Found problems: 4275
Let $ f:\mathbb{C}\longrightarrow\mathbb{C} $ be an holomorphic function which has the property that there exist three positive real numbers $ a,b,c $ such that $ |f(z)|\geqslant a|z|^b , $ for any complex numbers $ z $ with $ |z|\geqslant c. $
Prove that $ f $ is polynomial with degree at least $ \lceil b\rceil . $
Prove that there exists a polynomial with integer coefficients satisfying the following conditions:
(a)$f(x)=0$ has no rational root.
(b) For any positive integer $n$, there always exists an integer $m$ such that $n\mid f(m)$.
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$, $b$, and $c$. Find
$$\max\{a+b-c,a-b+c,-a+b+c\}.$$
For which positive integers $n$ do there exist two polynomials $f$ and $g$ with integer coefficients of $n$ variables $x_1, x_2, \ldots , x_n$ such that the following equality is satisfied:
\[\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ? \]
Let $P(x)$ be a polynomial with the coefficients being $0$ or $1$ and degree $2023$. If $P(0)=1$, then prove that every real root of this polynomial is less than $\frac{1-\sqrt{5}}{2}$.
Find all real polynomials $P(x, y)$ such that $P(x+y, x-y) = 2P(x, y)$, for all $x, y$ in $R$.
For an integer $n \ge 2$, find all real numbers $x$ for which the polynomial $f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4$ takes its minimum value.
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that
\[ A^2B+BA^2=2ABA \]
Prove there exists $k\in \mathbb{N}$ such that
\[ (AB-BA)^k=\mathbf{0}_n\]
Here $\mathbf{0}_n$ is the null matrix of order $n$.
Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is divided by $1000$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in a set $X$.)
[i]Proposed by David Altizio[/i]
Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.
Call a nonempty set $V$ of nonzero integers \emph{victorious} if there exists a polynomial $P(x)$ with integer coefficients such that $P(0)=330$ and that $P(v)=2|v|$ holds for all elements $v\in V$. Find the number of victorious sets.
[i]Proposed by Yannick Yao[/i]
Show that there exist a polynomial $P(x)$ whose one cofficient is $\frac{1}{2016}$ and remaining cofficients are rational numbers, such that $P(x)$ is an integer for any integer $x$ .
Find all polynomials $P(x)$ with real coefficients that satisfy the identity $$P(x^3-2)=P(x)^3-2,$$ for every real number $x$.
Let $Q$ be a quadratic polynomial with a unique zero. Suppose $Q(12)=Q(16)$ and $Q(20)=24$. Compute $Q(28)$.
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.
Let $F$ be a finite field having an odd number $m$ of elements. Let $p(x)$ be an irreducible (i.e. nonfactorable) polynomial over $F$ of the form $$x^2+bx+c, ~~~~~~ b,c \in F.$$ For how many elements $k$ in $F$ is $p(x)+k$ irreducible over $F$?
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.
The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
$\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$
Consider a sequence of polynomials such that $P_0(x)=2,P_1(x)=x$ and for all $n\ge1$ \[P_{n+1}(x)+P_{n-1}(x)=xP_n(x).\]
a) Determine the polynomial \[Q_n(x)=P^2_n(x)-xP_n(x)P_{n-1}(x)+P^2_{n-1}(x)\] for $n=1972.$
b) Express the polynomial \[\bigl(P_{n+1}(x)-P_{n-1}(x)\bigr)^2\] in terms of $P_n(x),Q_n(x).$
Let $a, b$, and $c$ be real numbers such that
\[\frac{1}{bc-a^2} + \frac{1}{ca-b^2}+\frac{1}{ab-c^2} = 0.\]
Prove that
\[\frac{a}{(bc-a^2)^2} + \frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2} = 0.\]
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*}
P(0) = P(3) = \cdots &=& P(3n) = 2, \\
P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\
P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\
&& P(3n+1) = 730.\end{eqnarray*}
Determine $n$.
$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$. Show that $a_1 = a_3$.
The roots of $x^2+bx+c=0$ are both real and greater than $1$. Let $s=b+c+1$. Then $s:$
$ \textbf{(A)}\ \text{may be less than zero}\qquad\textbf{(B)}\ \text{may be equal to zero}\qquad$ $\textbf{(C)}\ \text{must be greater than zero}\qquad\textbf{(D)}\ \text{must be less than zero}\qquad $
$\textbf{(E)}\text{ must be between -1 and +1} $
a. View the second-degree quadratic equation $x^2+? x +? = 0$
Two players successively put an integer each at the location of a question mark. Show that the second player can always ensure that the quadratic gets two integer solutions.
Note: we say that the quadratic also has two integer solutions, even when they are equal (for example if they are both equal to $3$).
b.View the third-degree equation $x^3 +? x^2 +? x +? = 0$
Three players successively put an integer each at the location of a question mark. The equation appears to have three integer (possibly again the same) solutions. It is given that two players each put a $3$ in the place of a question mark. What number did the third player put? Determine that number and the place where it is placed and prove that only one number is possible.
Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$