Found problems: 3597
Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.
Let $f(x)=3x^3-5x^2+2x-6$. If the roots of $f$ are given by $\alpha$, $\beta$, and $\gamma$, find
\[
\left(\frac{1}{\alpha-2}\right)^2+\left(\frac{1}{\beta-2}\right)^2+\left(\frac{1}{\gamma-2}\right)^2.
\]
Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order).
a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully.
b) Prove that such a tasteful tiling is unique.
[i]Victor Wang.[/i]
Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.
Let \[p(x)=x^{2008}+x^{2007}+x^{2006}+\cdots+x+1,\] and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$.
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i]
[b][i]Original Formulation[/i][/b]
Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f( \sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes.
[i]Proposed by Soviet Union. [/i]
If you have an algorithm for finding all the real zeros of any cubic polynomial, how do you find the real solutions to $x = p(y), y = p(x)$, where $p$ is a cubic polynomial?
Prove that for each natural number $m$, there is a natural number $N$ such that for each $b$ that $2\leq b\leq1389$ sum of digits of $N$ in base $b$ is larger than $m$.
Let $f(x)=x^2+18$ have roots $r_1$ and $r_2$, and let $g(x)=x^2-8x+17$ have roots $r_3$ and $r_4$. If $h(x)=x^4+ax^3+bx^2+cx+d$ has roots $r_1+r_3$, $r_1+r_4$, $r_2+r_3$, and $r_2+r_4$, then find $h(4)$.
A [i]repunit[/i] is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$. Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
$a$ is irrational , but $a$ and $a^3-6a$ are roots of square polynomial with integer coefficients.Find $a$
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and
${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$
Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.
Let $n$ be a positive integer and $a_1,...,a_n$ be positive real numbers.
Let $g(x)$ denote the product $(x + a_1)\cdot ... \cdot (x + a_n)$ .
Let $a_0$ be a real number and let
$f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}$ .
Prove that all the coeffcients $b_1,b_2,..., b_{n+1}$ of the polynomial $f(x)$ are negative if and only if $a_0 > a_1 + a_2 +...+ a_n$.
Define a sequence of polynomials $P_0\left(x\right)=x$ and $P_k\left(x\right)=P_{k-1}\left(x\right)^2-\left(-1\right)^kk$ for each $k\geq1$. Also define $Q_0\left(x\right)=x$ and $Q_k\left(x\right)=Q_{k-1}\left(x\right)^2+\left(-1\right)^kk$ for each $k\geq1$. Compute the product of the distinct real roots of \[P_1\left(x\right)Q_1\left(x\right)P_2\left(x\right)Q_2\left(x\right)\cdots P_{2018}\left(x\right)Q_{2018}\left(x\right).\]
[i]2018 CCA Math Bonanza Tiebreaker Round #2[/i]
Find all real polynomials $f$ and $g$, such that:
\[(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1), \]
for all $x\in\mathbb{R}$.
For all $n>1$. Find all polynomials with complex coefficient and degree more than one such that $(p(x)-x)^2$ divides $p^n(x)-x$. ($p^0(x)=x , p^i(x)=p(p^{i-1}(x))$)
[i]Proposed by Navid Safaie[/i]
Let $P$ be a polynomial with integer coefficients. We say $P$ is [i]good [/i] if there exist infinitely many prime numbers $q$ such that the set $$X=\left\{P(n) \mod q : \quad n\in \mathbb N\right\}$$ has at least $\frac{q+1}{2}$ members.
Prove that the polynomial $x^3+x$ is good.
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 $n\mid f(m)$.
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 $n$.