Solve the equation $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$
Find all real numbers $m$ such that $$x^2+my^2-4my+6y-6x+2m+8 \ge 0$$ for every pair of real numbers $x$ and $y$.
Find all numbers $n \ge 3$ for which there exists real numbers $a_1, a_2, ..., a_{n+2}$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and\[a_{i}a_{i+1} + 1 = a_{i+2}\]for $i = 1, 2, ..., n.$
The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$.
For each ordered pair of real numbers $(x,y)$ satisfying\[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that\[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$.
Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.
Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$.
Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4$($a_{n-1}$ $+$ $a_{n-2}$ $+$ $a_{n-3}$) is divided by $11$. Find $a_{2018}$ • $a_{2020}$ • $a_{2022}$.
Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube.
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial $x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$ are all real can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$.
Find the number of functions $f$ from $\{0, 1, 2, 3, 4, 5, 6\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and $|x - y|$ $\leq$ $|f(x) - f(y)|$ $\leq$ $3|x - y|$ for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}$.
Compute: $1\times 2\times 3 + 2\times 3\times 4 + \cdots + 18\times 19\times 20$.
Let $x_1$ and $x_2$ be the two roots of equation $x^2 − 3x + 2 = 0$. Find the following values without computing $x_1$ and $x_2$ directly.
i) $x_1^4 + x_2^4$
ii) $x_1 - x_2$
(Note: for (i) above, how many different solutions can you find?)
Find the sum of all possible integer values of $a$ such that the equation $(a + 1)x^2-(a^2 + 1)x + (2a^2 − 6) = 0$ is solvable in integers.
A person eats $X ( > 1)$ cookies in $N$ days in the following way:
What is the smallest possible value of $X$?
Let sequence $\{x_n\}$ satisfy the relation $x_{n+2}=x_{n+1}+2x_n$ for $n\ge 1$ where $x_1=1$ and $x_2=3$.
Let sequence $\{y_n\}$ satisfy the relation $y_{n+2}=2y_{n+1}+3y_n$ for $n\ge 1$ where $y_1=7$ and $y_2=17$.
Show that these two sequences do not share any common term.
Let $n$ be a positive integer not less than $4$. Show that there exists a polynomial with integral coefficients $$f(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2}+\cdots + a_1 x + a_0$$
such that for any positive integer $m$ and any $k \ge 2$ distinct integers $r_1$, $r_2$, $\cdots$, $r_k$, it always hold that $f(m)\ne f(r_1)f(r_2)\cdots f(r_k)$.