Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pariwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum_{i=1}^\infty C_{n-2}C_{n-1} = 6p$. Find $p$.
The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$.
For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$. Let $x_1, x_2, ..., x_{215}$ be positive real numbers such that $\sum_{i=1}^{215} x_i=1$ and $\sum_{i \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}$. The maximum possible value of $x_2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Given $7x + 13 = 328$, what is the value of $14x + 13$?
If $\frac{x + 5}{x-2} = \frac{2}{3}$, what is the value of $x$?
Ross and Max have a combined weight of 184 pounds. Ross and Seth have a combined weight of 197 pounds. Max and Seth have a combined weight of 189 pounds. How many pounds does Ross weigh?
A taxi charges \$3.25 for the first mile and \$0.45 for each additional 14 mile thereafter. At most, how many miles can a passenger travel using \$13.60? Express your answer as a mixed number.
A function $f (x)$ is defined for all positive integers. If $f (a) + f (b) = f (ab)$ for any two positive integers $a$ and $b$ and $f (3) = 5$, what is $f (27)$?
The digits of a 3-digit integer are reversed to form a new integer of greater value. The product of this new integer and the original integer is 91,567. What is the new integer?
The function $f (n) = a\cdot n! + b$, where $a$ and $b$ are positive integers, is defined for all positive integers. If the range of $f$ contains two numbers that differ by 20, what is the least possible value of $f (1)$?
Starting at the origin, a bug crawls 1 unit up, 2 units right, 3 units down and 4 units left. From this new point, the bug repeats this entire sequence of four moves 2015 more times, for a total of 2016 times. The coordinates of the bug’s final location are $(a, b)$. What is the value of $a + b$?
If $(x^2 + 3x + 6)(x^2 + ax + b) = x^4 + mx^2 + n$ for integers $a, b, m$ and $n$, what is the product of $m$ and $n$?
If $x$ is a number such that $3^x + 3^{x+2} = 9^x + 9^{x+2}$, then what is the value of $3^x$? Express your answer as a common fraction.
For each positive integer $n$, $a_n = 9n + 2$ and $b_n = 7n + 3$. If the values common to both sequences are written as a sequence, the $n^{th}$ term of that sequence can be expressed as $pn + q$. What is the value of $p − q$?
The function $f (n) = a ⋅ n! + b$, where a and b are positive integers, is defined for all positive integers. If the range of $f$ contains two numbers that differ by 20, what is the least possible value of $f (1)$?
Compute the least possible non-zero value of $A^2+B^2+C^2$ such that $A, B,$ and $C$ are integers satisfying $A\log16+B\log18+C\log24=0$.
In $\triangle{LEO}$, point $J$ lies on $\overline{LO}$ such that $\overline{JE}\perp\overline{EO}$, and point $S$ lies on $\overline{LE}$ such that $\overline{JS}\perp\overline{LE}$. Given that $JS=9, EO=20,$ and $JO+SE=37$, compute the perimeter of $\triangle{LEO}$.
Let $P(x)$ be the polynomial $x^3 + Ax^2 +Bx+C$ for some constants $A, B,$ and $C$. There exists constant $D$ and $E$ such that for all $x$, $P(x+1)=x^3 + Dx^2 + 54x +37$ and $P(x+2)=x^3 + 26x + Ex+115$. Compute the ordered triple $(A, B, C)$.
Find the largest of three prime divisors of $13^4+16^5-172^2$.
During the annual frog jumping contest at the county fair, the height of the frog's jump, in feet, is given by $$f(x)= -
\frac{1}{3}x^2+\frac{4}{3}x\:$$ What was the maximum height reached by the frog?
Let $x, y,$ and $z$ be some real numbers such that: $x+2y-z=6$ and $x-y+2z=3$. Find the minimal value of $x^2 + y^2 + z^2$.
Let $x$ be a negative real number. Find the maximum value of $y=x+\frac{4}{x} +2007$.
Let real numbers $a$ and $b$ satisfy $a^2 + ab + b^2 = 1$. Find the range of $a^2 - ab + b^2$.
Let the sum of first $n$ terms of arithmetic sequence $\{a_n\}$ be $S_n$, and the sum of first $n$ terms of arithmetic sequence $\{b_n\}$ be $T_n$. If $\frac{S_n}{T_n}=\frac{2n}{3n+7}$, compute the value of $\frac{a_8}{b_6}$.
Suppose every term in the sequence
$$1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, \cdots$$
is either $1$ or $2$. If there are exactly $(2k-1)$ twos between the $k^{th}$ one and the $(k+1)^{th}$ one, find the sum of its first $2014$ terms.