Found problems: 3597
Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have:
$$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$
$$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$
Then we have :
$$bP(\frac{a}{c})=dQ(\frac{a}{c})$$
(Two polynomials are relatively prime if they don't have a common root)
Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]
Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where \[
A = \prod_{(a,b) \in S} a
, \quad
B = \prod_{(a,b) \in S} b
, \quad \text{and} \quad
C = \sum_{(a,b) \in S} ab.
\][i]Proposed by Evan Chen[/i]
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 $x = 7$?
Let $F(n)$ be the set of polynomials $P(x) = a_0+a_1x+\cdots+a_nx^n$, with $a_0, a_1, . . . , a_n \in \mathbb R$ and $0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.$ Prove that if $f \in F(m)$ and $g \in F(n)$, then $fg \in F(m + n).$
Let $p$ be a prime number and $k$ be a positive integer. Let \[t=\sum_{i=0}^\infty\bigg\lfloor\frac{k}{p^i}\bigg\rfloor.\]a) Let $f(x)$ be a polynomial of degree $k$ with integer coefficients such that its leading coefficient is $1$ and its constant is divisible by $p.$ prove that there exists $n\in\mathbb{N}$ for which $p\mid f(n),$ but $p^{t+1}\nmid f(n).$
b) Prove that the statement above is sharp, i.e. there exists a polynomial $g(x)$ of degree $k,$ integer coefficients, leading coefficient $1$ and constant divisible by $p$ such that if $p\mid g(n)$ is true for a certain $n\in\mathbb{N},$ then $p^t\mid g(n)$ also holds.
[i]Proposed by Kristóf Szabó, Budapest[/i]
Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.
Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:
$\textbf{(A) }3\qquad
\textbf{(B) }4\qquad
\textbf{(C) }5\qquad
\textbf{(D) }6\qquad
\textbf{(E) }11$
Prove that the number of odd coefficients in the polynomial $(1+x)^n$ is a power of $2$ for every positive integer $N$
Let $A,B,C\in \mathcal{M}_3(\mathbb{R})$ such that $\det A=\det B=\det C$ and $\det(A+iB)=\det(C+iA)$. Prove that $\det (A+B)=\det (C+A)$.
A polynomial $P$ has integral coefficients, and it has at least 9 different integral roots. Let $n$ be an integer such that $|P(n)| < 2017$. Prove that $P(n) = 0$.
The polynomial P(x,y) is such that for every integer n >= 0 each of the polynomials P(x,n) and P(n,y) either is a constant zero or has a degree not greater than n. Is it possible that P(x,x) has an odd degree?
$p(x)\in \mathbb{C}[x]$ is a polynomial such that:
$\forall z\in \mathbb{C}, |z|=1\Longrightarrow p(z)\in \mathbb{R}$
Prove that $p(x)$ is constant.
Let $P(X),Q(X)$ be monic irreducible polynomials with rational coefficients. suppose that $P(X)$ and $Q(X)$ have roots $\alpha$ and $\beta$ respectively, such that $\alpha + \beta $ is rational. Prove that $P(X)^2-Q(X)^2$ has a rational root.
[i]Bogdan Enescu[/i]
(a) Find the value of the real number $k$, for which the polynomial $P(x)=x^3-kx+2$ has the number $2$ as a root. In addition, for the value of $k$ you will find, write this polynomial as the product of two polynomials with integer coefficients.
(b) If the positive real numbers $a,b$ satisfy the equation $$2a+b+\frac{4}{ab}=10,$$ find the maximum possible value of $a$.
Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.
Let $W(x)$ be a polynomial of integer coefficients such that for any pair of different rational number $r_1$, $r_2$ dependence $W(r_1) \neq W(r_2)$ is true. Decide, whether the assuptions imply that for any pair of different real numbers $t_1$, $t_2$ dependence $W(t_1) \neq W(t_2)$ is true.
Let $ k \in \mathbb{N}$. A polynomial is called [i]$ k$-valid[/i] if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.)
[b]a.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often.
[b]b.)[/b] For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .
Let us consider a polynomial $P(x)$ with integers coefficients satisfying
$$P(-1)=-4,\ P(-3)=-40,\text{ and } P(-5)=-156.$$
What is the largest possible number of integers $x$ satisfying
$$P(P(x))=x^2?$$
Suppose that $p>3$ is prime. Prove that the products of the primitive roots of $p$ between $1$ and $p-1$ is congruent to $1$ modulo $p$.
For each pair of integers \( j, k \geq 2 \), define the function \( f_{jk} : \mathbb{R} \to \mathbb{R} \) given by
\[
f_{jk}(x) = 1 - (1 - x^j)^k.
\]
(a) Prove that for any integers \( j, k \geq 2 \), there exists a unique real number \( p_{jk} \in (0, 1) \) such that \( f_{jk}(p_{jk}) = p_{jk} \). Furthermore, defining \( \lambda_{jk} := f'_{jk}(p_{jk}) \), prove that \( \lambda_{jk} > 1 \).
(b) Prove that \( p^j_{jk} = 1 - p_{kj} \) for any integers \( j, k \geq 2 \).
(c) Prove that \( \lambda_{jk} = \lambda_{kj} \) for any integers \( j, k \geq 2 \).
Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.
Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.
Denote by $F_0(x)$, $F_1(x)$, $\ldots$ the sequence of Fibonacci polynomials, which satisfy the recurrence $F_0(x)=1$, $F_1(x)=x$, and $F_n(x)=xF_{n-1}(x)+F_{n-2}(x)$ for all $n\geq 2$. It is given that there exist unique integers $\lambda_0$, $\lambda_1$, $\ldots$, $\lambda_{1000}$ such that \[x^{1000}=\sum_{i=0}^{1000}\lambda_iF_i(x)\] for all real $x$. For which integer $k$ is $|\lambda_k|$ maximized?
Find the remainder upon division by $x^2-1$ of the determinant
$$\begin{vmatrix}
x^3+3x & 2 & 1 & 0
\\ x^2+5x & 3 & 0 & 2
\\x^4 +x^2+1 & 2 & 1 & 3
\\x^5 +1 & 1 & 2 & 3
\\ \end{vmatrix}$$
Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$