ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: How many positive integer solutions are there to \( w + x + y + z = 20 \) where \( w + x \ge 5 \) and \( y + z \ge 5 \)?
Answer: 781
Problem: Determine all positive integers $n$ with at least 4 factors such that $n$ is the sum of the squares of its 4 smallest factors.
Answer: 130
Problem: Daniel chooses some distinct subsets of $\{1, \dots, 2019\}$ such that any two distinct subsets chosen are disjoint. Compute the maximum possible number of subsets he can choose.
Answer: 2020
Problem: The sequence \( \{c_{n}\} \) is determined by the following equation:  
\[ c_{n} = (n+1) \int_{0}^{1} x^{n} \cos \pi x\, dx \quad (n = 1, 2, \ldots). \]  
Let \( \lambda \) be the limit value \( \lim_{n \to \infty} c_{n}. \)  
Find \( \lim_{n \to \infty} \frac{c_{n+1} - \lambda}{c_{n} - \lambda}. \)
Answer: 1
Problem: Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes.
Answer: 43
Problem: Triangle $\triangle ABC$ has circumcenter $O$ and orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to $BC$, and suppose $AD = 12$. If $BD = \frac{1}{4} BC$ and $OH \parallel BC$, compute $AB^2$.
Answer: 160
Problem: Tanya wrote numbers in the form $n^7 - 1$ for $n = 2, 3, \ldots$ and noticed that for $n = 8$, she obtained a number divisible by $337$. For what minimal $n$ did she get a number divisible by $2022$?
Answer: 79
Problem: Let $V$ be the set of vertices of a regular 25-sided polygon with center at point $C$. How many triangles have vertices in $V$ and contain the point $C$ in the interior of the triangle?
Answer: 650
Problem: Meredith drives in a series of directions:

1. 5 miles to the northeast.
2. 15 miles to the southeast.
3. 25 miles to the southwest.
4. 35 miles to the northwest.
5. 20 miles to the northeast.

Calculate how many miles Meredith is from her starting point after completing her journey.
Answer: 20
Problem: The Flyfishing Club is choosing officers. There are $23$ members in the club: $14$ boys and $9$ girls. In how many ways can they choose a President and a Vice President if one of them must be a boy and the other must be a girl (either office can be held by the boy or the girl)?
Answer: 252
Problem: In duck language, only the letters $q$, $a$, and $k$ are used. There is no word with two consonants in succession, as ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there? 

In duck language, the letter $a$ is a vowel, while $q$ and $k$ are consonants.
Answer: 21
Problem: You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out, you must open all the doors and disarm all the mines. In the room, there is a panel with $3$ buttons, which conveniently contains an instruction manual:

- The red button arms a mine.
- The yellow button disarms two mines and closes a door.
- The green button opens two doors.

Initially, $3$ doors are closed, and $3$ mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
Answer: 9
Problem: Let $\triangle ABC$ be a triangle where $\angle ACB = 40^\circ$ and $\angle BAC = 60^\circ$. Let $D$ be a point inside the segment $BC$ such that $CD = \frac{AB}{2}$, and let $M$ be the midpoint of the segment $AC$. Find the measure of the angle $\angle CMD$ in degrees.
Answer: 40
Problem: A frog is standing in the center of a $3 \times 3$ grid of lilypads. Each minute, the frog chooses a square that shares exactly one side with its current square uniformly at random and jumps onto the lilypad on the chosen square. The frog stops jumping once it reaches a lilypad on a corner of the grid. What is the expected number of times the frog jumps?
Answer: 3
Problem: Let $a, b, c, d$ be real numbers satisfying the system of equations:
\[
\begin{align*}
(a+b)(c+d) &= 2, \\
(a+c)(b+d) &= 3, \\
(a+d)(b+c) &= 4.
\end{align*}
\]
Find the minimum value of $a^2+b^2+c^2+d^2$.
Answer: 7
Problem: Find the least six-digit palindrome that is a multiple of $45$. Note that a palindrome is a number that reads the same forward and backwards, such as $1441$ or $35253$.
Answer: 504405
Problem: Find the smallest positive integer such that when the last digit is moved to the front, the new number is $\frac{3}{2}$ times the original number.
Answer: 285714
Problem: For how many positive integers $x$ less than $4032$ is $x^2 - 20$ divisible by $16$ and $x^2 - 16$ divisible by $20$?
Answer: 403
Problem: One day, I went for a walk in the morning at $x$ minutes past 5 o'clock, where $x$ is a two-digit number. When I returned, it was $y$ minutes past 6 o'clock. I noticed the following:

1. I walked for exactly $x$ minutes.
2. $y$ was a two-digit number obtained by reversing the digits of $x$.

How many minutes did I walk?
Answer: 42
Problem: Let \( a_k = 0.\overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \overbrace{0 \ldots 0}^{k - 1 \: 0's} 1 \).

The value of \( \sum_{k = 1}^\infty a_k \) can be expressed as a rational number \( \frac{p}{q} \) in simplest form. Find \( p + q \).
Answer: 37
Problem: How many integers can be expressed in the form: $\pm 1 \pm 2 \pm 3 \pm 4 \pm \cdots \pm 2018$?
Answer: 2037172
Problem: In a $2 \times 8$ squared board, you want to color each square red or blue such that on each $2 \times 2$ sub-board there are at least $3$ squares painted blue. In how many ways can this coloring be done?

**Note:** A $2 \times 2$ board is a square made up of $4$ squares that have a common vertex.
Answer: 341
Problem: Define the sequence of positive integers $\{a_n\}$ as follows:
- Let $a_1=1$, $a_2=3$.
- For each $n>2$, let $a_n$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding $2$ (in base $n$).

For example:
- $a_2=3_{10}=11_2$, so $a_3=11_3+2_3=6_{10}$.

Express $a_{2013}$ in base $10$.
Answer: 23097
Problem: A rectangle $ABCD$ has side lengths $AB=6 \, \text{miles}$ and $BC=9\, \text{miles}$. A pigeon hovers at point $P$, which is 5 miles above some randomly chosen point inside $ABCD$. Given that the expected value of $AP^2 + CP^2 - BP^2 - DP^2$ can be expressed as $\frac{a}{b}$, what is $ab$?
Answer: 0
Problem: If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$,  $$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$  find the smallest $n$ such that $a_n < \frac{1}{2018}$.
Answer: 4036
Problem: Two concentric circles $\omega$ and $\Omega$ have radii 8 and 13, respectively. $AB$ is a diameter of $\Omega$, and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$?
Answer: 19
Problem: The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights.
Answer: 57
Problem: Nine people sit in three rows of three chairs each. The probability that two of them, Celery and Drum, sit next to each other in the same row is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Answer: 106
Problem: An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
Answer: 114
Problem: Given a $2018 \times 4$ grid, tint the cells with red and blue such that each row and each column has an equal number of red and blue cells. Determine the number of ways, $M$, to tint the grid with the mentioned requirement and find $M \pmod{2018}$.
Answer: 6
Problem: How many different (non-similar) triangles are there whose angles have an integer number of degrees?
Answer: 2700
Problem: How many ways are there to remove an $11 \times 11$ square from a $2011 \times 2011$ square so that the remaining part can be tiled with dominoes ($1 \times 2$ rectangles)?
Answer: 2002001
Problem: The diagram below shows twelve $30-60-90$ triangles placed in a circle such that the hypotenuse of each triangle coincides with the longer leg of the next triangle. The fourth and last triangle in this diagram are shaded. The ratio of the perimeters of these two triangles can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 337
Problem: Find the smallest two-digit positive integer that divides $201020112012$.
Answer: 12
Problem: If $x, y, z, p, q, r$ are real numbers such that:
\[ \frac{1}{x+p} + \frac{1}{y+p} + \frac{1}{z+p} = \frac{1}{p} \]
\[ \frac{1}{x+q} + \frac{1}{y+q} + \frac{1}{z+q} = \frac{1}{q} \]
\[ \frac{1}{x+r} + \frac{1}{y+r} + \frac{1}{z+r} = \frac{1}{r} \]
Find the numerical value of $\frac{1}{p} + \frac{1}{q} + \frac{1}{r}$.
Answer: 0
Problem: In one of the hotels on the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. The following information is known about the guests:

1. If two women know each other, then their husbands also know each other, and vice versa.
2. Each sauna can be used by either men only or women only (there is no limit on the number of people using a sauna at once, as long as they are of a single gender).
3. Each woman is only willing to share a sauna with women whom she knows.
4. Each man is only willing to share a sauna with men whom he does not know.

What is the greatest possible $k$ for which we can guarantee, without knowing the exact relationships between the couples, that all the guests can use the saunas simultaneously while respecting the restrictions above?
Answer: 2018
Problem: For a natural number $n$, define the function $f(n)$ as the sum of the digits of the number $n$. For example, $f(16) = 7$, $f(f(78)) = 6$, and $f(f(f(5978))) = 2$.

Find the smallest natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.
Answer: 20000000000000000000000
Problem: Let $x_1$ and $x_2$ be the roots of the equation $x^2 + 3x + 1 = 0$. Compute 
\[
\left(\frac{x_1}{x_2 + 1}\right)^2 + \left(\frac{x_2}{x_1 + 1}\right)^2
\]
Answer: 18
Problem: Find all integers $n$ such that the following number is a perfect square:
\[ N = n^4 + 6n^3 + 11n^2 + 3n + 31. \]
Answer: 10
Problem: A trifecta is an ordered triple of positive integers \((a, b, c)\) with \(a < b < c\) such that \(a\) divides \(b\), \(b\) divides \(c\), and \(c\) divides \(ab\). What is the largest possible sum \(a + b + c\) over all trifectas of three-digit integers?
Answer: 1736
Problem: Let $S_0 = 0$ and let $S_k$ equal $a_1 + 2a_2 + \ldots + ka_k$ for $k \geq 1$. Define $a_i$ to be $1$ if $S_{i-1} < i$ and $-1$ if $S_{i-1} \geq i$. What is the largest $k \leq 2010$ such that $S_k = 0$?
Answer: 1092
Problem: Compute the number of ordered pairs \((a,b)\) of positive integers satisfying \(a^b = 2^{100}\).
Answer: 9
Problem: What is the maximum possible value of $k$ for which $2013$ can be expressed as a sum of $k$ consecutive positive integers?
Answer: 61
Problem: For $f(x) = x^4 + |x|$, let $I_1 = \int_0^\pi f(\cos x)\, dx$ and $I_2 = \int_0^{\frac{\pi}{2}} f(\sin x)\, dx$. Find the value of $\frac{I_1}{I_2}$.
Answer: 2
Problem: Given an $8 \times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column?
Answer: 20736
Problem: Let $P(x)$ be a polynomial of degree 10 with non-negative integer coefficients. The remainder when $P(x)$ is divided by $(x - 1)$ is 3. How many such polynomials are there?
Answer: 66
Problem: How many positive integers $n$ are there such that $3 \leq n \leq 100$ and $x^{2^n} + x + 1$ is divisible by $x^2 + x + 1$?
Answer: 49
Problem: Let $ABC$ be an acute triangle with side lengths $AB = 7$, $BC = 12$, $AC = 10$, and let $\omega$ be its incircle. If $\omega$ is touching $AB$, $AC$ at $F, E$, respectively, and if $EF$ intersects $BC$ at $X$, suppose that the ratio in which the angle bisector of $\angle BAC$ divides the segment connecting the midpoint of $EX$ and $C$ is $\frac{a}{b}$, where $a, b$ are relatively prime integers. Find $a + b$.
Answer: 5
Problem: Compute the sum of the positive integers $n \le 100$ for which the polynomial $x^n + x + 1$ can be written as the product of at least 2 polynomials of positive degree with integer coefficients.
Answer: 1648
Problem: Find the number of squares in the sequence given by $a_0 = 91$ and $a_{n+1} = 10a_n + (-1)^n$ for $n \ge 0.$
Answer: 0
Problem: A company has $n$ employees. It is known that each of the employees works at least one of the $7$ days of the week, with the exception of an employee who does not work any of the $7$ days. Furthermore, for any two of these $n$ employees, there are at least $3$ days of the week in which one of the two works that day and the other does not (it is not necessarily the same employee who works those days). Determine the highest possible value of $n$.
Answer: 16
Problem: Let $ABCDEFGHIJKL$ be a regular dodecagon. Find $\frac{AB}{AF} + \frac{AF}{AB}$.
Answer: 4
Problem: Let $\omega_1$ and $\omega_2$ be two circles intersecting at points $P$ and $Q$. The tangent line closer to $Q$ touches $\omega_1$ and $\omega_2$ at $M$ and $N$ respectively. If $PQ = 3$, $QN = 2$, and $MN = PN$, what is $QM^2$?
Answer: 5
Problem: Let $f: \mathbb{N} \to \mathbb{N}$ satisfy $n = \sum_{d|n} f(d)$ for all $n \in \mathbb{N}$. What is the sum of all possible values of $f(100)$?
Answer: 40
Problem: Let $A = \{1, 2, \ldots, 100\}$ and $f(k)$, where $k \in \mathbb{N}$, be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k) = 50$?
Answer: 6
Problem: Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number of $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$.
Answer: 369
Problem: Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of \( \frac{\text{gcd}(m,n)}{\text{lcm}(m,n)} \) over all ordered pairs of positive integers \((m, n)\) where \(mn = k\). If $S$ can be written in simplest form as \( \frac{r}{s} \), compute \( r + s \).
Answer: 895
Problem: The sequence $\{a_n\}_{n\geq 1}$ is defined by the recurrence relation $a_{n+2} = 7a_{n+1} - a_n$ for positive integers $n$, with initial values $a_1 = 1$ and $a_2 = 8$. Another sequence, $\{b_n\}$, is defined by the recurrence relation $b_{n+2} = 3b_{n+1} - b_n$ for positive integers $n$, with initial values $b_1 = 1$ and $b_2 = 2$. Find $\gcd(a_{5000}, b_{501})$.
Answer: 89
Problem: Find the number of positive integers less than or equal to $2019$ that are no more than $10$ away from a perfect square.
Answer: 829
Problem: Let $a$ and $b$ be positive real numbers with $a > b$. Find the smallest possible value of 
$$S = 2a + 3 + \frac{32}{(a - b)(2b + 3)^2}.$$
Answer: 8
Problem: A paper cup has a base that is a circle with radius $r$, a top that is a circle with radius $2r$, and sides that connect the two circles with straight line segments. This cup has height $h$ and volume $V$. A second cup, exactly the same shape as the first, is held upright inside the first cup so that its base is a distance of $\frac{h}{2}$ from the base of the first cup. The volume of liquid that will fit inside the first cup and outside the second cup can be written as $\frac{m}{n} \cdot V$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 93
Problem: Determine the last two digits of the number $2^5 + 2^{5^2} + 2^{5^3} + \ldots + 2^{5^{1991}}$, written in decimal notation.
Answer: 12
Problem: Let $f(x) = 15x - 2016$. If $f(f(f(f(f(x))))) = f(x)$, find the sum of all possible values of $x$.
Answer: 144
Problem: Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Candles may be arbitrarily and instantly put out and relit. Compute the cost in cents of the cheapest set of big and small candles you need to measure exactly 1 minute.
Answer: 97
Problem: On $2020 \times 2021$ squares, cover the whole area with several L-Trominos and S-Tetrominos (Z-Tetrominos) along the square so that they do not overlap. The tiles can be rotated or flipped over. Find the minimum possible number of L-Trominos to be used.
Answer: 2020
Problem: A number is called *Norwegian* if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. 

(Note: The total number of positive divisors of a Norwegian number is allowed to be larger than 3.)
Answer: 1344
Problem: What is the smallest possible natural number $n$ for which the equation $x^2 - nx + 2014 = 0$ has integer roots?
Answer: 91
Problem: How many positive integers $n \leq 100$ satisfy $\lfloor n\pi \rfloor = \lfloor (n-1)\pi \rfloor + 3$? Here $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$; for example, $\lfloor \pi \rfloor = 3$.
Answer: 86
Problem: Suppose that $a, b, c$ are complex numbers satisfying the system of equations:  
\[a^2 = b - c,\]  
\[b^2 = c - a,\]  
\[c^2 = a - b.\]  
Compute all possible values of $a + b + c$.
Answer: 0
Problem: Compute the number of positive integers that divide at least two of the integers in the set $\{1^1, 2^2, 3^3, 4^4, 5^5, 6^6, 7^7, 8^8, 9^9, 10^{10}\}$.
Answer: 22
Problem: Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1 \cdot 2 \cdot 3 \cdot \ldots \cdot 2015 \cdot 2016$.
Answer: 47
Problem: Let $\triangle XOY$ be a right-angled triangle with $\angle XOY=90^\circ$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Find the length $XY$ given that $XN=22$ and $YM=31$.
Answer: 34
Problem: A rectangular tea bag $PART$ has a logo in its interior at the point $Y$. The distances from $Y$ to $PT$ and $PA$ are $12$ and $9$ respectively, and triangles $\triangle PYT$ and $\triangle AYR$ have areas $84$ and $42$ respectively. Find the perimeter of pentagon $PARTY$.
Answer: 78
Problem: The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion:
\[1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, \dots\]
Find the $2014$th number in this sequence.
Answer: 13
Problem: Let's call a natural number \textit{interesting} if any of its two consecutive digits form a number that is a multiple of $19$ or $21$. For example, the number $7638$ is interesting because $76$ is a multiple of $19$, $63$ is a multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers with $2022$ digits exist?
Answer: 9
Problem: Let $\triangle ABC$ be acute with $\angle BAC = 45^{\circ}$. Let $\overline{AD}$ be an altitude of $\triangle ABC$, let $E$ be the midpoint of $\overline{BC}$, and let $F$ be the midpoint of $\overline{AD}$. Let $O$ be the center of the circumcircle of $\triangle ABC$, let $K$ be the intersection of lines $DO$ and $EF$, and let $L$ be the foot of the perpendicular from $O$ to line $AK$. If $BL = 6$ and $CL = 8$, find $AL^2$.
Answer: 48
Problem: Find the minimum positive integer $n$ such that for any set $A$ with $n$ positive integers, there exists a subset of 15 elements whose sum is divisible by 15.
Answer: 29
Problem: $A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$?
Answer: 20
Problem: Determine the number of positive integers $n \leq 1000$ such that the sum of the digits of $5n$ and the sum of the digits of $n$ are the same.
Answer: 91
Problem: Find the product of all real $x$ for which 
\[ 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0. \]
Answer: -3
Problem: How many different solutions are there on the interval $[0, \pi]$ for the equation: $$6\sqrt{2} \sin x \cdot \tan x - 2\sqrt{2} \tan x + 3\sin x - 1 = 0$$?
Answer: 2
Problem: In the diagram, $ABCDEFG$ is a regular heptagon (a $7$-sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 727
Problem: Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.
Answer: 28
Problem: Evan, Larry, and Alex are drawing whales on the whiteboard. Evan draws 10 whales, Larry draws 15 whales, and Alex draws 20 whales. Michelle then starts randomly erasing whales one by one. The probability that she finishes erasing Larry's whales first can be expressed as $\frac{p}{q}$. Compute $p+q$.
Answer: 137
Problem: Five identical circles are placed in a line inside a larger circle as shown. If the shown chord has a length of $16$, find the radius of the large circle.
Answer: 10
Problem: A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle, and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as \[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \] for positive integers $m$, $n$, $p$, and $q$ such that $m$ and $n$ are relatively prime and $p$ is squarefree. Find $m+n+p+q$.
Answer: 11
Problem: Let $a_1 = 20$, $a_2 = 16$, and for $k \ge 3$, let $a_k = \sqrt[3]{k-a_{k-1}^3-a_{k-2}^3}$. Compute $a_1^3 + a_2^3 + \cdots + a_{10}^3$.
Answer: 8021
Problem: For a positive integer $n$, consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. 

For example, if $n = 117$, it has six divisors: $1, 3, 9, 13, 39, 117$. Two of these divisors, namely $3$ and $13$, have unit digits equal to $3$. Hence, for $n = 117$, $p = 33.33\ldots$.

Find the maximum possible value of $p$ for any positive integer $n$.
Answer: 50
Problem: Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\omega$. $X$, $Y$, and $Z$ are points on $\omega$ such that $X$ is on minor arc $AB$, $Y$ is on minor arc $CD$, and $Z$ is on minor arc $EF$, where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$). Compute the square of the smallest possible area of $\triangle XYZ$.
Answer: 7500
Problem: Find the greatest seven-digit integer divisible by $132$ whose digits, in order, are $2, 0, x, y, 1, 2, z$ where $x$, $y$, and $z$ are single digits.
Answer: 2093124
Problem: Tim has a working analog 12-hour clock with two hands that run continuously. He also has a clock that runs at half the correct rate. At noon one day, both clocks show the exact time. At any given instant, the hands on each clock form an angle between $0^\circ$ and $180^\circ$ inclusive. At how many times during that day are the angles on the two clocks equal?
Answer: 33
Problem: Find the minimum of $x + y + z$ where $x$, $y$, and $z$ are real numbers satisfying the following conditions:
- $x \geq 4$
- $y \geq 5$
- $z \geq 6$
- $x^2 + y^2 + z^2 \geq 90$
Answer: 16