Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $9{:}99$ just before midnight, $0{:}00$ at midnight, $1{:}25$ at the former $3{:}00$ AM, and $7{:}50$ at the former $6{:}00$ PM. After the conversion, a person who wanted to wake up at the equivalent of the former $6{:}36$ AM would set his new digital alarm clock for $A{:}BC,$ where $A,$ $B,$ and $C$ are digits. Find $100A + 10B + C.$
Positive integers $a$ and $b$ satisfy the condition $$
\log_2(\log_{2^a}(\log_{2^b}(2^{1000}))) = 0.
$$ Find the sum of all possible values of $a + b.$
A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\frac{T}{2}$ seconds after it is lit, the candle’s height in centimeters will be $h.$ Find $10h.$
In the Cartesian plane let $A = (1, 0)$ and $B = (2, 2\sqrt{3}).$ Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P = (x, y)$ be the center of $\triangle ABC.$ Then $x \cdot y$ can be written as $\frac{p\sqrt{q}}{r},$ where $p$ and $r$ are relatively prime positive integers and $q$ is an integer that is not divisible by the square of any prime. Find $p + q + r.$
In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}.$ Then $\sin(\angle DAE)$ can be expressed in the form $\frac{a\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a + b + c.$
A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar reassignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.
A hexagon that is inscribed in a circle has side lengths $22,$ $22,$ $20,$ $22,$ $22,$ and $20$ in that order. The radius of the circle can be written as $p + \sqrt{q},$ where $p$ and $q$ are positive integers. Find $p + q.$
A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at least once. For example, a $1 \times 1$ red tile followed by a $2 \times 1$ green tile, a $1 \times 1$ green tile, a $2 \times 1$ blue tile, and a $1 \times 1$ green tile is a valid tiling. Note that if the $2 \times 1$ blue tile is replaced by two $1 \times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000.$
Given a circle of radius $\sqrt{13},$ let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A.$ A line passing through the point $A$ intersects the circle at points $K$ and $L.$ The maximum possible area for $\triangle BKL$ can be written in the form $\frac{a - b\sqrt{c}}{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d.$
Let $A = \{1, 2, 3, 4, 5, 6, 7\},$ and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000.$
Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c,$ where $a,$ $b,$ and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13.$
In $\triangle ABC,$ $AC = BC,$ and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD.$ Let $E$ be the midpoint of $\overline{AD}.$ Given that $CE = \sqrt{7}$ and $BE = 3,$ the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n.$
For positive integers $n$ and $k,$ let $f(n, k)$ be the remainder when $n$ is divided by $k,$ and for $n \gt 1$ let $$
F(n) = \max_{1 \le k \le \frac{n}{2}} f(n, k).
$$ Find the remainder when $\sum_{n = 20}^{100} F(n)$ is divided by $1000.$
Let $A,$ $B,$ $C$ be angles of a triangle with $A$ and $C$ acute and $B$ greater than a right angle satisfying $$
\begin{aligned} &\cos^2 A + \cos^2 B \\ &\quad {}+ 2 \sin A \sin B \cos C = \frac{15}{8} \end{aligned}
$$ and $$
\begin{aligned} &\cos^2 B + \cos^2 C \\ &\quad {}+ 2 \sin B \sin C \cos A = \frac{14}{9}. \end{aligned}
$$ There are positive integers $p,$ $q,$ $r,$ and $s$ for which $$
\begin{aligned} &\cos^2 C + \cos^2 A \\ &\quad {}+ 2 \sin C \sin A \cos B \\ &= \frac{p - q\sqrt{r}}{s}, \end{aligned}
$$ where $p + q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p + q + r + s.$