Found problems: 3597
Find all polynomials $ p\in\mathbb Z[x]$ such that $ (m,n)\equal{}1\Rightarrow (p(m),p(n))\equal{}1$
Consider a polynomial
\[f(x)=x^{2012}+a_{2011}x^{2011}+\dots+a_1x+a_0.\]
Albert Einstein and Homer Simpson are playing the following game. In turn, they choose one of the coefficients $a_0,a_1,\dots,a_{2011}$ and assign a real value to it. Albert has the first move. Once a value is assigned to a coefficient, it cannot be changed any more. The game ends after all the coefficients have been assigned values.
Homer's goal is to make $f(x)$ divisible by a fixed polynomial $m(x)$ and Albert's goal is to prevent this.
(a) Which of the players has a winning strategy if $m(x)=x-2012$?
(b) Which of the players has a winning strategy if $m(x)=x^2+1$?
[i]Proposed by Fedor Duzhin, Nanyang Technological University.[/i]
Source: 1976 Euclid Part B Problem 4
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The remainder when $f(x)=x^5-2x^4+ax^3-x^2+bx-2$ is divided by $x+1$ is $-7$. When $f(x)$ is divided by $x-2$ the remainder is $32$. Determine the remainder when $f(x)$ is divided by $x-1$.
For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: \[ x_{n+1}=x_n(x_n+{1\over n}). \] Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$.
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
We say that $1\leq a\leq101$ is a quadratic polynomial residue modulo $101$ with respect to a quadratic polynomial $f(x)$ with integer coefficients if there exists an integer $b$ such that $101 \mid a-f(b)$. For a quadratic polynomial $f$, we define its quadratic residue set as the set of quadratic residues modulo $101$ with respect to $f(x)$. Compute the number of quadratic residue sets.
[i]Proposed by Michael Ren[/i]
Let $Q(x)$ be a non-zero polynomial and $k$ be a natural number. Prove that the polynomial $P(x) = (x-1)^kQ(x)$ has at least $k+1$ non-zero coefficients.
Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting
\begin{align*}
p_{n+1}(x)=(p_n(x))^2-\alpha
\end{align*}
How many distinct real roots does $p_{10}(x)$ have?
$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.
Determine all polynomials P(x) with real coefficients such that
[(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.
Let $G$ be the set of polynomials of the form
\[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\]
where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?
${ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 $
Let $n$ be a positive even integer, and let $c_1, c_2, \dots, c_{n-1}$ be real numbers satisfying \[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \] Prove that \[
2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2
\] has no real roots.
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
[asy]//Choice A
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125;
}
real g(real x) {
return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(A)}$", (-5,4.5));
[/asy]
[asy]//Choice B
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2;
}
real g(real x) {
return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(B)}$", (-5,4.5));
[/asy]
[asy]//Choice C
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.21875 x^2+0.28125 x+0.5;
}
real g(real x) {
return -0.375 x^2-0.75 x+0.5;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(C)}$", (-5,4.5));
[/asy]
[asy]//Choice D
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875;
}
real g(real x) {
return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625;
}
real z=3.14;
draw(graph(f,-z, z), heavygray);
draw(graph(g,-z, z), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(D)}$", (-5,4.5));
[/asy]
[asy]//Choice E
size(100);defaultpen(linewidth(0.7)+fontsize(8));
real end=4.5;
draw((-end,0)--(end,0), EndArrow(5));
draw((0,-end)--(0,end), EndArrow(5));
real ticks=0.2, four=3.7, r=0.1;
draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks));
label("$x$", (4,0), N);
label("$y$", (0,4), W);
label("$-4$", (-4,-ticks), S);
label("$-1$", (-1,-ticks), S);
label("$1$", (1,-ticks), S);
label("$4$", (4,-ticks), S);
real f(real x) {
return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598;
}
real g(real x) {
return -0.166667 x^3+0.125 x^2+0.479167 x-0.375;
}
draw(graph(f,-four, four), heavygray);
draw(graph(g,-four, four), black);
clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle);
label("$\textbf{(E)}$", (-5,4.5));
[/asy]
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$.
[i]Proposed by fattypiggy123[/i]
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$:
$(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$.
The scores of this problem were:
one time 17/20 (by the runner-up)
one time 4/20 (by Andrei Negut)
one time 1/20 (by the winner)
the rest had zero... just to give an idea of the difficulty.
Let $A_{i},B_{i},S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that [list][*]not all $A_{i}$ have a common real eigenvector, [*]$A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, [*]$A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$.[/list] Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.
Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd
[b]Problem 3[/b]
Let $ P(x) = x^{2015} -2x^{2014}+1$ and $ Q(x) = x^{2015} -2x^{2014}-1$. Determine for each of the polynomials
$P$ and $Q$ whether it is a divisor of some nonzero polynomial $c_0 + c_{1}x +\ldots + c_{n}x^n$
n whose coefficients $c_i$ are all in the set $ \{ -1, 1\}$.
For arbitrary non-constant polynomials $f_1(x),\ldots,f_{2018}(x)\in\mathbb Z[x]$, is it always possible to find a polynomial $g(x)\in\mathbb Z[x]$ such that
$$f_1(g(x)),\ldots,f_{2018}(g(x))$$are all reducible.
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \minus{} 1 \qquad \textbf{(E)}\ \text{an undetermined constant}$
Find real $a,b$ such that polynomial $P(x)=x^{n+1}+ax+b$ to be divisible by $(x-1)^2$. Then find the quotient $P(x):(x-1)^2 , n\in \mathbb{N}^*$