Found problems: 4275
Let \[N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1.\] How many positive integers are factors of $N$?
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 216 $
$p(x)$ is an irreducible polynomial in $\mathbb Q[x]$ that $\mbox{deg}\ p$ is odd. $q(x),r(x)$ are polynomials with rational coefficients that $p(x)|q(x)^2+q(x).r(x)+r(x)^2$. Prove that \[p(x)^2|q(x)^2+q(x).r(x)+r(x)^2\]
Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.
Find all real solutions of the following set of equations:
\[72x^3+4xy^2=11y^3\]
\[27x^5-45x^4y-10x^2y^3=\frac{-143}{32}y^5\]
$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.
For every integer $ n \ge 3$ and any given angle $ \alpha$ with $ 0 < \alpha < \pi$, let $ P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha$.
(a) Prove that there is a unique polynomial of the form $ f(x) \equal{} x^2 \plus{} ax \plus{} b$ which divides $ P_n(x)$ for every $ n \ge 3$.
(b) Prove that there is no polynomial $ g(x) \equal{} x \plus{} c$ which divides $ P_n(x)$ for every $ n \ge 3$.
Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$
[b]Q.[/b] Find all polynomials $P: \mathbb{R \times R}\to\mathbb{R\times R}$ with real coefficients, such that $$P(x,y) = P(x+y,x-y), \ \forall\ x,y \in \mathbb{R}.$$
[i]Proposed by TuZo[/i]
Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\},$
\[\sum_{k=0}^{p-1}k!n^k \quad \text{is not divisible by }p.\]
Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]
Show that the solution set of the inequality
\[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4}
\]
is a union of disjoint intervals, the sum of whose length is 1988.
Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 4x^2 -3x + 2$. The sum $|x_1| + |x_2| + |x_3|$ is
(A): $4$ (B): $6$ (C): $8$ (D): $10$ (E): None of the above.
A polynomial $f(x)$ with integer coefficients is called $\textit{splitty}$ if and only if for every prime $p$, there exists polynomials $g_p, h_p$ with integer coefficients with degrees strictly smaller than the the degree of $f$, such that all coefficients of $f-g_ph_p$ are divisible by $p$. Compute the sum of all positive integers $n \leq 100$ such that $x^4+16x^2+n$ is $\textit{splitty}$.
The sequence $\{x_{n}\}_{n \ge 1}$ is defined by \[x_{1}=x_{2}=1, \; x_{n+2}= 14x_{n+1}-x_{n}-4.\] Prove that $x_{n}$ is always a perfect square.
Find all Polynomial $P(x)$ and $Q(x)$ with Integer Coefficients satisfied the equation:
\[Q(a+b) = \frac{P(a) - P(b)}{a - b}\]
$\forall a, b \in \mathbb{Z}^+$ and $a>b$
Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.
Show that the roots of $x^4 +ax^3 +bx^2 +cx +d$, if suitably numbered, satisfy the relation $\frac{r_1 }{r_2 } = \frac{ r_3 }{r _4},$ provided $a^2 d=c^2 \ne 0.$
Let $f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0$ be a polynomial with real coefficients $(n \ge 2)$. Suppose all roots of $f$ are real. Prove that the absolute value of each root is at most $\sqrt{\frac{2(1-n)}n a_{n-2}}$.
Determine all polynomials $P(x)$ with real coeffcients such that $(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$.
Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $|BC|+|AD| = 7$, $|AB| = 9$ and $|BC| = 14$. What is the ratio of the area of the triangle formed by $CD$, angle bisector of $\widehat{BCD}$ and angle bisector of $\widehat{CDA}$ over the area of the trapezoid?
$
\textbf{a)}\ \dfrac{9}{14}
\qquad\textbf{b)}\ \dfrac{5}{7}
\qquad\textbf{c)}\ \sqrt 2
\qquad\textbf{d)}\ \dfrac{49}{69}
\qquad\textbf{e)}\ \dfrac 13
$
$ F(x)$ is polynomial with real coefficients. $ F(x) \equal{} x^{4}\plus{}a_{1}x^{3}\plus{}a_{2}x^{2}\plus{}a_{1}x^{1}\plus{}a_{0}$. $ M$ is local maximum and $ m$ is minimum. Prove that $ \frac{3}{10}(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}< M\minus{}m < 3(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}$
Does there exist functions $ f,g: \mathbb{R}\to\mathbb{R}$ such that $ f(g(x)) \equal{} x^2$ and $ g(f(x)) \equal{} x^k$ for all real numbers $ x$
a) if $ k \equal{} 3$?
b) if $ k \equal{} 4$?
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$.
a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$.
b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]
Prove that: for all positive integers $n\ge 2,$ the polynomial$$(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1$$
is irreducible in $\mathbb{Q}[x].$