A straight river that is 264 meters wide flows from west to east at a rate of 14 meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at 80 meters per minute, and Sherry swims at 60 meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find $D$.
Let $x_1 \le x_2 \le \dots \le x_{100}$ be real numbers such that $|x_1| + |x_2| + \dots + |x_{100}| = 1$ and $x_1 + x_2 + \dots + x_{100} = 0$. Among all such 100-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Find the number of ordered pairs of integers $(a, b)$ such that the sequence
$3, 4, 5, a, b, 30, 40, 50$
is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.
Let $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, $i$ be distinct integers from $1$ to $9$. The minimum possible positive value of
$$
\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}
$$
can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles.
Find the number of positive integers $n \le 600$ whose value can be uniquely determined when the values of $\lfloor \frac{n}{4} \rfloor$, $\lfloor \frac{n}{5} \rfloor$, and $\lfloor \frac{n}{6} \rfloor$ are given, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to the real number $x$.
Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18$. Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega$. Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A$, $\omega_B$, and $\omega_C$ meet in six points—two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC$, and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC$. The side length of the smaller equilateral triangle can be written as $\sqrt{a} - \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$.
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, A_3, \dots, A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \dots, B_n$ lie on $\ell_B$. Additionally, when segments $A_iB_j$ are drawn for all $i = 1, 2, 3, \dots, m$ and $j = 1, 2, 3, \dots, n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m = 7$ and $n = 5$. The figure shows that there are 8 regions when $m = 3$ and $n = 2$.
Ellina has twelve blocks, two each of red (R), blue (B), yellow (Y), green (G), orange (O), and purple (P). Call an arrangement of blocks *even* if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement RBBYGGYROPPO is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\frac{m}{n}$, where *m* and *n* are relatively prime positive integers. Find $m + n$.
Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A$, $B$, and $C$, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560$. Find $AC^2$.
Let $ABCD$ be a parallelogram with $\angle BAD \lt 90^\circ$. A circle tangent to sides $DA$, $AB$, and $BC$ intersects diagonal $AC$ at points $P$ and $Q$ with $AP \lt AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ 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 $m+n$.
Let $a$, $b$, $x$, and $y$ be real numbers with $a \gt 4$ and $b \gt 1$ such that
$$
\frac{x^2}{a^2} + \frac{y^2}{a^2 - 16} = \frac{(x - 20)^2}{b^2 - 1} + \frac{(y - 11)^2}{b^2} = 1.
$$
Find the least possible value for $a + b$.
Let $a$, $b$, $x$, and $y$ be real numbers with $a \gt 4$ and $b \gt 1$ such that
$$
\frac{x^2}{a^2} + \frac{y^2}{a^2 - 16} = \frac{(x - 20)^2}{b^2 - 1} + \frac{(y - 11)^2}{b^2} = 1.
$$
Find the least possible value for $a + b$.
For any finite set $X$, let $|X|$ denote the number of elements in $X$. Define
$$
S_n = \sum |A \cap B|,
$$
where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\{1, 2, 3, \dots, n\}$ with $|A| = |B|$. For example, $S_2 = 4$ because the sum is taken over the pairs of subsets
$$
(A, B) \in \{(\emptyset, \emptyset), (\{1\}, \{1\}), (\{1\}, \{2\}), (\{2\}, \{1\}), (\{2\}, \{2\}), (\{1, 2\}, \{1, 2\})\},
$$
giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4$. Let $\frac{S_{2022}}{S_{2021}} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p+q$ is divided by 1000.
There is a polynomial $P(x)$ with integer coefficients such that
$$
P(x) = \frac{(x^{2310} - 1)^6}{(x^{105} - 1)(x^{70} - 1)(x^{42} - 1)(x^{30} - 1)}
$$
holds for every $0 \lt x \lt 1$. Find the coefficient of $x^{2022}$ in $P(x)$.
Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\overline{abcd}$, where at least one of the digits $a, b, c$, or $d$ is nonzero. Let $N$ be the number of distinct numerators obtained when numbers in $S$ are written as fractions in lowest terms. For example, both $4$ and $410$ are counted among the distinct numerators for numbers in $S$ because $0.\overline{3636} = \frac{4}{11}$ and $0.\overline{1230} = \frac{410}{3333}$. Find the remainder when $N$ is divided by 1000.
For positive integers $a, b$, and $c$ with $a \lt b \lt c$, consider collections of postage stamps in denominations $a, b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to 1000 cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c) = 97$ for some choice of $a$ and $b$.
Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon.
Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations
$$
\begin{aligned}
\sqrt{2x - xy} + \sqrt{2y - xy} &= 1 \\
\sqrt{2y - yz} + \sqrt{2z - yz} &= \sqrt{2} \\
\sqrt{2z - zx} + \sqrt{2x - zx} &= \sqrt{3}.
\end{aligned}
$$
Then $[(1-x)(1-y)(1-z)]^2$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
