Found problems: 3597
Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?
Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.
Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$. Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$. We say that $G$ is a [i]gingado [/i] of $a$.
For example, as $2$ is root of $P(x)=x^2-x-2$, $G=|1|+|-1|+|-2|=4$, we say that $4$ is a [i]gingado[/i] of $2$. What is the fourth largest real number $a$ such that $3$ is a [i]gingado [/i] of $a$?
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.
A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line.
$f(x)$ is a monic polynomial of degree $2$ with integer coefficients such that $f(x)$ doesn't have any real roots and also $f(0)$ is a square-free integer (and is not $1$ or $-1$). Prove that for every integer $n$ the polynomial $f(x^n)$ is irreducible over $\mathbb Z[x]$.
[i]proposed by Mohammadmahdi Yazdi[/i]
Let $u$ be the positive root of the equation $x^2+x-4=0$. The polynomial
$$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$.
1) Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$.
2) Find the minimum possible value of $a_0+a_1+...+a_n$.
Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$.
[i]Nanang Susyanto, Jogjakarta[/i]
Find all polynomials $P_n(x)$ of the form
$$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\ldots+a_1x+(-1)^nn(n+1),$$with integer coefficients, such that its roots $x_1,x_2,\ldots,x_n$ satisfy $k\le x_k\le k+1$ for $k=1,2,\ldots,n$.
Let $f_1(x)$ be a polynomial of degree $2$ with the leading coefficient positive and $f_{n+1}(x) =f_1(f_n(x))$ for $n\ge 1.$ Prove that if the equation $f_2(x)=0$ has four different non-positive real roots, then for arbitrary $n$ then $f_n(x)$ has $2^n$ different real roots.
Let $ A_1,A_2,\dots,A_n$ be idempotent matrices with real entries. Prove that:
\[ \mbox{N}(A_1)\plus{}\mbox{N}(A_2)\plus{}\dots\plus{}\mbox{N}(A_n)\geq \mbox{rank}(I\minus{}A_1A_2\dots A_n)\]
$ \mbox{N}(A)$ is $ \mbox{dim}(\mbox{ker(A)})$
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that
\[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\]
\[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\]
What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. If $P(3) = 89$, what is the value of $P(10)$?
Let \[f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}\] be a polynomial with coefficients satisfying the conditions: \[0\le a_{i}\le a_{0},\quad i=1,2,\ldots,n.\] Let $b_{0},b_{1},\ldots,b_{2n}$ be the coefficients of the polynomial
\begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right)\\ &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*}
Prove that $b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}$.
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Consider the polynomial $$P\left(t\right)=t^3-29t^2+212t-399.$$ Find the product of all positive integers $n$ such that $P\left(n\right)$ is the sum of the digits of $n$.
Call a polynomial $f(x)$ [i]excellent[/i] if its coefficients are all in [0, 1) and $f(x)$ is an integer for all integers $x$.
a) Compute the number of excellent polynomials with degree at most 3.
b) Compute the number of excellent polynomials with degree at most $n$, in terms of $n$.
c) Find the minimum $n\ge3$ for which there exists an excellent polynomial of the form $\frac{1}{n!}x^n+g(x)$, where $g(x)$ is a polynomial of degree at most $n-3$.
For each integer $n > 0$ show that there is a polynomial $p(x)$ such that $p(2 cos x) = 2 cos nx$.
Let $n$ be a natural number and the polynomial, $P(x)=x^n+n$.
$(a)$ Is it possible that for some odd number $n$ , the polynomial $P(x)$ is composite for all natural numbers $x$.
$(b)$ Is it possible that for some even number $n$ , the polynomial $P(x)$ is composite for all natural numbers $x$.
Reason your answers.
Find the largest $a$ for which there exists a polynomial
$$P(x) =a x^4 +bx^3 +cx^2 +dx +e$$
with real coefficients which satisfies $0\leq P(x) \leq 1$ for $-1 \leq x \leq 1.$
Let $f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ be a polynomial of degree $n \geq 3$ with real coefficients.Suppose all roots of $f(z) =0$ lie in the half plane ${\ z \in \mathbb{C} : Re(z) < 0 \}}$. Prove that $a_k a_{k+3} < a_{k+1}a_{k+2}$ for $k = 0,1,2,3,.... n-3$
Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]
Let $f(x)$ be a non-constant polynomial with integer coefficients and $n,k$ be natural numbers. Show that there exist $n$ consecutive natural numbers $a,a+1,\ldots,a+n-1$ such that the numbers $f(a),f(a+1),\ldots,f(a+n-1)$ all have at least $k$ prime factors. (We say that the number $p_1^{\alpha_1}\cdots p_s^{\alpha_s}$ has $\alpha_1+\ldots+\alpha_s$ prime factors.)
Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.