Found problems: 3597
Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.
Find all polynomials $Q(x)= ax^2+bx+c$ with integer coefficients for which there exist three different prime numbers $p_1, p_2, p_3$ such that $|Q(p_1)| = |Q(p_2)| = |Q(p_3)| = 11$.
Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$
Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$.
(A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)
Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$.
[i]B. Berceanu[/i]
Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.
Give an example of a second degree polynomial $P\in \mathbb{R}[x]$ such that
\[\forall x\in \mathbb{R}\text{ with } |x|\leqslant 1: \;
\left|P(x)+\frac{1}{x-4}\right|
\leqslant 0.01.\]
Are there linear polynomials with this property?
[i]Octavian Stănășilă[/i]
Let $ p$ be a prime and let $ d \in \left\{0,\ 1,\ \ldots,\ p\right\}$. Prove that
\[ \sum_{k \equal{} 0}^{p \minus{} 1}{\binom{2k}{k \plus{} d}}\equiv r \pmod{p},
\]where $ r \equiv p\minus{}d \pmod 3$, $ r\in\{\minus{}1,0,1\}$.
Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
The equation \[ x^{10}+(13x-1)^{10}=0 \] has 10 complex roots $r_1, \overline{r_1}, r_2, \overline{r_2}, r_3, \overline{r_3}, r_4, \overline{r_4}, r_5, \overline{r_5},$ where the bar denotes complex conjugation. Find the value of \[ \frac 1{r_1\overline{r_1}}+\frac 1{r_2\overline{r_2}}+\frac 1{r_3\overline{r_3}}+\frac 1{r_4\overline{r_4}}+\frac 1{r_5\overline{r_5}}. \]
Are there positive real numbers $x$, $y$ and $z$ such that
$ x^4 + y^4 + z^4 = 13\text{,} $
$ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $
$ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $
[i]Proposed by Matko Ljulj.[/i]
The polynomials $P_n(x,y), n=1,2,... $ are defined by \[P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)\] Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n $.
Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities
$$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$
Prove that the degrees of the three polynomials are all even.
Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that \[\frac{a^{m}+a-1}{a^{n}+a^{2}-1}\] is itself an integer.
Let $p(z)$ be a polynomial of degree $n>0$ with complex coefficients. Prove that there are at least $n+1$ complex numbers $z$ for which $p(z)\in \{0,1\}$.
Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$.
Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$.
[i]Proposed by Aaron Lin
[/i]
Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if
a) $ n$ is a positive integer not divisible by the square of a prime.
b) $ n$ is a positive integer not divisible by the cube of a prime.
Let $f(x)$ be a monic polynomial of degree $2023$ with positive integer coefficients.
Show that for any sufficiently large integer $N$ and any prime number $p>2023N$, the product
\[f(1)f(2)\dots f(N)\]
is at most $\binom{2023}{2}$ times divisible by $p$.
[i]Proposed by Ashwin Sah[/i]
Prove that the set of all linearly combinations (with real coefficients) of the system of polynomials $ \{ x^n\plus{}x^{n^2} \}_{n\equal{}0}^{\infty}$ is dense in $ C[0,1]$.
[i]J. Szabados[/i]
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
Find all polynomials $P(x, y)$ with real coefficients which for all real numbers $x$ and $y$ satisfy $P(x + y, x - y) = 2P(x, y)$.
Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.
A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?
For a polynomial $P$ and a positive integer $n$, define $P_n$ as the number of positive integer pairs $(a,b)$ such that $a<b \leq n$ and $|P(a)|-|P(b)|$ is divisible by $n$. Determine all polynomial $P$ with integer coefficients such that $P_n \leq 2021$ for all positive integers $n$.