Maya lists all the positive divisors of $2010^2.$ She then randomly selects two distinct divisors from this list. Let $p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $p$ can be expressed in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
Suppose that $y = \frac{3}{4}x$ and $x^y = y^x.$ The quantity $x + y$ can be expressed as a rational number $\frac{r}{s},$ where $r$ and $s$ are relatively prime positive integers. Find $r + s.$
Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac{4}{7}.$ Jackie flips the three coins, and then Phil flips the three coins. Let $\frac{m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
Positive integers $a,$ $b,$ $c,$ and $d$ satisfy $a \gt b \gt c \gt d,$ $a + b + c + d = 2010,$ and $a^2 - b^2 + c^2 - d^2 = 2010.$ Find the number of possible values of $a.$
Let $P(x)$ be a quadratic polynomial with real coefficients satisfying $$
\begin{aligned} x^2 - 2x + 2 &\le P(x) \\ &\le 2x^2 - 4x + 3 \end{aligned}
$$ for all real numbers $x,$ and suppose $P(11) = 181.$ Find $P(16).$
Define an ordered triple $(\mathcal{A}, \mathcal{B}, \mathcal{C})$ of sets to be *minimally intersecting* if $|\mathcal{A} \cap \mathcal{B}| = |\mathcal{B} \cap \mathcal{C}| = |\mathcal{C} \cap \mathcal{A}| = 1$ and $\mathcal{A} \cap \mathcal{B} \cap \mathcal{C} = \emptyset.$ For example, $(\{1, 2\}, \{2, 3\}, \{1, 3, 4\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $\{1, 2, 3, 4, 5, 6, 7\}.$ Find the remainder when $N$ is divided by $1000.$
Note: $|\mathcal{S}|$ represents the number of elements in the set $\mathcal{S}.$
For a real number $a,$ let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a.$ Let $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x, y)$ such that $$
\lfloor x \rfloor^2 + \lfloor y \rfloor^2 = 25.
$$ The region $\mathcal{R}$ is completely contained in a disk of radius $r$ (a disk is the union of a circle and its interior). The minimum value of $r$ can be written as $\frac{\sqrt{m}}{n},$ where $m$ and $n$ are integers and $m$ is not divisible by the square of any prime. Find $m + n.$
Let $(a, b, c)$ be a real solution of the system of equations $$
\begin{aligned} x^3 - xyz &= 2, \\ y^3 - xyz &= 6, \\ z^3 - xyz &= 20. \end{aligned}
$$ The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
Let $N$ be the number of ways to write $2010$ in the form $$
\begin{aligned} 2010 &= a_3 \cdot 10^3 + a_2 \cdot 10^2 \\ &\quad {}+ a_1 \cdot 10 + a_0, \end{aligned}
$$ where the $a_i$’s are integers, and $0 \le a_i \le 99.$ An example of such a representation is $1 \cdot 10^3$ $+ 3 \cdot 10^2$ $+ 67 \cdot 10^1$ $+ 40 \cdot 10^0.$ Find $N.$
Let $\mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $|8 - x| + y \le 10$ and $3y - x \ge 15.$ When $\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15,$ the volume of the resulting solid is $\frac{m\pi}{n\sqrt{p}},$ where $m,$ $n,$ and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m + n + p.$
Let $m \ge 3$ be an integer and let $S = \{3, 4, 5, \ldots, m\}.$ Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a,$ $b,$ and $c$ (not necessarily distinct) such that $ab = c.$
Note: a partition of $S$ is a pair of sets $A,$ $B$ such that $A \cap B = \emptyset$ and $A \cup B = S.$
Rectangle $ABCD$ and a semicircle with diameter $\overline{AB}$ are coplanar and have nonoverlapping interiors. Let $\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $\ell$ meets the semicircle, segment $\overline{AB},$ and segment $\overline{CD}$ at distinct points $N,$ $U,$ and $T,$ respectively. Line $\ell$ divides region $\mathcal{R}$ into two regions with areas in the ratio $1 : 2.$ Suppose that $AU = 84,$ $AN = 126,$ and $UB = 168.$ Then $DA$ can be represented as $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n.$
In $\triangle ABC$ with $AB = 12,$ $BC = 13,$ and $AC = 15,$ let $M$ be a point on $\overline{AC}$ such that the incircles of $\triangle ABM$ and $\triangle BCM$ have equal radii. Let $p$ and $q$ be positive relatively prime integers such that $\frac{AM}{CM} = \frac{p}{q}.$ Find $p + q.$