Found problems: 4275
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]
Determine all real numbers $a, b,c$ such that the polynomial $f(x) = ax^2 + bx + c$ satisfi es simultaneously the folloving conditions $\begin{cases} |f(x)| \le 1 \text{ for } |x | \le 1 \\
f(x) \ge 7 \text{ for } x \ge 2 \end{cases} $
Let $\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}$ be such that the polynomials
$f_1(x) =\prod_{i=1}^d(x-\alpha_i)$ and $f_2(x) =\prod_{i=1}^e(x-\beta_i)$
have integer coefficients.
Suppose that there exist polynomials $g_1, g_2 \in\mathbb{Z}[x]$ such that $f_1g_1 +f_2g_2 = 1$.
Prove that $\left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j) \right|=1$
If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is
$\textbf{(A) }4\qquad\textbf{(B) }3\qquad\textbf{(C) }2\qquad\textbf{(D) }1\qquad \textbf{(E) }0$
Let $S$ be a randomly chosen $6$-element subset of the set $\{0,1,2,\ldots,n\}.$ Consider the polynomial $P(x)=\sum_{i\in S}x^i.$ Let $X_n$ be the probability that $P(x)$ is divisible by some nonconstant polynomial $Q(x)$ of degree at most $3$ with integer coefficients satisfying $Q(0) \neq 0.$ Find the limit of $X_n$ as $n$ goes to infinity.
On each face of two dice some positive integer is written. The two dice are thrown and the numbers on the top face are added. Determine whether one can select the integers on the faces so that the possible sums are $2,3,4,5,6,7,8,9,10,11,12,13$, all equally likely?
Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.
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:
$\textbf{(A) }3\qquad
\textbf{(B) }4\qquad
\textbf{(C) }5\qquad
\textbf{(D) }6\qquad
\textbf{(E) }11$
Denote by $d(n)$ be the biggest prime divisor of $|n|>1$. Find all polynomials with integer coefficients satisfy;
$$P(n+d(n))=n+d(P(n)) $$
for the all $|n|>1$ integers such that $P(n)>1$ and $d(P(n))$ can be defined.
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$ .