A polynomial $p$ with real coefficients satisfies $p(x+1)-p(x)=x^{100}$ for all $x \in \mathbb{R}.$ Prove that $p(1-t) \ge p(t)$ for $0 \le t \le 1/2.$

Do there exist polynomials $f(x)$, $g(x)$ with real coefficients and a positive integer $k$ satisfying the following condition? (Here, the equation $x^2 = 0$ is considered to have $1$ distinct real roots. The equation $0 = 0$ has infinitely many distinct real roots.) For any real numbers $a, b$ with $(a,b) \neq (0,0)$, the number of distinct real roots of $a f(x) + b g(x) = 0$ is $k$.

Let $f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ be a polynomial of degree $n \geq 3$ with real coefficients.Suppose all roots of $f(z) =0$ lie in the half plane ${\ z \in \mathbb{C} : Re(z) < 0 \}}$. Prove that $a_k a_{k+3} < a_{k+1}a_{k+2}$ for $k = 0,1,2,3,.... n-3$

Determine the largest positive integer $n$ such that the following statement is true: There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.

$P(z)=a_nz^n+\dots+a_1z+z_0$, with $a_n\neq 0$ is a polynomial with complex coefficients, such that when $|z|=1$, $|P(z)|\leq 1$. Prove that for any $0\leq k\leq n-1$, $|a_k|\leq 1-|a_n|^2$. [i]Proposed by Yijun Yao[/i]

Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$. a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$. b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]

Suppose the polynomial $ x^{n} \plus{} a_{n \minus{} 1}x^{n \minus{} 1} \plus{} ... \plus{} a_{1} \plus{} a_{0}$ can be factorized as $ (x \plus{} r_{1})(x \plus{} r_{2})...(x \plus{} r_{n})$, with $ r_{1}, r_{2}, ..., r_{n}$ real numbers. Show that $ (n \minus{} 1)a_{n \minus{} 1}^{2}\geq\ 2na_{n \minus{} 2}$

Let $ p(x) $ be a quadratic polynomial for which : \[ |p(x)| \leq 1 \qquad \forall x \in \{-1,0,1\} \] Prove that: \[ \ |p(x)|\leq\frac{5}{4} \qquad \forall x \in [-1,1]\]

Let $m, n$ be positive integers, define: $f(x)=(x-1)(x^2-1)\cdots(x^m-1)$, $g(x)=(x^{n+1}-1)(x^{n+2}-1)\cdots(x^{n+m}-1)$. Show that there exists a polynomial $h(x)$ of degree $mn$ such that $f(x)h(x)=g(x)$, and its $mn+1$ coefficients are all positive integers.

Find all polynomials P(x) such that P(x)+P(1/x)=x+1/x

Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have $$f(x+P(x)f(y)) = (y+1)f(x)$$ (a) Prove that $P$ has degree at most 1. (b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$

Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.

Let $d$ and $m$ be two fixed positive integers. Pinocchio and Geppetto know the values of $d$ and $m$ and play the following game: In the beginning, Pinocchio chooses a polynomial $P$ of degree at most $d$ with integer coefficients. Then Geppetto asks him questions of the following form "What is the value of $P(n)$?'' for $n \in \mathbb{Z}$. Pinocchio usually says the truth, but he can lie up to $m$ times. What is, as a function of $d$ and $m$, the minimal number of questions that Geppetto needs to ask to be sure to determine $P$, no matter how Pinocchio chooses to reply?

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.

Decompose the space of homogeneous polynomials of degree $5$ in $(x, y, z)$ into irreducible subspaces invariant with respect to the rotation group $SO(3)$.

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?

Let $ \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $ f(x), g(x) \in\mathbb{Z}[x]$ be nonconstant polynomials such that $ g(x)$ divides $ f(x)$ in $ \mathbb{Z}[x]$. Prove that if the polynomial $ f(x)\minus{}2008$ has at least 81 distinct integer roots, then the degree of $ g(x)$ is greater than 5.

Let $k$ be a real number, such that the equation $kx^2 + k = 3x^2 + 2-2kx$ has two real solutions different. Determine all possible values of $k$, such that the sum of the roots of the equation is equal to the product of the roots of the equation increased by $k$.

The polynomials $P(z)$ and $Q(z)$ with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials $$P(z)+1 \;\; \text{and} \;\; Q(z)+1.$$ Prove that $P(z)=Q(z).$

Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$

Find all positive integers $d$ with the following property: there exists a polynomial $P$ of degree $d$ with integer coefficients such that $\left|P(m)\right|=1$ for at least $d+1$ different integers $m$.