Found problems: 4275
Show that the polynomial over variables $x,y,z$
\[
x^4(x-y)(x-z) + y^4(y-z)(y-x) + z^4(z-x)(z-y)
\]
can't be written as a finite sum of squares of real polynomials over $x,y,z$.
The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely:
$ \textbf{(A)}\ 0, 12, -12\qquad\textbf{(B)}\ 0, 12\qquad\textbf{(C)}\ 12, -12\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 0 $
The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$.
[hide="comment"]
Hi guys ,
Here is a nice problem:
Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$
Here are some futher question proposed by me :Prove or disprove that :
1) $gcd(n,a_n)=1$
2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$
Thanks kiu si u
[i]Edited by Orl.[/i]
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(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite.
(b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.
Given $k\ge 2$, for which polynomials $P\in \mathbb{Z}[X]$ does there exist a function $h:\mathbb{N}\rightarrow\mathbb{N}$ with $h^{(k)}(n)=P(n)$?
Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and
\[ f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right).\]
Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x\plus{}c) \equal{} f(x)$ for all $ x \in \mathbb{R}$).
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
Let $p$ be a prime number and let $\mathbb{F}_p$ be the finite field with $p$ elements. Consider an automorphism $\tau$ of the polynomial ring $\mathbb{F}_p[x]$ given by
\[\tau(f)(x)=f(x+1).\]
Let $R$ denote the subring of $\mathbb{F}_p[x]$ consisting of those polynomials $f$ with $\tau(f)=f$. Find a polynomial $g \in \mathbb{F}_p[x]$ such that $\mathbb{F}_p[x]$ is a free module over $R$ with basis $g,\tau(g),\dots,\tau^{p-1}(g)$.
In a game, Andi and a computer take turns. At the beginning, the computer shows a polynomial $x^2 + mx + n$ where $m,n \in \mathbb{Z}$, such that it doesn't have real roots. Andi then begins the game. On his turn, Andi may change a polynomial in the form $x^2 + ax + b$ into either $x^2 + (a+b)x + b$ or $x^2 + ax + (a+b)$. However, Andi may only choose a polynomial that has real roots. On the computer's turn, it simply switches the coefficient of $x$ and the constant of the polynomial. Andi loses if he can't continue to play. Find all $(m,n)$ such that Andi always loses (in finitely many turns).
Consider the set of all polynomials of degree less than or equal to $4$ with rational coefficients.
a) Prove that it has a vector space structure over the field of numbers rational.
b) Prove that the polynomials $1, x - 2, (x -2)^2, (x - 2)^3$ and $(x -2)^4$ form a base of this space.
c) Express the polynomial $7 + 2x - 45x^2 + 3x^4$ in the previous base.
Let $n$ be a positive integer, with prime factorization $$n = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$$ for distinct primes $p_1, \ldots, p_r$ and $e_i$ positive integers. Define $$rad(n) = p_1p_2\cdots p_r,$$ the product of all distinct prime factors of $n$.
Find all polynomials $P(x)$ with rational coefficients such that there exists infinitely many positive integers $n$ with $P(n) = rad(n)$.
Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that
\[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0.
\]
Determine when equality occurs.
Determine all pairs of positive integers $(x, y)$ such that:
$$x^4 - 6x^2 + 1 = 7\cdot 2^y$$
Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$.
[i]Aaron Pixton.[/i]
Write the following polynomial as a product of irreducible polynomials in $\mathbb{Z}[X]$
\[ f(X) = X^{2005} - 2005 X + 2004 . \]Justify your answer.
How many positive integers $n$ are there such that $n+2$ divides $(n+18)^2$?
Do there exist non-zero reals $a$, $b$, $c$ such that, for any $n>3$, there exists a polynomial $P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c$, which has exactly $n$ (not necessary distinct) integral roots?
[i]N. Agakhanov, I. Bogdanov[/i]
Let $f = f(x,y,z)$ be a polynomial in three variables $x$, $y$, $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$. Show that there exist three polynomials $A$, $B$, $C$ in these same three variables such that $A + B + C = 0$ and \[ f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). \] Is there any polynomial $f$ for which these $A$, $B$, $C$ are uniquely determined?
Let $ n$ be the number of integer values of $ x$ such that $ P \equal{} x^4 \plus{} 6x^3 \plus{} 11x^2 \plus{} 3x \plus{} 31$ is the square of an integer. Then $ n$ is:
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 0$
Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.
Find all poltnomials $ P(x)$ with real coefficients satisfying: For all $ a>1995$, the number of real roots of $ P(x)\equal{}a$ (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.
A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains:
[list]$f(7) = 77$
$f(b) = 85$, where $b$ is Beth's age,
$f(c) = 0$, where $c$ is Charles' age.[/list]
How old is each child?
Find all $n$ such that there is a real polynomial $f(x)$ of degree $n$ such that $f(x) \ge f'(x)$ for all real $x$.
We call a positive integer $n$ $\textit{sixish}$ if $n=p(p+6)$, where $p$ and $p+6$ are prime numbers. For example, $187=11\cdot17$ is sixish, but $475=19\cdot25$ is not sixish. Define a function $f$ on the positive integers such that $f(n)$ is the sum of the squares of the positive divisors of $n$. For example, $f(10)=1^2+2^2+5^2+10^2=130$.
(a) Find, with proof, an irreducible polynomial function $g(x)$ with integer coefficients such that $f(n)=g(n)$ for all sixish $n$. ("Irreducible" means that $g(x)$ cannot be factored as the product of two polynomials of smaller degree with integer coefficients.)
(b) We call a positive integer $n$ $\textit{pseudo-sixish}$ if $n$ is not sixish but nonetheless $f(n)=g(n)$, where $g(n)$ is the polynomial function that you found in part (a). Find, with proof, all pseudo-sixish positive integers.
Suppose that $P(x)=a_1x+a_2x^2+\ldots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\ldots$ for all $x.$ Prove that $b_k$ is nonzero for all $k \geq 0.$