Found problems: 252
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that
(1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$;
(2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees;
(3) for any integers $ x, |f(x)|$ isn't prime numbers.
Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$
Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.
Let $H$ be an $n\times n$ matrix all of whose entries are $\pm1$ and whose rows are mutually orthogonal. Suppose $H$ has an $a\times b$ submatrix whose entries are all $1.$ Show that $ab\le n.$
Let $x,y\in[0,\infty)$. Which of the following is true?
$\textbf{(A)}~\left|\log\left(1+x^2\right)-\log\left(1+y^2\right)\right|\le|x-y|$
$\textbf{(B)}~\left|\sin^2x-\sin^2y\right|\le|x-y|$
$\textbf{(C)}~\left|\tan^{-1}x-\tan^{-1}y\right|\le|x-y|$
$\textbf{(D)}~\text{None of the above}$
Let $x_0=2024^{2024}$ and $x_{n+1}=|x_n-\pi|$ for $n \ge 0$. Show that there exists a value of $n$ such that $x_{n+2}=x_n$.
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that
(1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$;
(2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees;
(3) for any integers $ x, |f(x)|$ isn't prime numbers.
Let $A$ be a $8\times 8$ array with entries from the set $\{-1,1\}$ such that any $2\times 2$ sub-square of the array has the absolute value of the sum of its element equal with 2. Prove that the array must have at least two identical lines.
Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$.
Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.
Find the number of integer solutions to $||x| - 2020| < 5$.
Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\ldots,x_n)$ such that
\[ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| \]
attains its minimum.
Given real numbers $x,y,z$ which satisfies
$$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$
Show that $max\{ |x|,|y|,|z|\} \le 1$.
Let $ a$, $ b$, $ c$ be three real numbers satisfying that \[ \left\{\begin{array}{c c c} \left(a\plus{}b\right)\left(b\plus{}c\right)\left(c\plus{}a\right)&\equal{}&abc\\ \left(a^3\plus{}b^3\right)\left(b^3\plus{}c^3\right)\left(c^3\plus{}a^3\right)&\equal{}&a^3b^3c^3\end{array}\right.\] Prove that $ abc\equal{}0$.
For any complex number $w = a + bi$, $|w|$ is defined to be the real number $\sqrt{a^2 + b^2}$. If $w = \cos{40^\circ} + i\sin{40^\circ}$, then
\[ |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \]
equals
$\textbf{(A)}\ \frac{1}{9}\sin{40^\circ} \qquad \textbf{(B)}\ \frac{2}{9}\sin{20^\circ} \qquad \textbf{(C)}\ \frac{1}{9}\cos{40^\circ} \qquad \textbf{(D)}\ \frac{1}{18}\cos{20^\circ} \qquad \textbf{(E)}\text{ none of these}$
Calculate the following indefinite integrals.
[1] $\int (x^2+4x-3)^2(x+2)dx$
[2] $\int \frac{\ln x}{x(\ln x+1)}dx$
[3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$
[4] $\int \frac{dx}{\sin x\cos ^ 2 x}$
[5] $\int \sqrt{1-3x}\ dx$
Describe the region $R$ consisting of the points $(a,b)$ of the cartesian plane for which both (possibly complex) roots of the polynomial $z^2+az+b$ have absolute value smaller than $1$.
Let $n$ and $k$ be integers with $k\leq n-2$. The absolute value of the sum of elements of any $k$-element subset of $\{a_1,a_2,\cdots,a_n\}$ is less than or equal to 1. Show that: If $|a_1|\geq1$, then for any $2\leq i \leq n$, we have:
$$|a_1|+|a_i|\leq2$$
Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$.
$ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $
There are $n$ points on a circle, such that each line segment connecting two points is either red or blue.
$P_iP_j$ is red if and only if $P_{i+1} P_{j+1}$ is blue, for all distinct $i, j$ in $\left\{1, 2, ..., n\right\}$.
(a) For which values of $n$ is this possible?
(b) Show that one can get from any point on the circle to any other point, by doing a maximum of 3 steps, where one step is moving from a point to another point through a red segment connecting these points.
Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations?
\begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*}
$\textbf{(A) } 1 \qquad
\textbf{(B) } 2 \qquad
\textbf{(C) } 3 \qquad
\textbf{(D) } 4 \qquad
\textbf{(E) } 8 $
Suppose that \[x_1+1=x_2+2=x_3+3=\cdots=x_{2008}+2008=x_1+x_2+x_3+\cdots+x_{2008}+2009.\]
Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\displaystyle\sum_{n=1}^{2008}x_n$.
Prove that there exists a nonzero complex number $c$ and a real number $d$ such that \[\left|\left|\dfrac1{1+z+z^2}\right|-\left|\dfrac1{1+z+z^2}-c\right|\right|=d\] for all $z$ with $|z|=1$ and $1+z+z^2\neq 0$. (Here, $|z|$ denotes the absolute value of the complex number $z$, so that $|a+bi|=\sqrt{a^2+b^2}$ for real numbers $a,b$.)
For all $x>0$ prove
$$\frac{\sin^2x-x}{\ln\left(\frac{\sin^2x}x\right)^{\sqrt x}}+\frac{\cos^2x-x}{\ln\left(\frac{\cos^2x}x\right)^{\sqrt x}}>|\sin x|+|\cos x|$$
[i]Proposed by Pirkulyiev Rovsen[/i]
If $x<0$, then $\left|x-\sqrt{(x-1)^2}\right|$ equals
$\textbf{(A) }1\qquad\textbf{(B) }1-2x\qquad\textbf{(C) }-2x-1\qquad\textbf{(D) }1+2x\qquad \textbf{(E) }2x-1$