Intereting Posts

Is the order of universal/existential quantifiers important?
Mollifiers: Approximation
Five squares in a box.
A Proof of Legendre's Conjecture
Online tool for making graphs (vertices and edges)?
How to prove $\inf(S)=-\sup(-S)$?
Determinant of a matrix with $t$ in all off-diagonal entries.
Evaluate $\int \cos(3x) \sin(2x) \, dx$.
Why do maximal-rank transformations of an infinite set $X$ generate the whole full transformation semigroup?
Modular Inverses
A closed set in a metric space is $G_\delta$
Is the integral closure of a Henselian DVR $A$ in a finite extension of its field of fractions finite over $A$?
meaning of powers on trig functions
Learning how to prove that a function can't proved total?
What About The Converse of Lagrange's Theorem?

Let $P(x)$ be a polynolmial with degree $2009$ and leading coefficient unity such that $P(0)=2008,P(1)=2007,P(2)=2006,\ldots,P(2008)=0$,then the value of $P(2009)=n!-a$ where $n!$ is $n$ factorial,$n,a$ are natural numbers.Find $n+a.$

If i let $P(x)=a_{2009}x^{2009}+a_{2008}x^{2008}+a_{2007}x^{2007}+\cdots+a_1 x+a_0$ with $a_{2009}=1$,then it is very difficult to find $P(2009)$ with the given data.What should i do?

- How do you define functions for non-mathematicians?
- Prove that $\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$
- $2^n=C_0+C_1+\dots+C_n$
- 2011 AIME Problem 12, probability round table
- Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$
- Reducing an indicator function summation into a simpler form.

- Pi Estimation using Integers
- Can a cubic equation have three complex roots?
- Counting Irreducible Polynomials
- Chebyshev's Theorem regarding real polynomials: Why do only the Chebyshev polynomials achieve equality in this inequality?
- How to prove that $(x-1)^2$ is a factor of $x^4 - ax^2 + (2a-4)x + (3-a)$ for $a\in\mathbb R$?
- surjective, but not injective linear transformation
- How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?
- Why does this distribution of polynomial roots resemble a collection of affine IFS fractals?
- Number of real roots of $2 \cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}$
- Irreducible polynomial.

You want a way to find $P(2009)$ in terms of a factorial, but it is an extra hint! The question tells you that $P(2009)$ can be viewed as a sum of a factorial and another number! This suggest the polynomial should look something like $(x-1)(x-2) \ldots$ now let’s give up heuristic reasoning and focus on algebra $\ldots$

Note that

\begin{align}

\forall x \in [0,2008] \quad &q(x) = p(x) – (x-2008) = 0\\

&\Rightarrow q(x) = ax(x-1)(x-2) \ldots (x-2008)\\

&\Rightarrow p(x) = q(x) + (x-2008)\\

&\Rightarrow p(x) = ax(x-1)(x-2) \ldots (x-2008) + (x-2008)

\end{align}

but we don’t know $a \ldots$ wait wait, *leading coefficient unity* tells us that $a = 1$! Hence we got a solution

$$p(2009) = a\cdot 2009 \cdot (2009-1) \cdot \ldots \cdot 1 + (2009 – 2008) = 1 \cdot 2009! + 1$$

thus

$$n = 2009, a = -1\\

\Rightarrow n+a = 2008$$

Somehow I think that’s from a mathematical contest in $2008 \ldots$

- Looking for a Calculus Textbook
- Homology of the loop space
- Simplifying $\sum_{k=0}^{\lfloor{\frac{n}{2}}\rfloor}\binom{n}{2k}2^{2k}$
- Is this determinant identity known?
- A real differentiable function is convex if and only if its derivative is monotonically increasing
- Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$
- Types of polynomial functions. Quadratic, cubic, quartic, quintic, …,?
- $\operatorname{span}(x^0, x^1, x^2,\cdots)$ and the vector space of all real valued continuous functions on $\Bbb R$
- Uniform convergence of sequence of convex functions
- Conformal mapping of disk, surjective, not injective
- Basis of a $2 \times 2$ matrix with trace $0$
- Find $ord_m b^2$ if $ord_m a = 10$ and $ab\equiv 1\pmod m$
- Binary long division for polynomials in CRC computation
- Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?
- Difference between Analytic and Holomorphic function