Found problems: 4275
Which of the following polynomials with integer coefficients has $\sin\left(10^{\circ}\right)$ as a root?
$\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$
Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots.
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.
$P(z)=a_nz^n+\dots+a_1z+z_0$, with $a_n\neq 0$ is a polynomial with complex coefficients, such that when $|z|=1$, $|P(z)|\leq 1$. Prove that for any $0\leq k\leq n-1$, $|a_k|\leq 1-|a_n|^2$.
[i]Proposed by Yijun Yao[/i]
Given fi nitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial $C(q)$ of degree two so that the following holds: for any $q\ge 1$ integer, if the half planes cover each point of the plane at least $q$ times, then the set of points covered exactly $q$ times is the union of at most $C(q)$ domains.
(translated by L. Erdős)
Find all polynomials $P$ with integer coefficients such that $P(P(x))-x$ is irreducible over $\mathbb{Z}[x]$.
In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?
In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.
Find all functions $f \colon \mathbb{R_{+}}\to \mathbb{R_{+}}$ satisfying : \[f ( f (x)-x) = 2x\] for all $x > 0$.
A polynomial whose roots are all equal to each other is called a [i]unicorn[/i]. Compute the number of distinct ordered triples $(M, P, G)$, where $M$, $P$, $G$ are complex numbers, such that the polynomials $z^3 + M z^2 + Pz + G$ and $z^3 + G z^2 + Pz + M$ are both unicorns.
Prove that for every positive integer n, there exists a polynomial p(x) with integer coefficients such that p(1), p(2),..., p(n-1), p(n) are distinct powers of 2.
Let $a$, $b$, and $c$ be the roots of the polynomial $$x^3+4x^2-7x-1.$$ Which of the following has roots $ab$, $bc$, and $ac$?
$\text{(A) }x^3-4x^2+7x-1\qquad\text{(B) }x^3-7x^2+4x-1\qquad\text{(C) }x^3+7x^2-4x-1\qquad\text{(D) }x^3-4x^2+7x+1\qquad\text{(E) }x^3+7x^2-4x+1$
Let $S$ be a finite set of integers. Prove that there exists a number $c$ depending on $S$ such that for each non-constant polynomial $f$ with integer coefficients the number of integers $k$ satisfying $f(k)\in S$ does not exceed $\max(\deg f,c)$.
Determine the maximum value of $\lambda$ such that if $f(x) = x^3 +ax^2 +bx+c$ is a cubic polynomial with all its roots nonnegative, then \[f(x)\geq\lambda(x -a)^3\] for all $x\geq0$. Find the equality condition.
Show that a polynomial of odd degree $2m+1$ over $\mathbb{Z}$, \[f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0},\] is irreducible if there exists a prime $p$ such that \[p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.\]
Suppose that $x^3-x+10^{-6}=0$. Suppose that $x_1<x_2<x_3$ are the solutions for $x$. Find the integers $(a,b,c)$ closest to $10^8x_1$, $10^8x_2$, and $10^8x_3$ respectively.
Let $S$ be the set of monic polynomials in $x$ of degree 6 all of whose roots are members of the set $\{ -1, 0, 1\}$. Let $P$ be the sum of the polynomials in $S$. What is the coefficient of $x^4$ in $P(x)$?
A positive interger $n$ is called [i][u]rising[/u][/i] if its decimal representation $a_ka_{k-1}\cdots a_0$ satisfies the condition $a_k\le a_{k-1}\le\cdots \le a_0$. Polynomial $P$ with real coefficents is called [i][u]interger-valued[/u][/i] if for all integer numbers $n$, $P(n)$ takes interger values. $P(n)$ is called [i][u]rising-valued[/u][/i] if for all [i]rising[/i] numbers $n$, $P(n)$ takes integer values.
Does it necessarily mean that, "every [i]rising-valued[/i] $P$ is also [i]interger-valued[/i] $P$"?
Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.
Let $P(x),Q(x)$ be monic polynomials with integer coeeficients. Let $a_n=n!+n$ for all natural numbers $n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for all positive integer $n$ then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n\neq0$.
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[i]Hint (given in question): Try applying division algorithm for polynomials [/i]
Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.
Find all polynomials $P(x)$ with integer coefficients, such that for any positive integer $n$ number $P(n)$ is a positive integer and a divisor of $n!$.
[i]Proposed by Mykyta Kharin[/i]
A sequence of numbers is defined by $ u_1\equal{}a, u_2\equal{}b$ and $ u_{n\plus{}1}\equal{}\frac{u_n\plus{}u_{n\minus{}1}}{2}$ for $ n \ge 2$. Prove that $ \displaystyle\lim_{n\to\infty}u_n$ exists and express its value in terms of $ a$ and $ b$.
Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find the second-smallest possible value of $P(0).$
[i]Proposed by Andrew Wu[/i]
Prove that for all $n \in \mathbb N$ the following is true:
\[2^n \prod_{k=1}^n \sin \frac{k \pi}{2n+1} = \sqrt{2n+1}\]