Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle $\mathcal{C}$ with radius $30.$ Let $K$ be the area of the region inside $\mathcal{C}$ and outside all of the six circles in the ring. Find $\lfloor K\rfloor.$ (The notation $\lfloor K\rfloor$ denotes the greatest integer that is less than or equal to $K.$)
For each positive integer $k,$ let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k.$ For example, $S_3$ is the sequence $1,$ $4,$ $7,$ $\ldots.$ For how many values of $k$ does $S_k$ contain the term $2005?$
How many positive integers have exactly three proper divisors, each of which is less than $50?$ (A *proper divisor* of a positive integer $n$ is a positive integer divisor of $n$ other than $n$ itself.)
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are $5$ members left over. The director finds that if they are arranged in a rectangular formation with $7$ more rows than columns, the desired result can be obtained. Find the maximum number of members this band can have.
Robert has $4$ indistinguishable gold coins and $4$ indistinguishable silver coins. Each coin has an engraving of a face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the $8$ coins.
Let $P$ be the product of the nonreal roots of $x^4 - 4x^3 + 6x^2 - 4x = 2005.$ Find $\lfloor P\rfloor.$ (The notation $\lfloor P\rfloor$ denotes the greatest integer that is less than or equal to $P.$)
In quadrilateral $ABCD,$ $BC = 8,$ $CD = 12,$ $AD = 10,$ and $m\angle A = m\angle B = 60^\circ.$ Given that $AB = p + \sqrt{q},$ where $p$ and $q$ are positive integers, find $p + q.$
The equation $$
2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1
$$ has three real roots. Given that their sum is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$
Twenty-seven unit cubes are each painted orange on a set of four faces so that the two unpainted faces share an edge. The $27$ cubes are then randomly arranged to form a $3 \times 3 \times 3$ cube. Given that the probability that the entire surface of the larger cube is orange is $\frac{p^a}{q^b r^c},$ where $p,$ $q,$ and $r$ are distinct primes and $a,$ $b,$ and $c$ are positive integers, find $a + b + c + p + q + r.$
Triangle $ABC$ lies in the Cartesian plane and has area $70.$ The coordinates of $B$ and $C$ are $(12, 19)$ and $(23, 20),$ respectively, and the coordinates of $A$ are $(p, q).$ The line containing the median to side $\overline{BC}$ has slope $-5.$ Find the largest possible value of $p + q.$
A semicircle with diameter $d$ is contained in a square whose sides have length $8.$ Given that the maximum value of $d$ is $m - \sqrt{n},$ where $m$ and $n$ are integers, find $m + n.$
For positive integers $n,$ let $\tau(n)$ denote the number of positive integer divisors of $n,$ including $1$ and $n.$ For example, $\tau(1) = 1$ and $\tau(6) = 4.$ Define $S(n)$ by $$
S(n) = \tau(1) + \tau(2) + \cdots + \tau(n).
$$ Let $a$ denote the number of positive integers $n \le 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \le 2005$ with $S(n)$ even. Find $|a - b|.$
A particle moves in the Cartesian plane from one lattice point to another according to the following rules:
• From any lattice point $(a, b),$ the particle may move only to $(a+1, b),$ $(a, b+1),$ or $(a+1, b+1).$
• There are no right angle turns in the particle’s path. That is, the sequence of points visited contains neither a subsequence of the form $(a, b),$ $(a+1, b),$ $(a+1, b+1)$ nor a subsequence of the form $(a, b),$ $(a, b+1),$ $(a+1, b+1).$
How many different paths can the particle take from $(0, 0)$ to $(5, 5)?$
Consider the points $A(0, 12),$ $B(10, 9),$ $C(8, 0),$ and $D(-4, 7).$ There is a unique square $\mathcal{S}$ such that each of the four points is on a different side of $\mathcal{S}.$ Let $K$ be the area of $\mathcal{S}.$ Find the remainder when $10K$ is divided by $1000.$
In $\triangle ABC,$ $AB = 20.$ The incircle of the triangle divides the median containing $C$ into three segments of equal length. Given that the area of $\triangle ABC$ is $m\sqrt{n},$ where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime, find $m + n.$