Found problems: 3597
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]
Find all monic polynomials $P,Q$ which are non-constant, have real coefficients and they satisfy $2P(x)=Q(\frac{(x+1)^2}{2})-Q(\frac{(x-1)^2}{2})$ and $P(1)=1$ for all real $x$.
Let $P,Q,R$ be polynomials of degree at least $1$ with integer coefficients such that for any real number $x$ holds: $P(Q(x))\equal{}Q(R(x))\equal{}R(P(x))$. Show that the polynomials $P,Q,R$ are equal.
Let $n$ be an even positive integer. An $n$-degree monic polynomial $P(x)$ has $n$ real roots (not necessarily distinct). Suppose $y$ is a positive real number such that for any real number $t<y$, we have $P(t)>0$. Prove that
\[P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.\]
Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1\equal{}1, a_2>1,$ and $ a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2}$ for $ n \ge 1$.
Let $P(x)=ax^3+(b-a)x^2-(c+b)x+c$ and $Q(x)=x^4+(b-1)x^3+(a-b)x^2-(c+a)x+c$ be polynomials of $x$ with $a,b,c$ non-zero real numbers and $b>0$.If $P(x)$ has three distinct real roots $x_0,x_1,x_2$ which are also roots of $Q(x)$ then:
A)Prove that $abc>28$,
B)If $a,b,c$ are non-zero integers with $b>0$,find all their possible values.
Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.
Determine all polynomials $P$ with integer coefficients for which there exists an integer $a_n$ such that $P(a_n)=n^n$ for all positive integers $n$.
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares:
a) in exactly one way;
b) in exactly two ways?
Note: two splittings that differ only in the order of summands are considered to be the same.
Sergey Markelov
The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$.
Prove that $ \alpha\beta$ is not a perfect square.
A polynomial with rational coefficients is called [i]integer[/i], if it takes integer values for all integer values of the variable. For an integer polynomial $P$, consider the sequence $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$
a) Prove that this sequence is periodic, the period of which is some power of two (i.e. for some integer $k$ and for all natural $i$, the $i$-th and ($i+2^k$)th members of the sequence are equal).
b) Prove that for any periodic sequence consisting of $(- 1)$ and $ 1$ and with a period of some power of two, there exists a integer, polynomial P for which this sequence is $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$
Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?
For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.
Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.
Find such real $p, q, a, b$, that for all $x$ an equality is held: $$(2x-1)^{20} - (ax+b)^{20} = (x^2+px+q)^{10}$$
Let $ P(x)$ be a polynomial with integer coefficients. Prove that there exist two polynomials $ Q(x)$ and $ R(x)$, again with integer coefficients, such that
[b](i)[/b] $ P(x) \cdot Q(x)$ is a polynomial in $ x^2$ , and
[b](ii)[/b] $ P(x) \cdot R(x)$ is a polynomial in $ x^3$.
Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$, $p$ and $q$, there exist not necessarily distinct integers $a \neq 0$, $b$, $c$ belonging to $S$, such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$. Determine the maximum number of elements that $S$ can have.
The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.
Let $S_n $ be sum of squares of the coefficient of the polynomial $(1+x)^n$. Prove that $S_{2n} +1$ is not divisible by $3.$
Polynomial $P(x)$ with integer coefficients is given. For some positive integer $n$ numbers $P(0),P(1),\dots,P(2^n+1)$ are all divisible by $2^{2^n}$. Prove that values of $P(x)$ in all integer points are divisible by $2^{2^n}$.
Let \(P(x)=x^{2}+ax+b\) be a quadratic polynomial where \(a\) is real and \(b \neq 2\), is rational. Suppose \(P(0)^{2},P(1)^{2},P(2)^{2}\) are integers, prove that \(a\) and \(b\) are integers.
For each positive integer $ k$, find the smallest number $ n_{k}$ for which there exist real $ n_{k}\times n_{k}$ matrices $ A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold:
(1) $ A_{1}^{2}= A_{2}^{2}= \ldots = A_{k}^{2}= 0$,
(2) $ A_{i}A_{j}= A_{j}A_{i}$ for all $ 1 \le i, j \le k$, and
(3) $ A_{1}A_{2}\ldots A_{k}\ne 0$.