ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: For each positive integer $N$, an Eden sequence from $\{1,2,3, \ldots, N\}$ is defined to be a sequence that satisfies the following conditions:

(i) each of its terms is an element of the set of consecutive integers $\{1,2,3, \ldots, N\}$,

(ii) the sequence is increasing, and

(iii) the terms in odd numbered positions are odd and the terms in even numbered positions are even.

For example, the four Eden sequences from $\{1,2,3\}$ are

$$
\begin{array}{llll}
1 & 3 & 1,2 & 1,2,3
\end{array}
$$
For each positive integer $N$, define $e(N)$ to be the number of Eden sequences from $\{1,2,3, \ldots, N\}$. If $e(17)=4180$ and $e(20)=17710$, determine $e(18)$ and $e(19)$.
Answer: 6764$,$10945
Problem: If $a$ is chosen randomly from the set $\{1,2,3,4,5\}$ and $b$ is chosen randomly from the set $\{6,7,8\}$, what is the probability that $a^{b}$ is an even number?
Answer: \frac{2}{5}
Problem: A bag contains some blue and some green hats. On each turn, Julia removes one hat without looking, with each hat in the bag being equally likely to be chosen. If it is green, she adds a blue hat into the bag from her supply of extra hats, and if it is blue, she adds a green hat to the bag. The bag initially contains 4 blue hats and 2 green hats. What is the probability that the bag again contains 4 blue hats and 2 green hats after two turns?
Answer: \frac{11}{18}
Problem: Suppose that, for some angles $x$ and $y$,

$$
\begin{aligned}
& \sin ^{2} x+\cos ^{2} y=\frac{3}{2} a \\
& \cos ^{2} x+\sin ^{2} y=\frac{1}{2} a^{2}
\end{aligned}
$$

Determine the possible value(s) of $a$.
Answer: 1
Problem: The sequence $2,5,10,50,500, \ldots$ is formed so that each term after the second is the product of the two previous terms. The 15 th term ends with exactly $k$ zeroes. What is the value of $k$ ?
Answer: 233
Problem: If $\log _{2} x-2 \log _{2} y=2$, determine $y$, as a function of $x$
Answer: \frac{1}{2},\sqrt{x}
Problem: Define $f(x)=\sin ^{6} x+\cos ^{6} x+k\left(\sin ^{4} x+\cos ^{4} x\right)$ for some real number $k$.
Determine all real numbers $k$ for which $f(x)$ is constant for all values of $x$.
Answer: -\frac{3}{2}
Problem: Define $f(x)=\sin ^{6} x+\cos ^{6} x+k\left(\sin ^{4} x+\cos ^{4} x\right)$ for some real number $k$.
If $k=-0.7$, determine all solutions to the equation $f(x)=0$.
Answer: x=\frac{1}{6} \pi+\pi k, \frac{1}{3} \pi+\pi k, \frac{2}{3} \pi+\pi k, \frac{5}{6} \pi+\pi k
Problem: Define $f(x)=\sin ^{6} x+\cos ^{6} x+k\left(\sin ^{4} x+\cos ^{4} x\right)$ for some real number $k$.
Determine all real numbers $k$ for which there exists a real number $c$ such that $f(c)=0$.
Answer: [-1,-\frac{1}{2}]
Problem: Hexagon $A B C D E F$ has vertices $A(0,0), B(4,0), C(7,2), D(7,5), E(3,5)$, $F(0,3)$. What is the area of hexagon $A B C D E F$ ?
Answer: 29
Problem: A list $a_{1}, a_{2}, a_{3}, a_{4}$ of rational numbers is defined so that if one term is equal to $r$, then the next term is equal to $1+\frac{1}{1+r}$. For example, if $a_{3}=\frac{41}{29}$, then $a_{4}=1+\frac{1}{1+(41 / 29)}=\frac{99}{70}$. If $a_{3}=\frac{41}{29}$, what is the value of $a_{1} ?$
Answer: \frac{7}{5}
Problem: A hollow cylindrical tube has a radius of $10 \mathrm{~mm}$ and a height of $100 \mathrm{~mm}$. The tube sits flat on one of its circular faces on a horizontal table. The tube is filled with water to a depth of $h \mathrm{~mm}$. A solid cylindrical rod has a radius of $2.5 \mathrm{~mm}$ and a height of $150 \mathrm{~mm}$. The rod is inserted into the tube so that one of its circular faces sits flat on the bottom of the tube. The height of the water in the tube is now $64 \mathrm{~mm}$. Determine the value of $h$.
Answer: 60
Problem: A function $f$ has the property that $f\left(\frac{2 x+1}{x}\right)=x+6$ for all real values of $x \neq 0$. What is the value of $f(4) ?$
Answer: \frac{13}{2}
Problem: Determine all real numbers $a, b$ and $c$ for which the graph of the function $y=\log _{a}(x+b)+c$ passes through the points $P(3,5), Q(5,4)$ and $R(11,3)$.
Answer: \frac{1}{3},-2,5
Problem: A computer is programmed to choose an integer between 1 and 99, inclusive, so that the probability that it selects the integer $x$ is equal to $\log _{100}\left(1+\frac{1}{x}\right)$. Suppose that the probability that $81 \leq x \leq 99$ is equal to 2 times the probability that $x=n$ for some integer $n$. What is the value of $n$ ?
Answer: 9
Problem: What is the smallest positive integer $x$ for which $\frac{1}{32}=\frac{x}{10^{y}}$ for some positive integer $y$ ?
Answer: 3125
Problem: Determine all possible values for the area of a right-angled triangle with one side length equal to 60 and with the property that its side lengths form an arithmetic sequence.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, $3,5,7,9$ are the first four terms of an arithmetic sequence.)
Answer: 2400, 1350, 864
Problem: Amrita and Zhang cross a lake in a straight line with the help of a one-seat kayak. Each can paddle the kayak at $7 \mathrm{~km} / \mathrm{h}$ and swim at $2 \mathrm{~km} / \mathrm{h}$. They start from the same point at the same time with Amrita paddling and Zhang swimming. After a while, Amrita stops the kayak and immediately starts swimming. Upon reaching the kayak (which has not moved since Amrita started swimming), Zhang gets in and immediately starts paddling. They arrive on the far side of the lake at the same time, 90 minutes after they began. Determine the amount of time during these 90 minutes that the kayak was not being paddled.
Answer: 50
Problem: Determine all pairs $(x, y)$ of real numbers that satisfy the system of equations

$$
\begin{aligned}
x\left(\frac{1}{2}+y-2 x^{2}\right) & =0 \\
y\left(\frac{5}{2}+x-y\right) & =0
\end{aligned}
$$
Answer: (0,0),(0, \frac{5}{2}),(\frac{1}{2}, 0),(-\frac{1}{2}, 0),(\frac{3}{2}, 4),(-1, \frac{3}{2})
Problem: Determine all real numbers $x>0$ for which

$$
\log _{4} x-\log _{x} 16=\frac{7}{6}-\log _{x} 8
$$
Answer: 2^{-2 / 3}$, $8
Problem: The string $A A A B B B A A B B$ is a string of ten letters, each of which is $A$ or $B$, that does not include the consecutive letters $A B B A$.

The string $A A A B B A A A B B$ is a string of ten letters, each of which is $A$ or $B$, that does include the consecutive letters $A B B A$.

Determine, with justification, the total number of strings of ten letters, each of which is $A$ or $B$, that do not include the consecutive letters $A B B A$.
Answer: 631
Problem: Let $k$ be a positive integer with $k \geq 2$. Two bags each contain $k$ balls, labelled with the positive integers from 1 to $k$. André removes one ball from each bag. (In each bag, each ball is equally likely to be chosen.) Define $P(k)$ to be the probability that the product of the numbers on the two balls that he chooses is divisible by $k$.
Calculate $P(10)$.
Answer: \frac{27}{100}
Problem: In an arithmetic sequence, the first term is 1 and the last term is 19 . The sum of all the terms in the sequence is 70 . How many terms does the sequence have? (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 is an arithmetic sequence with four terms.)
Answer: 7
Problem: Suppose that $a(x+b(x+3))=2(x+6)$ for all values of $x$. Determine $a$ and $b$.
Answer: -2,-2
Problem: An integer $n$, with $100 \leq n \leq 999$, is chosen at random. What is the probability that the sum of the digits of $n$ is 24 ?
Answer: \frac{1}{90}
Problem: The parabola $y=x^{2}-2 x+4$ is translated $p$ units to the right and $q$ units down. The $x$-intercepts of the resulting parabola are 3 and 5 . What are the values of $p$ and $q$ ?
Answer: 3,4
Problem: If $\log _{2} x,\left(1+\log _{4} x\right)$, and $\log _{8} 4 x$ are consecutive terms of a geometric sequence, determine the possible values of $x$.

(A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a constant. For example, $3,6,12$ is a geometric sequence with three terms.)
Answer: 64,\frac{1}{4}
Problem: Determine the two pairs of positive integers $(a, b)$ with $a<b$ that satisfy the equation $\sqrt{a}+\sqrt{b}=\sqrt{50}$.
Answer: (2,32), (8,18)
Problem: Consider the system of equations:

$$
\begin{aligned}
c+d & =2000 \\
\frac{c}{d} & =k
\end{aligned}
$$

Determine the number of integers $k$ with $k \geq 0$ for which there is at least one pair of integers $(c, d)$ that is a solution to the system.
Answer: 20
Problem: Determine all real numbers $x$ for which $2 \log _{2}(x-1)=1-\log _{2}(x+2)$.
Answer: \sqrt{3}
Problem: Consider the function $f(x)=x^{2}-2 x$. Determine all real numbers $x$ that satisfy the equation $f(f(f(x)))=3$.
Answer: 3,1,-1,1+\sqrt{2}, 1-\sqrt{2}
Problem: Suppose that $x$ satisfies $0<x<\frac{\pi}{2}$ and $\cos \left(\frac{3}{2} \cos x\right)=\sin \left(\frac{3}{2} \sin x\right)$.

Determine all possible values of $\sin 2 x$, expressing your answers in the form $\frac{a \pi^{2}+b \pi+c}{d}$ where $a, b, c, d$ are integers.
Answer: \frac{\pi^{2}-9}{9}
Problem: For positive integers $a$ and $b$, define $f(a, b)=\frac{a}{b}+\frac{b}{a}+\frac{1}{a b}$.

For example, the value of $f(1,2)$ is 3 .
Determine the value of $f(2,5)$.
Answer: 3
Problem: For positive integers $a$ and $b$, define $f(a, b)=\frac{a}{b}+\frac{b}{a}+\frac{1}{a b}$.

For example, the value of $f(1,2)$ is 3 .
Determine all positive integers $a$ for which $f(a, a)$ is an integer.
Answer: 1
Problem: Amir and Brigitte play a card game. Amir starts with a hand of 6 cards: 2 red, 2 yellow and 2 green. Brigitte starts with a hand of 4 cards: 2 purple and 2 white. Amir plays first. Amir and Brigitte alternate turns. On each turn, the current player chooses one of their own cards at random and places it on the table. The cards remain on the table for the rest of the game. A player wins and the game ends when they have placed two cards of the same colour on the table. Determine the probability that Amir wins the game.
Answer: \frac{7}{15}
Problem: Consider the sequence $t_{1}=1, t_{2}=-1$ and $t_{n}=\left(\frac{n-3}{n-1}\right) t_{n-2}$ where $n \geq 3$. What is the value of $t_{1998}$ ?
Answer: \frac{-1}{1997}
Problem: The $n$th term of an arithmetic sequence is given by $t_{n}=555-7 n$.

If $S_{n}=t_{1}+t_{2}+\ldots+t_{n}$, determine the smallest value of $n$ for which $S_{n}<0$.
Answer: 158
Problem: If $x$ and $y$ are real numbers, determine all solutions $(x, y)$ of the system of equations

$$
\begin{aligned}
& x^{2}-x y+8=0 \\
& x^{2}-8 x+y=0
\end{aligned}
$$
Answer: (-1,-9),(4+2 \sqrt{2}, 8),(4-2 \sqrt{2}, 8)
Problem: The equations $x^{2}+5 x+6=0$ and $x^{2}+5 x-6=0$ each have integer solutions whereas only one of the equations in the pair $x^{2}+4 x+5=0$ and $x^{2}+4 x-5=0$ has integer solutions.
Determine $q$ in terms of $a$ and $b$.
Answer: \frac{a b}{2}
Problem: Determine all values of $k$, with $k \neq 0$, for which the parabola

$$
y=k x^{2}+(5 k+3) x+(6 k+5)
$$

has its vertex on the $x$-axis.
Answer: -1,-9
Problem: The function $f(x)$ satisfies the equation $f(x)=f(x-1)+f(x+1)$ for all values of $x$. If $f(1)=1$ and $f(2)=3$, what is the value of $f(2008)$ ?
Answer: -1
Problem: The numbers $a, b, c$, in that order, form a three term arithmetic sequence (see below) and $a+b+c=60$.

The numbers $a-2, b, c+3$, in that order, form a three term geometric sequence. Determine all possible values of $a, b$ and $c$.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, $3,5,7$ is an arithmetic sequence with three terms.

A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a constant. For example, $3,6,12$ is a geometric sequence with three terms.)

Present your answer in the form of coordinates (e.g. (1, 2, 3) for a=1, b=2, c=3).
Answer: (27,20,13), (18,20,22)
Problem: The average of three consecutive multiples of 3 is $a$.

The average of four consecutive multiples of 4 is $a+27$.

The average of the smallest and largest of these seven integers is 42 .

Determine the value of $a$.
Answer: 27
Problem: Billy and Crystal each have a bag of 9 balls. The balls in each bag are numbered from 1 to 9. Billy and Crystal each remove one ball from their own bag. Let $b$ be the sum of the numbers on the balls remaining in Billy's bag. Let $c$ be the sum of the numbers on the balls remaining in Crystal's bag. Determine the probability that $b$ and $c$ differ by a multiple of 4 .
Answer: \frac{7}{27}
Problem: The equation $2^{x+2} 5^{6-x}=10^{x^{2}}$ has two real solutions. Determine these two solutions.
Answer: 2,-\log _{10} 250
Problem: Determine all real solutions to the system of equations

$$
\begin{aligned}
& x+\log _{10} x=y-1 \\
& y+\log _{10}(y-1)=z-1 \\
& z+\log _{10}(z-2)=x+2
\end{aligned}
$$

and prove that there are no more solutions.
Answer: 1,2,3
Problem: The positive integers 34 and 80 have exactly two positive common divisors, namely 1 and 2. How many positive integers $n$ with $1 \leq n \leq 30$ have the property that $n$ and 80 have exactly two positive common divisors?
Answer: 9
Problem: A function $f$ is defined so that

- $f(1)=1$,
- if $n$ is an even positive integer, then $f(n)=f\left(\frac{1}{2} n\right)$, and
- if $n$ is an odd positive integer with $n>1$, then $f(n)=f(n-1)+1$.

For example, $f(34)=f(17)$ and $f(17)=f(16)+1$.

Determine the value of $f(50)$.
Answer: 3
Problem: The perimeter of equilateral $\triangle P Q R$ is 12. The perimeter of regular hexagon $S T U V W X$ is also 12. What is the ratio of the area of $\triangle P Q R$ to the area of $S T U V W X$ ?
Answer: \frac{2}{3}
Problem: For how many integers $k$ with $0<k<18$ is $\frac{5 \sin \left(10 k^{\circ}\right)-2}{\sin ^{2}\left(10 k^{\circ}\right)} \geq 2$ ?
Answer: 13
Problem: Eight people, including triplets Barry, Carrie and Mary, are going for a trip in four canoes. Each canoe seats two people. The eight people are to be randomly assigned to the four canoes in pairs. What is the probability that no two of Barry, Carrie and Mary will be in the same canoe?
Answer: \frac{4}{7}
Problem: Diagonal $W Y$ of square $W X Y Z$ has slope 2. Determine the sum of the slopes of $W X$ and $X Y$.
Answer: -\frac{8}{3}
Problem: Determine all values of $x$ such that $\log _{2 x}(48 \sqrt[3]{3})=\log _{3 x}(162 \sqrt[3]{2})$.
Answer: \sqrt{6}
Problem: In an infinite array with two rows, the numbers in the top row are denoted $\ldots, A_{-2}, A_{-1}, A_{0}, A_{1}, A_{2}, \ldots$ and the numbers in the bottom row are denoted $\ldots, B_{-2}, B_{-1}, B_{0}, B_{1}, B_{2}, \ldots$ For each integer $k$, the entry $A_{k}$ is directly above the entry $B_{k}$ in the array, as shown:

| $\ldots$ | $A_{-2}$ | $A_{-1}$ | $A_{0}$ | $A_{1}$ | $A_{2}$ | $\ldots$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $\ldots$ | $B_{-2}$ | $B_{-1}$ | $B_{0}$ | $B_{1}$ | $B_{2}$ | $\ldots$ |

For each integer $k, A_{k}$ is the average of the entry to its left, the entry to its right, and the entry below it; similarly, each entry $B_{k}$ is the average of the entry to its left, the entry to its right, and the entry above it.
In one such array, $A_{0}=A_{1}=A_{2}=0$ and $A_{3}=1$.

Determine the value of $A_{4}$.
Answer: 6
Problem: The populations of Alphaville and Betaville were equal at the end of 1995. The population of Alphaville decreased by $2.9 \%$ during 1996, then increased by $8.9 \%$ during 1997 , and then increased by $6.9 \%$ during 1998 . The population of Betaville increased by $r \%$ in each of the three years. If the populations of the towns are equal at the end of 1998, determine the value of $r$ correct to one decimal place.
Answer: 4.2
Problem: Determine the coordinates of the points of intersection of the graphs of $y=\log _{10}(x-2)$ and $y=1-\log _{10}(x+1)$.
Answer: (4, \log _{10} 2)
Problem: Charlie was born in the twentieth century. On his birthday in the present year (2014), he notices that his current age is twice the number formed by the rightmost two digits of the year in which he was born. Compute the four-digit year in which Charlie was born.
Answer: 1938
Problem: Let $A, B$, and $C$ be randomly chosen (not necessarily distinct) integers between 0 and 4 inclusive. Pat and Chris compute the value of $A+B \cdot C$ by two different methods. Pat follows the proper order of operations, computing $A+(B \cdot C)$. Chris ignores order of operations, choosing instead to compute $(A+B) \cdot C$. Compute the probability that Pat and Chris get the same answer.
Answer: \frac{9}{25}
Problem: Bobby, Peter, Greg, Cindy, Jan, and Marcia line up for ice cream. In an acceptable lineup, Greg is ahead of Peter, Peter is ahead of Bobby, Marcia is ahead of Jan, and Jan is ahead of Cindy. For example, the lineup with Greg in front, followed by Peter, Marcia, Jan, Cindy, and Bobby, in that order, is an acceptable lineup. Compute the number of acceptable lineups.
Answer: 20
Problem: In triangle $A B C, a=12, b=17$, and $c=13$. Compute $b \cos C-c \cos B$.
Answer: 10
Problem: The sequence of words $\left\{a_{n}\right\}$ is defined as follows: $a_{1}=X, a_{2}=O$, and for $n \geq 3, a_{n}$ is $a_{n-1}$ followed by the reverse of $a_{n-2}$. For example, $a_{3}=O X, a_{4}=O X O, a_{5}=O X O X O$, and $a_{6}=O X O X O O X O$. Compute the number of palindromes in the first 1000 terms of this sequence.
Answer: 667
Problem: Compute the smallest positive integer $n$ such that $214 \cdot n$ and $2014 \cdot n$ have the same number of divisors.
Answer: 19133
Problem: Let $N$ be the least integer greater than 20 that is a palindrome in both base 20 and base 14 . For example, the three-digit base-14 numeral (13)5(13) ${ }_{14}$ (representing $13 \cdot 14^{2}+5 \cdot 14^{1}+13 \cdot 14^{0}$ ) is a palindrome in base 14 , but not in base 20 , and the three-digit base-14 numeral (13)31 14 is not a palindrome in base 14 . Compute the base-10 representation of $N$.
Answer: 105
Problem: $\quad$ Compute the greatest integer $k \leq 1000$ such that $\left(\begin{array}{c}1000 \\ k\end{array}\right)$ is a multiple of 7 .
Answer: 979
Problem: An integer-valued function $f$ is called tenuous if $f(x)+f(y)>x^{2}$ for all positive integers $x$ and $y$. Let $g$ be a tenuous function such that $g(1)+g(2)+\cdots+g(20)$ is as small as possible. Compute the minimum possible value for $g(14)$.
Answer: 136
Problem: Let $T=(0,0), N=(2,0), Y=(6,6), W=(2,6)$, and $R=(0,2)$. Compute the area of pentagon $T N Y W R$.
Answer: 20
Problem: Let $T=20$. The lengths of the sides of a rectangle are the zeroes of the polynomial $x^{2}-3 T x+T^{2}$. Compute the length of the rectangle's diagonal.
Answer: 20 \sqrt{7}
Problem: Let $T=20 \sqrt{7}$. Let $w>0$ be a real number such that $T$ is the area of the region above the $x$-axis, below the graph of $y=\lceil x\rceil^{2}$, and between the lines $x=0$ and $x=w$. Compute $\lceil 2 w\rceil$.
Answer: 10
Problem: Compute the least positive integer $n$ such that $\operatorname{gcd}\left(n^{3}, n !\right) \geq 100$.
Answer: 8
Problem: Let $T=8$. At a party, everyone shakes hands with everyone else exactly once, except Ed, who leaves early. A grand total of $20 T$ handshakes take place. Compute the number of people at the party who shook hands with Ed.
Answer: 7
Problem: Let $T=7$. Given the sequence $u_{n}$ such that $u_{3}=5, u_{6}=89$, and $u_{n+2}=3 u_{n+1}-u_{n}$ for integers $n \geq 1$, compute $u_{T}$.
Answer: 233
Problem: In each town in ARMLandia, the residents have formed groups, which meet each week to share math problems and enjoy each others' company over a potluck-style dinner. Each town resident belongs to exactly one group. Every week, each resident is required to make one dish and to bring it to his/her group.

It so happens that each resident knows how to make precisely two dishes. Moreover, no two residents of a town know how to make the same pair of dishes. Shown below are two example towns. In the left column are the names of the town's residents. Adjacent to each name is the list of dishes that the corresponding resident knows how to make.

| ARMLton |  |
| :--- | :--- |
| Resident | Dishes |
| Paul | pie, turkey |
| Arnold | pie, salad |
| Kelly | salad, broth |


| ARMLville |  |
| :--- | :--- |
| Resident | Dishes |
| Sally | steak, calzones |
| Ross | calzones, pancakes |
| David | steak, pancakes |

The population of a town $T$, denoted $\operatorname{pop}(T)$, is the number of residents of $T$. Formally, the town itself is simply the set of its residents, denoted by $\left\{r_{1}, \ldots, r_{\mathrm{pop}(T)}\right\}$ unless otherwise specified. The set of dishes that the residents of $T$ collectively know how to make is denoted $\operatorname{dish}(T)$. For example, in the town of ARMLton described above, pop(ARMLton) $=3$, and dish(ARMLton) $=$ \{pie, turkey, salad, broth\}.

A town $T$ is called full if for every pair of dishes in $\operatorname{dish}(T)$, there is exactly one resident in $T$ who knows how to make those two dishes. In the examples above, ARMLville is a full town, but ARMLton is not, because (for example) nobody in ARMLton knows how to make both turkey and salad.

Denote by $\mathcal{F}_{d}$ a full town in which collectively the residents know how to make $d$ dishes. That is, $\left|\operatorname{dish}\left(\mathcal{F}_{d}\right)\right|=d$.
Compute $\operatorname{pop}\left(\mathcal{F}_{17}\right)$.
Answer: 136
Problem: In each town in ARMLandia, the residents have formed groups, which meet each week to share math problems and enjoy each others' company over a potluck-style dinner. Each town resident belongs to exactly one group. Every week, each resident is required to make one dish and to bring it to his/her group.

It so happens that each resident knows how to make precisely two dishes. Moreover, no two residents of a town know how to make the same pair of dishes. Shown below are two example towns. In the left column are the names of the town's residents. Adjacent to each name is the list of dishes that the corresponding resident knows how to make.

| ARMLton |  |
| :--- | :--- |
| Resident | Dishes |
| Paul | pie, turkey |
| Arnold | pie, salad |
| Kelly | salad, broth |


| ARMLville |  |
| :--- | :--- |
| Resident | Dishes |
| Sally | steak, calzones |
| Ross | calzones, pancakes |
| David | steak, pancakes |

The population of a town $T$, denoted $\operatorname{pop}(T)$, is the number of residents of $T$. Formally, the town itself is simply the set of its residents, denoted by $\left\{r_{1}, \ldots, r_{\mathrm{pop}(T)}\right\}$ unless otherwise specified. The set of dishes that the residents of $T$ collectively know how to make is denoted $\operatorname{dish}(T)$. For example, in the town of ARMLton described above, pop(ARMLton) $=3$, and dish(ARMLton) $=$ \{pie, turkey, salad, broth\}.

A town $T$ is called full if for every pair of dishes in $\operatorname{dish}(T)$, there is exactly one resident in $T$ who knows how to make those two dishes. In the examples above, ARMLville is a full town, but ARMLton is not, because (for example) nobody in ARMLton knows how to make both turkey and salad.

Denote by $\mathcal{F}_{d}$ a full town in which collectively the residents know how to make $d$ dishes. That is, $\left|\operatorname{dish}\left(\mathcal{F}_{d}\right)\right|=d$.
Let $n=\operatorname{pop}\left(\mathcal{F}_{d}\right)$. In terms of $n$, compute $d$.
Answer: d=\frac{1+\sqrt{1+8 n}}{2}
Problem: A student computed the repeating decimal expansion of $\frac{1}{N}$ for some integer $N$, but inserted six extra digits into the repetend to get $.0 \overline{0231846597}$. Compute the value of $N$.
Answer: 606
Problem: Let $n$ be a four-digit number whose square root is three times the sum of the digits of $n$. Compute $n$.
Answer: 2916
Problem: Compute the sum of the reciprocals of the positive integer divisors of 24.
Answer: \frac{5}{2}
Problem: There exists a digit $Y$ such that, for any digit $X$, the seven-digit number $\underline{1} \underline{2} \underline{3} \underline{X} \underline{5} \underline{Y} \underline{7}$ is not a multiple of 11. Compute $Y$.
Answer: 4