Suppose the graph of $f(x)=x^4 + ax^3 + bx^2 + cd + d$, where $a$, $b$, $c$, $d$ are all real constants, passes through three points $A \big(2,\frac{1}{2}\big)$, $B \big(3, \frac{1}{3}\big)$, and $C \big(4, \frac{1}{4}\big)$. Find the value of $f(1) + f(5)$.
Find a quadratic polynomial $f(x)=x^2 + mx +n$ such that $$f(a)=bc,\quad f(b) = ca,\quad f(c) = ab$$ where $a$, $b$, $c$ are three distinct real numbers.
If all coefficients of the polynomial $$f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_3x^3+x^2+x+1=0$$ are real numbers, prove that its roots cannot be all real.
Compute the value of $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}+\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
and simplify $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}\quad\text{and}\quad\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
Prove there cannot exist a $998$-degree polynomial with real number coefficients $P(x)$ such that $$[P(x)]^2-1=P(x^2+1)$$ holds for any $x\in\mathbb{C}$.
If $abc=1$, solve this equation $$\frac{2ax}{ab+a+1}+\frac{2bx}{bc+b+1}+\frac{2cx}{ca+c+1}=1$$
How many pairs of ordered real numbers $(x, y)$ are there such that
$$
\left\{
\begin{array}{ccl}
\mid x\mid + y &=& 12\\
x + \mid y \mid &=&6
\end{array}
\right.
$$
Let $a$, $b$, and $c$ be three distinct numbers such that $$\frac{a+b}{a-b}=\frac{b+c}{2(b-c)}=\frac{c+a}{3(c-a)}$$
Prove that $8a + 9b + 5c = 0$.
Solve this equation in real numbers: $$\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\frac{1}{2}\times(x+y+z)$$
Let $a$, $b$, and $c$ be the lengths of $\triangle{ABC}$'s three sides. Compute the area of $\triangle{ABC}$ if the following relations hold: $$\frac{2a^2}{1+a^2}=b,\qquad \frac{2b^2}{1+b^2}=c,\qquad \frac{2c^2}{1+c^2}=a$$
Let real numbers $x$, $y$, and $z$ satisfy $0 < x, y, z < 1$. Prove $$x(1-y)+y(1-z)+z(1-x)< 1$$
Show that $$\frac{(x+a)(x+b)}{(c-a)(c-b)}+\frac{(x+b)(x+c)}{(a-b)(a-c)}+\frac{(x+c)(x+a)}{(b-c)(b-a)}=1$$ without expanding the left side of the equation.
Find the range of function $f(x)=3^{-|\log_2x|}-4|x-1|$.
Find the minimal value of $y=\sqrt{x^2+2x+5}+\sqrt{x^2-4x+5}$.
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?
If $17!=355687ab8096000$ where $a$ and $b$ are two missing single digits. Find $a$ and $b$.
Show that neither $385^{97}$ nor $366^{17}$ can be expressed as the sum of cubes of some consecutive integers.
An integer $N$ is selected at random in the range $1\leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. If $S$ denotes the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find $S$.
Let $x$ be an integer and $p$ is a prime divisor of $(x^6 + x^5 + \cdots + 1)$. Show that $p=7$ or $p\equiv 1\pmod{7}$.
A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, find the probability that Sharon can discard one of her cards and $\textit{still}$ have at least one card of each color and at least one card with each number.
Find the minimum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$.
Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a > 1$ and $b > 1$ such that $(a^b + b^a)$ is divisible by $(a + b)$.