Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals: $\textbf{(A)}\ \dfrac{1}{2}f(n) \qquad \textbf{(B)}\ f(n)\qquad \textbf{(C)}\ 2f(n)+1 \qquad \textbf{(D)}\ f^2(n) \qquad \textbf{(E)}\ \dfrac{1}{2}(f^2(n)-1)$

Suppose $z=a+bi$ is a solution of the polynomial equation $c_4z^4+ic_3z^3+c_2z^2+ic_1z+c_0=0$, where $c_0$, $c_1$, $c_2$, $c_3$, $a$, and $b$ are real constants and $i^2=-1$. Which of the following must also be a solution? $\textbf{(A) } -a-bi\qquad \textbf{(B) } a-bi\qquad \textbf{(C) } -a+bi\qquad \textbf{(D) }b+ai \qquad \textbf{(E) } \text{none of these}$

Let $f=f_0+f_1z+f_2z^2+\ldots+f_{2n}z^{2n}$ and $f_k=f_{2n-k}$ for each $k$. Prove that $f(z)=z^ng(z+z^{-1})$, where $g$ is a polynomial of degree $n$.

Consider the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ with real coefficients $a, b$. Determine the number of distinct real roots and their multiplicities for various values of $a$ and $b$. Display your result graphically in the $(a, b)$ plane.

For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.

Prove that there is no polynomial $P$ with integer coefficients such that $P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$.

Investigate the singular points of the differential form $dt = dx/y$ on the compact Riemann surface $y^2/2 + U(x) = E$, where $U$ is a polynomial and $E$ is not a critical value.

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set $A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$ $B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$ Prove that $2|A|\geq |B|$.

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

A polynomial $f(x)$ with integer coefficients is said to be [i]tri-divisible[/i] if $3$ divides $f(k)$ for any integer $k$. Determine necessary and sufficient conditions for a polynomial to be tri-divisible.

Do there exist two polynomials $P(x)$ and $Q(x)$ with integer coefficients such that $$(P-Q)(x), \,\,\,\, P(x) \,\,\,\, and \,\,\,\,(P+Q)(x)$$ are squares of polynomials (and $Q$ is not equal to $cP$, where $c$ is a real number)? (V Prasolov)

Determine all polynomials $P{}$ with integer coefficients, satisfying $0 \leqslant P (n) \leqslant n!$ for all non-negative integers $n$. [i]Andrei Chirita[/i]

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f$.

Let $n$ be a positive integer such that $n^5+n^3+2n^2+2n+2$ is a perfect cube. Prove that $2n^2+n+2$ is not a perfect cube. [i]Proposed by Anastasija Trajanova[/i]

Find at least one polynomial $P(x)$ of degree 2001 such that $P(x)+P(1- x)=1$ holds for all real numbers $x$.

Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.

The real root of the equation $8x^3 - 3x^2 - 3x - 1 = 0$ can be written in the form $\frac{\sqrt[3]a + \sqrt[3]b + 1}{c}$, where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.

A sequence of real numbers $a_0, a_1, \dots$ satisfies the condition\[\sum\limits_{n=0}^{m}a_n\cdot(-1)^n\cdot\dbinom{m}{n}=0\]for all large enough positive integers $m$. Prove that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\ge0$.

Find all polynomials $P(x)$ with real coefficieints such that $P\left(x^2\right)=P(x)P(x-1)$ for all $x\in\mathbb R$.

Let $P(x) \in \mathbb{Z}[x]$ be a polynomial. Determine all polynomials $Q(x) \in \mathbb{Z}[x]$, such that for every positive integer $n$, there exists a polynomial $R_n(x) \in \mathbb{Z}[x]$ satisfies $$Q(x)^{2n} - 1 = R_n(x)\left(P(x)^{2n} - 1\right).$$

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

The sum of the numerical coefficients in the expansion of the binomial $ (a\plus{}b)^8$ is: $ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 7$

The roots of $ Ax^2 \plus{} Bx \plus{} C \equal{} 0$ are $ r$ and $ s$. For the roots of \[ x^2 \plus{} px \plus{} q \equal{} 0 \] to be $ r^2$ and $ s^2$, $ p$ must equal: $ \textbf{(A)}\ \frac{B^2 \minus{} 4AC}{A^2}\qquad \textbf{(B)}\ \frac{B^2 \minus{} 2AC}{A^2}\qquad \textbf{(C)}\ \frac{2AC \minus{} B^2}{A^2}\qquad \\ \textbf{(D)}\ B^2 \minus{} 2C\qquad \textbf{(E)}\ 2C \minus{} B^2$