Found problems: 3597
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
Let $f: \mathbb{R}\to\mathbb{R}$ be a function such that $(f(x))^{n}$ is a polynomial for every integer $n\geq 2$. Is $f$ also a polynomial?
Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial
\[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\]
with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality
\[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]
Prove that there does not exist a polynomial $P (x)$ with integer coefficients and a natural number $m$ such that
$$x^m + x + 2 = P(P(x))$$
holds for all integers $x$.
Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that
\[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \]
for all real number $x$.
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 $n\ge0$.
If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$?
(Note: The roots of the polynomial do not have to be different from each other.)
Let $S$ be a set of integers containing the numbers $0$ and $1996$. Suppose further that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. Prove that $-2$ belongs to $S$.
Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.
Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows.
(1) $g_n(x)=(1+x)^n$
(2) $g_n(x)=\sin n\pi x$
(3) $g_n(x)=e^{nx}$
Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$.
[i]From the folklore[/i]
$P(x)$ is a polynomial with real coefficients such that $P(a_1) = 0, P(a_{i+1}) = a_i$ ($i = 1, 2,\ldots$) where $\{a_i\}_{i=1,2,\ldots}$ is an infinite sequence of distinct natural numbers. Determine the possible values of degree of $P(x)$.
Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.
Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions:
[b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2}
x^2 + a_{2n}, a_0 > 0$;
[b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left(
\begin{array}{c}
2n\\
n\end{array} \right) a_0 a_{2n}$;
[b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of
\[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \]
Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$
Let $ f $ be a polynom of degree $ 3 $ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $ a,b $ and two roots $ x,y $ of $ f $ such that $ ax+by $ is rational, then all roots of $ f $ are rational.
Let a polynomial $ P(x) \equal{} rx^3 \plus{} qx^2 \plus{} px \plus{} 1$ $ (r > 0)$ such that the equation $ P(x) \equal{} 0$ has only one real root. A sequence $ (a_n)$ is defined by $ a_0 \equal{} 1, a_1 \equal{} \minus{} p, a_2 \equal{} p^2 \minus{} q, a_{n \plus{} 3} \equal{} \minus{} pa_{n \plus{} 2} \minus{} qa_{n \plus{} 1} \minus{} ra_n$.
Prove that $ (a_n)$ contains an infinite number of nagetive real numbers.
Let $N$ denote the number of ordered 2011-tuples of positive integers $(a_1,a_2,\ldots,a_{2011})$ with $1\le a_1,a_2,\ldots,a_{2011} \le 2011^2$ such that there exists a polynomial $f$ of degree $4019$ satisfying the following three properties:
[list] [*] $f(n)$ is an integer for every integer $n$; [*] $2011^2 \mid f(i) - a_i$ for $i=1,2,\ldots,2011$; [*] $2011^2 \mid f(n+2011) - f(n)$ for every integer $n$. [/list]
Find the remainder when $N$ is divided by $1000$.
[i]Victor Wang[/i]
Let $n$ be an integer greater than $1$. Let $x\in\mathbb R$.
(a) Evaluate $S(x,n)=\sum(x+p)(x+q)$, where the summation is over all pairs $(p,q)$ of different numbers from $\{1,2,\ldots,n\}$.
(b) Do there exist integers $x,n$ for which $S(x,n)=0$?
The roots of the equation $x^3 - x + 1 = 0$ are $a, b, c$. Find $a^8 + b^8 + c^8$.
Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that
\[ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) \]
for every real number $x$ and every positive integer $n$.
Let $f (x) = x^2 + px + q$, where $p, q$ are integers.
Prove that there is an integer $m$ such that $f (m) = f (2015) \cdot f (2016)$.
Let $ a_1,a_{2},\ldots ,a_{100}$ be nonnegative integers satisfying the inequality
\[a_1\cdot (a_1-1)\cdot\ldots\cdot (a_1-20)+a_2\cdot (a_2-1)\cdot\ldots\cdot (a_2-20)+\\ \ldots+a_{100}\cdot (a_{100}-1)\cdot\ldots\cdot (a_{100}-20)\le 100\cdot 99\cdot 98\cdot\ldots\cdot 79.\]
Prove that $a_1+a_2+\ldots+a_{100}\le 9900$.