 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{None}$

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

Suppose $P (x)$ is a polynomial with real coefficients such that $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$.

Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation: $P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.