Found problems: 4275
Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.
What is the polynomial of smallest degree that passes through $(-2, 2), (-1, 1), (0, 2),(1,-1)$, and $(2, 10)$?
Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?
[b]b.)[/b] For which $n>2$ is there exactly one set having this property?
Determine all polynomials $ P_n(x)\equal{}x^n\plus{}a_1 x^{n\minus{}1}\plus{}...\plus{}a_{n\minus{}1} x\plus{}a_n$ with integer coefficients whose $ n$ zeros are precisely the numbers $ a_1,...,a_n$ (counted with their respective multiplicities).
Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r^2_1+\dots+r^2_5$?
The polynomial $P(x) = x^3 + \sqrt{6} x^2 - \sqrt{2} x - \sqrt{3}$ has three distinct real roots. Compute the sum of all $0 \le \theta < 360$ such that $P(\tan \theta^\circ) = 0$.
[i]Proposed by Lewis Chen[/i]
Let $k$ be a constant number larger than $1$. Find all polynomials $P(x)$ such that $P({x^k}) = {\left( {P(x)} \right)^k}$ for all real $x$.
The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S =\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.
A polynomial $p$ can be written as
\begin{align*}
p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d.
\end{align*}
Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$.
Let $ f$ be a polynomial of degree 2 with integer coefficients. Suppose that $ f(k)$ is divisible by 5 for every integer $ k$. Prove that all coefficients of $ f$ are divisible by 5.
Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum
$$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.
Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Find the least natural number $k$, depending on $n$, such that the above polynomials $p_1, p_2, \cdots, p_k$ exist.
Find all polynomials $P(x)$ with real coefficients such that \[xP\bigg(\frac{y}{x}\bigg)+yP\bigg(\frac{x}{y}\bigg)=x+y\] for all nonzero real numbers $x$ and $y$.
Let $n$ be a given positive integer. Solve the system
\[x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,\]
\[x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}\]
in the set of nonnegative real numbers.
Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$
Given a polynomial $P$, assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$.
Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where $ f(X)$ and $ g(X)$ are integer polynomials.
[i]Mircea Becheanu[/i].
Do there exist $2022$ polynomials with real coefficients, each of degree equal to $2021$, so that the $2021 \cdot 2022 + 1$ coefficients in their product are equal?
Prove that if the numbers $p_1, p_2, q_1, q_2$ satisfy the condition $$(q_1 - q_2)^2 + (p_1 - p_2)(p_1q_2 -p_2q_1)<0$$
then the square polynomials $x^2 + p_1x + q_1$ and $x^2 + p_2x + q_2$ have real roots, and between the roots of each there is a root of another one.
Let $n$ be a positive integer. Prove that the polynomial
\[P(x)= \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+...+x+1 \]
Does not have any rational root.
Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different.
For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$.
Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.
Find the polynomials $ f(x)$ having the following properties:
(i) $ f(0) \equal{} 1$, $ f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0$
(ii) $ f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0$
Given are integers $a, b, c$ and an odd prime $p.$ Prove that $p$ divides $x^2 + y^2 + ax + by + c$ for some integers $x$ and $y.$
[i](A. Golovanov )[/i]
If $a,b,c,d$ are Distinct Real no. such that
$a = \sqrt{4+\sqrt{5+a}}$
$b = \sqrt{4-\sqrt{5+b}}$
$c = \sqrt{4+\sqrt{5-c}}$
$d = \sqrt{4-\sqrt{5-d}}$
Then $abcd = $