Found problems: 4275
Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of
\[ |(x-y)(y-z)(z-x) | \]
Find all polynomial $P(x)$ with degree $\leq n$and non negative coefficients such that $$P(x)P(\frac{1}{x})\leq P(1)^2$$ for all positive $x$. Here $n$ is a natuaral number
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.
Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.
Suppose $p(x)$ is a polynomial with integer coefficients. It is known that $p(a) - p(b) = 1$ (where $a$ and $b$ are integers). Prove that $a$ and $b$ differ by $1$ .
(Folklore)
What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ?
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
The polynomial $p(x)$ is of degree $9$ and $p(x)-1$ is exactly divisible by $(x-1)^{5}$.
Given that $p(x) + 1$ is exactly divisible by $(x+1)^{5}$, find $p(x)$.
There are some counters in some cells of $100\times 100$ board. Call a cell [i]nice[/i] if there are an even number of counters in adjacent cells. Can exactly one cell be [i]nice[/i]?
[i]K. Knop[/i]
Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions:
[b]a)[/b] $ f''>0\implies f-\alpha >0 $
[b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $
Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$.
[img]http://i14.tinypic.com/4tk6mnr.png[/img]
a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero.
b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.
For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}). \]
Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$.
(a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$.
(b) Show that $ a_{2008} \neq 0$
Let $m=\frac{-1+\sqrt{17}}{2}$. Let the polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is given, where $n$ is a positive integer, the coefficients $a_0,a_1,a_2,...,a_n$ are positive integers and $P(m) =2018$ . Prove that the sum $a_0+a_1+a_2+...+a_n$ is divisible by $2$ .
The polynomial $P(X)$ is defined by $P(X)=(X+2X^{2}+\ldots +nX^{n})^{2}=a_{0}+a_{1}X+\ldots +a_{2n}X^{2n}$. Prove that $a_{n+1}+a_{n+2}+\ldots +a_{2n}=\frac{n(n+1)(5n^{2}+5n+2)}{24}$.
Let $S$ be a finite set of polynomials in two variables, $x$ and $y$. For $n$ a positive integer, define $ \Omega _ { n } ( S ) $ to be the collection of all expressions $ p _ { 1 } p _ { 2 } \dots p _ { k } ,$ where $p_i \in S$ and $1\leq k \leq n$. Let $d_n(S)$ indicate the maximum number of linearly independent polynomials in $ \Omega _ { n } ( S ) $. For example, $ \Omega _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = \left\{ x ^ { 2 } , y , x ^ { 2 } y , x ^ { 4 } , y ^ { 2 } \right\} $ and $d _ { 2 } \left( \left\{ x ^ { 2 } , y \right\} \right) = 5 $
(a) Find $ d _ { 2 } ( \{ 1 , x , x + 1 , y \} ) $.
(b) Find a closed formula in $n$ for $ d _ { n } ( \{ 1 , x , y \} ) $.
(c) Calculate the least upper bound over all such sets of $ \overline{\text{lim}} _ { n \rightarrow \infty } \frac { \log d _ { n } ( S ) } { \log n } $ ($ \overline{\text{lim}} _ { n \rightarrow \infty } a _ { n } = \lim _ { n \rightarrow \infty } ( \sup \left\{ a _ { n } , a _ { n + 1 } , \ldots \right\} $, where sup means supremum or least upper bound.)
Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.
Consider the polynomial $P(x)=X^{2n}-X^{2n-1}+\dots-x+1$, where
$n\in{N^*}$. Find the remainder of the division of polynomial
$P(x^{2n+1})$ by $P(x)$.
Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property:
For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.
Let $ f(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{100}$. Find $ f'(1)$.
Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$
a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number.
b) $x^{2^n}+1,$ with $n$ is a positive integer.
Let \( P(x), Q(x) \) be non-constant real polynomials, such that for all positive integer \( m \), there exists a positive integer \( n \) satisfy \( P(m) = Q(n) \). Prove that
(1) If \(\deg Q \mid \deg P\), then there exists real polynomial \( h(x) \) \( x \), satisfy \( P(x) = Q(h(x)) \) holds for all real number $x.$
(2) \(\deg Q \mid \deg P\).
Let $P(x)$ be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of $P(x)$ are less than $n- \frac{1}{2}$ , polynomial $x-n+1$ does not divide $P(x)$, and $P(n)$ is a prime number. Prove that the polynomial $P(x)$ is irreducible (over $Z[x]$).
Let $P(z)$ and $Q(z)$ be complex-variable polynomials, with degree not less than $1$. Let
\[P_k = \{z \in \mathbb C | P(z) = k \}, Q_k = \{ z \in \mathbb C | Q(z) = k \}.\]
Let also $P_0 = Q_0$ and $P_1 = Q_1$. Prove that $P(z) \equiv Q(z).$
Find all monic polynomials $P(x),Q(x)$ with integer coefficients such that $Q(0) =0$ and $P(Q(x)) = (x-1)(x-2)...(x-15)$.
Let $a,b, c, d,e$ be the roots of $p(x) = 2x^5 - 3x^3 + 2x -7$. Find the value of
$$(a^3 - 1)(b^3 - 1)(c^3 - 1)(d^3 - 1)(e^3 - 1).$$