Practice (25)
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Solve the following system in integers:
$$
\left\{
\begin{array}{ll}
x_1 + x_2 + \cdots + x_n &= n \\
x_1^2 + x_2^2 + \cdots + x_n^2 &= n \\
\cdots\\
x_1^n + x_2^n + \cdots + x_n^n &= n
\end{array}
\right.
$$
Let $a_1, a_2, \cdots, a_{100}, b_1, b_2, \cdots, b_{100}$ be distinct real numbers. They are used to fill a $100 \times 100$ grids by putting the value of $(a_i + b_j)$ in the cell $(i, j)$ where $1 \le i, j \le 100$. Let $A_i$ be the product of all the numbers in column $i$, and $B_i$ be the product of all the numbers in row $i$. Show that if every $A_i$ equals to 1, then every $B_j$ equals to -1.
Find the range of the function $$y=x+\sqrt{x^2 -3x+2}$$
For any non-negative real numbers $x$ and $y$, the function $f(x+y^2)=f(x) + 2[f(y)]^2$ always holds, $f(x)\ge 0$, $f(1)\ne 0$. Find the value of $f(2+\sqrt{3})$.
Which pair contains same functions:
(A) $f(x)=\sqrt{(x-1)^2}$ and $g(x)=x-1$
(B) $f(x)=\sqrt{x^2 -1}$ and $g(x)=\sqrt{x+1}\cdot\sqrt{x-1}$
(C) $f(x)=(\sqrt{x -1})^2$ and $g(x)=\sqrt{(x-1)^2}$
(D) $f(x)\displaystyle\sqrt{\frac{x^2-1}{x+1}}$ and $g(x)=\displaystyle\frac{\sqrt{x^2-1}}{\sqrt{x+1}}$
Select all correct answers.
Let function $f(x)$ is defined as the following: $$ f(x)= \left\{ \begin{array}{ll} x+2 &, \text{if } x \le -1\\ 2x &, \text{if } -1 < x < 2\\ \displaystyle\frac{x^2}{2} &, \text{if } x \ge 2 \end{array} \right. $$ (A) Compute $f(f(f(-\frac{7}{4})))$ (B) If $f(a)=3$, find the value of $a$
Let $a$ and $k$ be two positive integers, and function $f(x)=3x+1$. If $f(x)$'s domain is $\{1, 2, 3, k\}$ and range is $\{4, 7, a^4, a^2 + 3a\}$, find the value of $a$ and $k$.
Which of the following function is the same as $y=\sqrt{-2x^3}$?
(A) $y=x\sqrt{-2x}\qquad$ (B) $y=-x\sqrt{-2x}\qquad$ (C) $y=-\sqrt{2x^3}\qquad$ (D) $y=x^2\sqrt{-2/x}$
Let $f(x)$ be an odd function and $g(x)$ be an even function. If $f(x)+g(x)=\frac{1}{x-1}$, find $f(x)$ and $g(x)$.
If $f\Big(\displaystyle\frac{x+1}{x}\Big)=\displaystyle\frac{x^2+x+1}{x^2}$, find $f(x)$.
If $f(x)$ is an odd function defined on $\mathbb{R}$, compute $f(0)$.
Let function $f(x)$ satisfy $f(a)+f(b)=f(ab)$, and $f(2)=2$ and $f(3)=3$. Compute $f(72)$.
A function $f$ has its domain equal to the set of integers $\{0, 1, ..., 11\}$, and $f(n)\ge 0$ for all such $n$, and $f$ satisfies: $f(0) = 0$, $f(6) = 1$. If $x \ge 0$, $y\ge 0$, and $x + y\le 11$, then $f(x + y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$. Find $f(2)^2 + f(10)^2$.
For nonnegative integer $n$, the following are true:
$f(0) = 0$
$f(1) = 1$
$f(n) = f(n-\frac{m(m-1)}{2})-f(\frac{m(m+1)}{2} -n)$ for integer $m$ satisfying $m \ge 2$ and $\frac{m(m-1)}{2} < n \le \frac{m(m+1)}{2}$.
Find the smallest $n$ such that $f(n) = 4$.
Consider all functions $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $$f(f(x)+2x+20)=15$$ Call an integer $n$ $\textit{good}$ if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n)=m$. Find the sum of all good integers $x$.
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n)) + f(n) = 2n + 3$ for all nonnegative integers $n$. Find $f(2014)$.
Let $f(x)= \sqrt{ax^2+bx}$. For how many real values of $a$ is there at least one positive value of $b$ for which the domain of $f$ and the range of $f$ are the same set?
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?
Determine all polynomials such that $P(0) = 0$ and $P(x^2 + 1) = P(x)^2 + 1$.
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$
Let even function $f(x)$ and odd function $g(x)$ satisfy the relationship of $f(x)+g(x)=\sqrt{1+x+x^2}$. Find $f(3)$.
Let $f\Big(\dfrac{1}{x}\Big)=\dfrac{1}{x^2+1}$. Compute $$f\Big(\dfrac{1}{2013}\Big)+f\Big(\dfrac{1}{2012}\Big)+f\Big(\dfrac{1}{2011}\Big)+\cdots +f\Big(\dfrac{1}{2}\Big)+f(1)+f(2)+\cdots +f(2011)+f(2012)+f(2013)$$
Let $f(x)=x^{-\frac{k^2}{2}+\frac{3}{2}k+1}$ be an odd function where $k$ is an integer. If $f(x)$ is monotonically increasing when $x\in(0,+\infty)$, find all the possible values of $k$.
Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$, and $(P(2))^2 = P(3)$. Then $P\big(\frac72\big)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.