Found problems: 3597
Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows.
(1) $g_n(x)=(1+x)^n$
(2) $g_n(x)=\sin n\pi x$
(3) $g_n(x)=e^{nx}$
A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$
Suppose that $F,G,H$ are polynomials of degree at most $2n+1$ with real coefficients such that:
i) For all real $x$ we have $F(x)\le G(x)\le H(x)$.
ii) There exist distinct real numbers $x_1,x_2,\ldots ,x_n$ such that $F(x_i)=H(x_i)\quad\text{for}\ i=1,2,3,\ldots ,n$.
iii) There exists a real number $x_0$ different from $x_1,x_2,\ldots ,x_n$ such that $F(x_0)+H(x_0)=2G(x_0)$.
Prove that $F(x)+H(x)=2G(x)$ for all real numbers $x$.
Find the remainder after division of the polynomial $x+x^3 +x^9 +x^{27} +x^{81} +x^{243}$ by $x-1$.
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
A polynomial $c_dx^d+c_{d-1}x^{d-1}+\dots+c_1x+c_0$ with degree $d$ is [i]reflexive[/i] if there is an integer $n\ge d$ such that $c_i=c_{n-i}$ for every $0\le i\le n$, where $c_i=0$ for $i>d$. Let $\ell\ge 2$ be an integer and $p(x)$ be a polynomial with integer coefficients. Prove that there exist reflexive polynomials $q(x)$, $r(x)$ with integer coefficients such that
\[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]
Let $P(x)=x^{100}+20x^{99}+198x^{98}+a_{97}x^{97}+\ldots+a_1x+1$ be a polynomial where the $a_i~(1\le i\le97)$ are real numbers. Prove that the equation $P(x)=0$ has at least one nonreal root.
Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers?
[i]Proposed by Finland.[/i]
The polynomial $P(x)=x^3+ax+1$ has exactly one solution on the interval $[-2,0)$ and has exactly one solution on the interval $(0,1]$ where $a$ is a real number. Which of the followings cannot be equal to $P(2)$?
$
\textbf{(A)}\ \sqrt{17}
\qquad\textbf{(B)}\ \sqrt[3]{30}
\qquad\textbf{(C)}\ \sqrt{26}-1
\qquad\textbf{(D)}\ \sqrt {30}
\qquad\textbf{(E)}\ \sqrt [3]{10}
$
You are given $N$ such that $ n \ge 3$. We call a set of $N$ points on a plane acceptable if their abscissae are unique, and each of the points is coloured either red or blue. Let's say that a polynomial $P(x)$ divides a set of acceptable points either if there are no red dots above the graph of $P(x)$, and below, there are no blue dots, or if there are no blue dots above the graph of $P(x)$ and there are no red dots below. Keep in mind, dots of both colors can be present on the graph of $P(x)$ itself. For what least value of k is an arbitrary set of $N$ points divisible by a polynomial of degree $k$?
Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and
\[\gcd(P(0), P(1), P(2), \ldots ) = 1.\]
Show there are infinitely many $n$ such that
\[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]
Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that:
(i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again,
(ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps,
(iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.
Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$.
[i]Daniel Jinga[/i]
For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.)
[i]Aprameya Tripathy[/i]
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $ 3$ rows of small congruent equilateral triangles, with $ 5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $ 2003$ small equilateral triangles?
[asy]unitsize(15mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
pair Ap=(0,0), Bp=(1,0), Cp=(2,0), Dp=(3,0), Gp=dir(60);
pair Fp=shift(Gp)*Bp, Ep=shift(Gp)*Cp;
pair Hp=shift(Gp)*Gp, Ip=shift(Gp)*Fp;
pair Jp=shift(Gp)*Hp;
pair[] points={Ap,Bp,Cp,Dp,Ep,Fp,Gp,Hp,Ip,Jp};
draw(Ap--Dp--Jp--cycle);
draw(Gp--Bp--Ip--Hp--Cp--Ep--cycle);
for(pair p : points)
{
fill(circle(p, 0.07),white);
}
pair[] Cn=new pair[5];
Cn[0]=centroid(Ap,Bp,Gp);
Cn[1]=centroid(Gp,Bp,Fp);
Cn[2]=centroid(Bp,Fp,Cp);
Cn[3]=centroid(Cp,Fp,Ep);
Cn[4]=centroid(Cp,Ep,Dp);
label("$1$",Cn[0]);
label("$2$",Cn[1]);
label("$3$",Cn[2]);
label("$4$",Cn[3]);
label("$5$",Cn[4]);
for (pair p : Cn)
{
draw(circle(p,0.1));
}[/asy]
$ \textbf{(A)}\ 1,\!004,\!004 \qquad
\textbf{(B)}\ 1,\!005,\!006 \qquad
\textbf{(C)}\ 1,\!507,\!509 \qquad
\textbf{(D)}\ 3,\!015,\!018 \qquad
\textbf{(E)}\ 6,\!021,\!018$
Find all possible values of $ x_0$ and $ x_1$ such that the sequence defined by:
$ x_{n\plus{}1}\equal{}\frac{x_{n\minus{}1} x_n}{3x_{n\minus{}1}\minus{}2x_n}$ for $ n \ge 1$
contains infinitely many natural numbers.
Let $P(x)$ be a polynomial with integer coefficient such that $P(1) = 10$ and $P(-1) = 22$.
(a) Give an example of $P(x)$ such that $P(x) = 0$ has an integer root.
(b) Suppose that $P(0) = 4$, prove that $P(x) = 0$ does not have an integer root.
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.
$\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.
Let $ P(x)$ be a polynomial with integer coefficients such that \[ P(m_1) \equal{} P(m_2) \equal{} P(m_3) \equal{} P(m_4) \equal{} 7\] for given distinct integers $ m_1,m_2,m_3,$ and $ m_4.$ Show that there is no integer m such that $ P(m) \equal{} 14.$
a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$. Prove that $a, b, c$ and $d$ are all divisible by $5$.
b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$. Prove that $a, b, c, d$ and $e$ are all divisible by $7$.
Let $n$ be an odd number larger than 1, and $f(x)$ is a polynomial with degree $n$ such that $f(k)=2^k$ for $k=0,1,\cdots,n$. Prove that there is only finite integer $x$ such that $f(x)$ is the power of two.
For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.
Prove that, for any positive integer $n$, there exists a polynomial $p(x)$ of degree at most $n$ whose coefficients are all integers such that, $p(k)$ is divisible by $2^n$ for every even integer $k$, and $p(k) -1$ is divisible by $2^n$ for every odd integer $k$.
Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.