ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A point is selected at random from the interior of a right triangle with legs of length $2 \sqrt{3}$ and 4 . Let $p$ be the probability that the distance between the point and the nearest vertex is less than 2. Then $p$ can be written in the form $a+\sqrt{b} \pi$, where $a$ and $b$ are rational numbers. Compute $(a, b)$.
Answer: (\frac{1}{4}, \frac{1}{27})
Problem: The square $A R M L$ is contained in the $x y$-plane with $A=(0,0)$ and $M=(1,1)$. Compute the length of the shortest path from the point $(2 / 7,3 / 7)$ to itself that touches three of the four sides of square $A R M L$.
Answer: \frac{2}{7} \sqrt{53}
Problem: For each positive integer $k$, let $S_{k}$ denote the infinite arithmetic sequence of integers with first term $k$ and common difference $k^{2}$. For example, $S_{3}$ is the sequence $3,12,21, \ldots$ Compute the sum of all $k$ such that 306 is an element of $S_{k}$.
Answer: 326
Problem: Compute the sum of all values of $k$ for which there exist positive real numbers $x$ and $y$ satisfying the following system of equations.

$$
\left\{\begin{aligned}
\log _{x} y^{2}+\log _{y} x^{5} & =2 k-1 \\
\log _{x^{2}} y^{5}-\log _{y^{2}} x^{3} & =k-3
\end{aligned}\right.
$$
Answer: \frac{43}{48}
Problem: Let $W=(0,0), A=(7,0), S=(7,1)$, and $H=(0,1)$. Compute the number of ways to tile rectangle $W A S H$ with triangles of area $1 / 2$ and vertices at lattice points on the boundary of WASH.
Answer: 3432
Problem: Compute $\sin ^{2} 4^{\circ}+\sin ^{2} 8^{\circ}+\sin ^{2} 12^{\circ}+\cdots+\sin ^{2} 176^{\circ}$.
Answer: \frac{45}{2}
Problem: Compute the area of the region defined by $x^{2}+y^{2} \leq|x|+|y|$.
Answer: 2+\pi
Problem: The arithmetic sequences $a_{1}, a_{2}, a_{3}, \ldots, a_{20}$ and $b_{1}, b_{2}, b_{3}, \ldots, b_{20}$ consist of 40 distinct positive integers, and $a_{20}+b_{14}=1000$. Compute the least possible value for $b_{20}+a_{14}$.
Answer: 10
Problem: Compute the ordered triple $(x, y, z)$ representing the farthest lattice point from the origin that satisfies $x y-z^{2}=y^{2} z-x=14$.
Answer: (-266,-3,-28)
Problem: The sequence $a_{1}, a_{2}, a_{3}, \ldots$ is a geometric sequence with $a_{20}=8$ and $a_{14}=2^{21}$. Compute $a_{21}$.
Answer: 1
Problem: Let $T=1$. Circles $L$ and $O$ are internally tangent and have radii $T$ and $4 T$, respectively. Point $E$ lies on circle $L$ such that $\overline{O E}$ is tangent to circle $L$. Compute $O E$.
Answer: 2 \sqrt{2}
Problem: Let $T=2 \sqrt{2}$. In a right triangle, one leg has length $T^{2}$ and the other leg is 2 less than the hypotenuse. Compute the triangle's perimeter.
Answer: 40
Problem: $\quad$ Let $T=40$. If $x+9 y=17$ and $T x+(T+1) y=T+2$, compute $20 x+14 y$.
Answer: 8
Problem: Let $T=8$. Let $f(x)=a x^{2}+b x+c$. The product of the roots of $f$ is $T$. If $(-2,20)$ and $(1,14)$ lie on the graph of $f$, compute $a$.
Answer: \frac{8}{5}
Problem: Let $T=\frac{8}{5}$. Let $z_{1}=15+5 i$ and $z_{2}=1+K i$. Compute the smallest positive integral value of $K$ such that $\left|z_{1}-z_{2}\right| \geq 15 T$.
Answer: 25
Problem: Let $T=25$. Suppose that $T$ people are standing in a line, including three people named Charlie, Chris, and Abby. If the people are assigned their positions in line at random, compute the probability that Charlie is standing next to at least one of Chris or Abby.
Answer: \frac{47}{300}
Problem: Let $A$ be the number you will receive from position 7 and let $B$ be the number you will receive from position 9. Let $\alpha=\sin ^{-1} A$ and let $\beta=\cos ^{-1} B$. Compute $\sin (\alpha+\beta)+\sin (\alpha-\beta)$.
Answer: \frac{94}{4225}
Problem: Let $T=13$. If $r$ is the radius of a right circular cone and the cone's height is $T-r^{2}$, let $V$ be the maximum possible volume of the cone. Compute $\pi / V$.
Answer: \frac{12}{169}
Problem: Let $T=650$. If $\log T=2-\log 2+\log k$, compute the value of $k$.
Answer: 13
Problem: Let $T=100$. Nellie has a flight from Rome to Athens that is scheduled to last for $T+30$ minutes. However, owing to a tailwind, her flight only lasts for $T$ minutes. The plane's speed is 1.5 miles per minute faster than what it would have been for the originally scheduled flight. Compute the distance (in miles) that the plane travels.
Answer: 650
Problem: Let $T=9$. Compute $\sqrt{\sqrt{\sqrt[T]{10^{T^{2}-T}}}}$.
Answer: 100
Problem: Let $T=3$. Regular hexagon $S U P E R B$ has side length $\sqrt{T}$. Compute the value of $B E \cdot S U \cdot R E$.
Answer: 9
Problem: Let $T=70$. Chef Selma is preparing a burrito menu. A burrito consists of: (1) a choice of chicken, beef, turkey, or no meat, (2) exactly one of three types of beans, (3) exactly one of two types of rice, and (4) exactly one of $K$ types of cheese. Compute the smallest value of $K$ such that Chef Selma can make at least $T$ different burrito varieties.
Answer: 3
Problem: Compute the smallest positive integer $N$ such that $20 N$ is a multiple of 14 and $14 N$ is a multiple of 20 .
Answer: 70
Problem: Call a positive integer fibbish if each digit, after the leftmost two, is at least the sum of the previous two digits. Compute the greatest fibbish number.
Answer: 10112369
Problem: An ARMLbar is a $7 \times 7$ grid of unit squares with the center unit square removed. A portion of an ARMLbar is a square section of the bar, cut along the gridlines of the original bar. Compute the number of different ways there are to cut a single portion from an ARMLbar.
Answer: 96
Problem: Regular hexagon $A B C D E F$ and regular hexagon $G H I J K L$ both have side length 24 . The hexagons overlap, so that $G$ is on $\overline{A B}, B$ is on $\overline{G H}, K$ is on $\overline{D E}$, and $D$ is on $\overline{J K}$. If $[G B C D K L]=\frac{1}{2}[A B C D E F]$, compute $L F$.
Answer: 18
Problem: Compute the largest base-10 integer $\underline{A} \underline{B} \underline{C} \underline{D}$, with $A>0$, such that $\underline{A} \underline{B} \underline{C} \underline{D}=B !+C !+D !$.
Answer: 5762
Problem: Let $X$ be the number of digits in the decimal expansion of $100^{1000^{10,000}}$, and let $Y$ be the number of digits in the decimal expansion of $1000^{10,000^{100,000}}$. Compute $\left\lfloor\log _{X} Y\right\rfloor$.
Answer: 13
Problem: Compute the smallest possible value of $n$ such that two diagonals of a regular $n$-gon intersect at an angle of 159 degrees.
Answer: 60
Problem: Compute the number of quadratic functions $f(x)=a x^{2}+b x+c$ with integer roots and integer coefficients whose graphs pass through the points $(0,0)$ and $(15,225)$.
Answer: 8
Problem: A bubble in the shape of a hemisphere of radius 1 is on a tabletop. Inside the bubble are five congruent spherical marbles, four of which are sitting on the table and one which rests atop the others. All marbles are tangent to the bubble, and their centers can be connected to form a pyramid with volume $V$ and with a square base. Compute $V$.
Answer: \frac{1}{54}
Problem: Compute the smallest positive integer base $b$ for which $16_{b}$ is prime and $97_{b}$ is a perfect square.
Answer: 53
Problem: For a positive integer $n$, let $C(n)$ equal the number of pairs of consecutive 1's in the binary representation of $n$. For example, $C(183)=C\left(10110111_{2}\right)=3$. Compute $C(1)+C(2)+$ $C(3)+\cdots+C(256)$.
Answer: 448
Problem: A set $S$ contains thirteen distinct positive integers whose sum is 120 . Compute the largest possible value for the median of $S$.
Answer: 11
Problem: Let $T=11$. Compute the least positive integer $b$ such that, when expressed in base $b$, the number $T$ ! ends in exactly two zeroes.
Answer: 5
Problem: Let $T=5$. Suppose that $a_{1}=1$, and that for all positive integers $n, a_{n+1}=$ $\left\lceil\sqrt{a_{n}^{2}+34}\right\rceil$. Compute the least value of $n$ such that $a_{n}>100 T$.
Answer: 491
Problem: Compute the smallest $n$ such that in the regular $n$-gon $A_{1} A_{2} A_{3} \cdots A_{n}, \mathrm{~m} \angle A_{1} A_{20} A_{13}<60^{\circ}$.
Answer: 37
Problem: Let $T=37$. A cube has edges of length $T$. Square holes of side length 1 are drilled from the center of each face of the cube through the cube's center and across to the opposite face; the edges of each hole are parallel to the edges of the cube. Compute the surface area of the resulting solid.
Answer: 8640
Problem: Let $T=8640$. Compute $\left\lfloor\log _{4}\left(1+2+4+\cdots+2^{T}\right)\right\rfloor$.
Answer: 4320
Problem: In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0$ is an integer and $n^{\prime}$ is an integer not divisible by 3 , then $d(n)=3^{-k}$. For example, $d(18)=1 / 9$ and $d(17)=1$. Notice that even though no houses have negative numbers, $d(n)$ is well-defined for negative values of $n$. For example, $d(-33)=1 / 3$ because $-33=3^{1} \cdot-11$. By definition, $d(0)=0$. Following the dictum "location, location, location," this Power Question will refer to "houses" and "house numbers" interchangeably.

Curiously, the arrangement of the houses is such that the distance from house $n$ to house $m$, written $d(m, n)$, is simply $d(m-n)$. For example, $d(3,4)=d(-1)=1$ because $-1=3^{0} \cdot-1$. In particular, if $m=n$, then $d(m, n)=0$.
Compute $d(6), d(16)$, and $d(72)$.
Answer: \frac{1}{3},1,\frac{1}{9}
Problem: In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0$ is an integer and $n^{\prime}$ is an integer not divisible by 3 , then $d(n)=3^{-k}$. For example, $d(18)=1 / 9$ and $d(17)=1$. Notice that even though no houses have negative numbers, $d(n)$ is well-defined for negative values of $n$. For example, $d(-33)=1 / 3$ because $-33=3^{1} \cdot-11$. By definition, $d(0)=0$. Following the dictum "location, location, location," this Power Question will refer to "houses" and "house numbers" interchangeably.

Curiously, the arrangement of the houses is such that the distance from house $n$ to house $m$, written $d(m, n)$, is simply $d(m-n)$. For example, $d(3,4)=d(-1)=1$ because $-1=3^{0} \cdot-1$. In particular, if $m=n$, then $d(m, n)=0$.
Of the houses with positive numbers less than 100, find, with proof, the house or houses which is (are) closest to City Hall.
Answer: 81
Problem: In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0$ is an integer and $n^{\prime}$ is an integer not divisible by 3 , then $d(n)=3^{-k}$. For example, $d(18)=1 / 9$ and $d(17)=1$. Notice that even though no houses have negative numbers, $d(n)$ is well-defined for negative values of $n$. For example, $d(-33)=1 / 3$ because $-33=3^{1} \cdot-11$. By definition, $d(0)=0$. Following the dictum "location, location, location," this Power Question will refer to "houses" and "house numbers" interchangeably.

Curiously, the arrangement of the houses is such that the distance from house $n$ to house $m$, written $d(m, n)$, is simply $d(m-n)$. For example, $d(3,4)=d(-1)=1$ because $-1=3^{0} \cdot-1$. In particular, if $m=n$, then $d(m, n)=0$.


The neighborhood of a house $n$, written $\mathcal{N}(n)$, is the set of all houses that are the same distance from City Hall as $n$. In symbols, $\mathcal{N}(n)=\{m \mid d(m)=d(n)\}$. Geometrically, it may be helpful to think of $\mathcal{N}(n)$ as a circle centered at City Hall with radius $d(n)$.
Suppose that $n$ is a house with $d(n)=1 / 27$. Determine the ten smallest positive integers $m$ (in the standard ordering of the integers) such that $m \in \mathcal{N}(n)$.
Answer: 27,54,108,135,189,216,270,297,351,378
Problem: In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0$ is an integer and $n^{\prime}$ is an integer not divisible by 3 , then $d(n)=3^{-k}$. For example, $d(18)=1 / 9$ and $d(17)=1$. Notice that even though no houses have negative numbers, $d(n)$ is well-defined for negative values of $n$. For example, $d(-33)=1 / 3$ because $-33=3^{1} \cdot-11$. By definition, $d(0)=0$. Following the dictum "location, location, location," this Power Question will refer to "houses" and "house numbers" interchangeably.

Curiously, the arrangement of the houses is such that the distance from house $n$ to house $m$, written $d(m, n)$, is simply $d(m-n)$. For example, $d(3,4)=d(-1)=1$ because $-1=3^{0} \cdot-1$. In particular, if $m=n$, then $d(m, n)=0$.


The neighborhood of a house $n$, written $\mathcal{N}(n)$, is the set of all houses that are the same distance from City Hall as $n$. In symbols, $\mathcal{N}(n)=\{m \mid d(m)=d(n)\}$. Geometrically, it may be helpful to think of $\mathcal{N}(n)$ as a circle centered at City Hall with radius $d(n)$.
Suppose that $d(17, m)=1 / 81$. Determine the possible values of $d(16, m)$.
Answer: 1
Problem: In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0$ is an integer and $n^{\prime}$ is an integer not divisible by 3 , then $d(n)=3^{-k}$. For example, $d(18)=1 / 9$ and $d(17)=1$. Notice that even though no houses have negative numbers, $d(n)$ is well-defined for negative values of $n$. For example, $d(-33)=1 / 3$ because $-33=3^{1} \cdot-11$. By definition, $d(0)=0$. Following the dictum "location, location, location," this Power Question will refer to "houses" and "house numbers" interchangeably.

Curiously, the arrangement of the houses is such that the distance from house $n$ to house $m$, written $d(m, n)$, is simply $d(m-n)$. For example, $d(3,4)=d(-1)=1$ because $-1=3^{0} \cdot-1$. In particular, if $m=n$, then $d(m, n)=0$.


The neighborhood of a house $n$, written $\mathcal{N}(n)$, is the set of all houses that are the same distance from City Hall as $n$. In symbols, $\mathcal{N}(n)=\{m \mid d(m)=d(n)\}$. Geometrically, it may be helpful to think of $\mathcal{N}(n)$ as a circle centered at City Hall with radius $d(n)$.


Unfortunately for new development, ARMLopolis is full: every nonnegative integer corresponds to (exactly one) house (or City Hall, in the case of 0). However, eighteen families arrive and are looking to move in. After much debate, the connotations of using negative house numbers are deemed unacceptable, and the city decides on an alternative plan. On July 17, Shewad Movers arrive and relocate every family from house $n$ to house $n+18$, for all positive $n$ (so that City Hall does not move). For example, the family in house number 17 moves to house number 35.
Ross takes a walk starting at his house, which is number 34 . He first visits house $n_{1}$, such that $d\left(n_{1}, 34\right)=1 / 3$. He then goes to another house, $n_{2}$, such that $d\left(n_{1}, n_{2}\right)=1 / 3$. Continuing in that way, he visits houses $n_{3}, n_{4}, \ldots$, and each time, $d\left(n_{i}, n_{i+1}\right)=1 / 3$. At the end of the day, what is his maximum possible distance from his original house? Justify your answer.
Answer: 1/3
Problem: In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$, where $k \geq 0$ is an integer and $n^{\prime}$ is an integer not divisible by 3 , then $d(n)=3^{-k}$. For example, $d(18)=1 / 9$ and $d(17)=1$. Notice that even though no houses have negative numbers, $d(n)$ is well-defined for negative values of $n$. For example, $d(-33)=1 / 3$ because $-33=3^{1} \cdot-11$. By definition, $d(0)=0$. Following the dictum "location, location, location," this Power Question will refer to "houses" and "house numbers" interchangeably.

Curiously, the arrangement of the houses is such that the distance from house $n$ to house $m$, written $d(m, n)$, is simply $d(m-n)$. For example, $d(3,4)=d(-1)=1$ because $-1=3^{0} \cdot-1$. In particular, if $m=n$, then $d(m, n)=0$.


The neighborhood of a house $n$, written $\mathcal{N}(n)$, is the set of all houses that are the same distance from City Hall as $n$. In symbols, $\mathcal{N}(n)=\{m \mid d(m)=d(n)\}$. Geometrically, it may be helpful to think of $\mathcal{N}(n)$ as a circle centered at City Hall with radius $d(n)$.

Later, ARMLopolis finally decides on a drastic expansion plan: now house numbers will be rational numbers. To define $d(p / q)$, with $p$ and $q$ integers such that $p q \neq 0$, write $p / q=3^{k} p^{\prime} / q^{\prime}$, where neither $p^{\prime}$ nor $q^{\prime}$ is divisible by 3 and $k$ is an integer (not necessarily positive); then $d(p / q)=3^{-k}$.
Compute $d(3 / 5), d(5 / 8)$, and $d(7 / 18)$.
Answer: \frac{1}{3}, 1, 9
Problem: Let $A R M L$ be a trapezoid with bases $\overline{A R}$ and $\overline{M L}$, such that $M R=R A=A L$ and $L R=$ $A M=M L$. Point $P$ lies inside the trapezoid such that $\angle R M P=12^{\circ}$ and $\angle R A P=6^{\circ}$. Diagonals $A M$ and $R L$ intersect at $D$. Compute the measure, in degrees, of angle $A P D$.
Answer: 48
Problem: A regular hexagon has side length 1. Compute the average of the areas of the 20 triangles whose vertices are vertices of the hexagon.
Answer: \frac{9 \sqrt{3}}{20}
Problem: Paul was planning to buy 20 items from the ARML shop. He wanted some mugs, which cost $\$ 10$ each, and some shirts, which cost $\$ 6$ each. After checking his wallet he decided to put $40 \%$ of the mugs back. Compute the number of dollars he spent on the remaining items.
Answer: 120
Problem: Let $x$ be the smallest positive integer such that $1584 \cdot x$ is a perfect cube, and let $y$ be the smallest positive integer such that $x y$ is a multiple of 1584 . Compute $y$.
Answer: 12
Problem: Emma goes to the store to buy apples and peaches. She buys five of each, hands the shopkeeper one $\$ 5$ bill, but then has to give the shopkeeper another; she gets back some change. Jonah goes to the same store, buys 2 apples and 12 peaches, and tries to pay with a single $\$ 10$ bill. But that's not enough, so Jonah has to give the shopkeeper another $\$ 10$ bill, and also gets some change. Finally, Helen goes to the same store to buy 25 peaches. Assuming that the price in cents of each fruit is an integer, compute the least amount of money, in cents, that Helen can expect to pay.
Answer: 1525
Problem: Circle $O$ has radius 6. Point $P$ lies outside circle $O$, and the shortest distance from $P$ to circle $O$ is 4. Chord $\overline{A B}$ is parallel to $\overleftrightarrow{O P}$, and the distance between $\overline{A B}$ and $\overleftrightarrow{O P}$ is 2 . Compute $P A^{2}+P B^{2}$.
Answer: 272
Problem: A palindrome is a positive integer, not ending in 0 , that reads the same forwards and backwards. For example, 35253,171,44, and 2 are all palindromes, but 17 and 1210 are not. Compute the least positive integer greater than 2013 that cannot be written as the sum of two palindromes.
Answer: 2019
Problem: Positive integers $x, y, z$ satisfy $x y+z=160$. Compute the smallest possible value of $x+y z$.
Answer: 50
Problem: Compute $\cos ^{3} \frac{2 \pi}{7}+\cos ^{3} \frac{4 \pi}{7}+\cos ^{3} \frac{8 \pi}{7}$.
Answer: -\frac{1}{2}
Problem: In right triangle $A B C$ with right angle $C$, line $\ell$ is drawn through $C$ and is parallel to $\overline{A B}$. Points $P$ and $Q$ lie on $\overline{A B}$ with $P$ between $A$ and $Q$, and points $R$ and $S$ lie on $\ell$ with $C$ between $R$ and $S$ such that $P Q R S$ is a square. Let $\overline{P S}$ intersect $\overline{A C}$ in $X$, and let $\overline{Q R}$ intersect $\overline{B C}$ in $Y$. The inradius of triangle $A B C$ is 10 , and the area of square $P Q R S$ is 576 . Compute the sum of the inradii of triangles $A X P, C X S, C Y R$, and $B Y Q$.
Answer: 14
Problem: Compute the sum of all real numbers $x$ such that

$$
\left\lfloor\frac{x}{2}\right\rfloor-\left\lfloor\frac{x}{3}\right\rfloor=\frac{x}{7}
$$
Answer: -21
Problem: Let $S=\{1,2, \ldots, 20\}$, and let $f$ be a function from $S$ to $S$; that is, for all $s \in S, f(s) \in S$. Define the sequence $s_{1}, s_{2}, s_{3}, \ldots$ by setting $s_{n}=\sum_{k=1}^{20} \underbrace{(f \circ \cdots \circ f)}_{n}(k)$. That is, $s_{1}=f(1)+$ $\cdots+f(20), s_{2}=f(f(1))+\cdots+f(f(20)), s_{3}=f(f(f(1)))+f(f(f(2)))+\cdots+f(f(f(20)))$, etc. Compute the smallest integer $p$ such that the following statement is true: The sequence $s_{1}, s_{2}, s_{3}, \ldots$ must be periodic after a certain point, and its period is at most $p$. (If the sequence is never periodic, then write $\infty$ as your answer.)
Answer: 140
Problem: Compute the smallest positive integer $n$ such that $n^{2}+n^{0}+n^{1}+n^{3}$ is a multiple of 13 .
Answer: 5
Problem: Let $T=T N Y W R$. Compute $2^{\log _{T} 8}-8^{\log _{T} 2}$.
Answer: 0
Problem: Let $T=T N Y W R$. At some point during a given week, a law enforcement officer had issued $T+2$ traffic warnings, 20 tickets, and had made $T+5$ arrests. How many more tickets must the officer issue in order for the combined number of tickets and arrests to be 20 times the number of warnings issued that week?
Answer: 15
Problem: $\quad$ Let $T=T N Y W R$. In parallelogram $A R M L$, points $P$ and $Q$ trisect $\overline{A R}$ and points $W, X, Y, Z$ divide $\overline{M L}$ into fifths (where $W$ is closest to $M$, and points $X$ and $Y$ are both between $W$ and $Z$ ). If $[A R M L]=T$, compute $[P Q W Z]$.
Answer: 7
Problem: Let $T=T N Y W R$. Compute the number of positive perfect cubes that are divisors of $(T+10) !$.
Answer: 36
Problem: Let $T=T N Y W R$. The graph of $y=x^{2}+2 x-T$ intersects the $x$-axis at points $A$ and $M$, which are diagonally opposite vertices of square $A R M L$. Compute $[A R M L]$.
Answer: 74
Problem: Let $S$ be the set of prime factors of the numbers you receive from positions 7 and 9 , and let $p$ and $q$ be the two least distinct elements of $S$, with $p<q$. Hexagon HEXAGO is inscribed in circle $\omega$, and every angle of $H E X A G O$ is $120^{\circ}$. If $H E=X A=G O=p$ and $E X=A G=O H=q$, compute the area of circle $\omega$.
Answer: \frac{67 \pi}{3}
Problem: Let $T=T N Y W R$. A group of $n$ friends goes camping; two of them are selected to set up the campsite when they arrive and two others are selected to take down the campsite the next day. Compute the smallest possible value of $n$ such that there are at least $T$ ways of selecting the four helpers.
Answer: 7
Problem: Let $T=T N Y W R$. The parabola $y=x^{2}+T x$ is tangent to the parabola $y=-(x-2 T)^{2}+b$. Compute $b$.
Answer: 184
Problem: Let $T=T N Y W R$. The first two terms of a sequence are $a_{1}=3 / 5$ and $a_{2}=4 / 5$. For $n>2$, if $n$ is odd, then $a_{n}=a_{n-1}^{2}-a_{n-2}^{2}$, while if $n$ is even, then $a_{n}=2 a_{n-2} a_{n-3}$. Compute the sum of the squares of the first $T-3$ terms of the sequence.
Answer: 8
Problem: Let $T=T N Y W R$. A regular $n$-gon has exactly $T$ more diagonals than a regular $(n-1)$-gon. Compute the value of $n$.
Answer: 19
Problem: Let $T=T N Y W R$. The sequence $a_{1}, a_{2}, a_{3}, \ldots$, is arithmetic with $a_{16}=13$ and $a_{30}=20$. Compute the value of $k$ for which $a_{k}=T$.
Answer: 17
Problem: Let $T=T N Y W R$. A rectangular prism has a length of 1 , a width of 3 , a height of $h$, and has a total surface area of $T$. Compute the value of $h$.
Answer: \frac{27}{2}