meetforever/GRPO_CL huggingface dataset:
user: You have a $10 \times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$, $\ldots$, $10$ from left to right across the top row, then $11$, $12$, $\ldots$, $20$ across the second row, and so on, ending with a $100$ in the bottom right square. You then write a second number in each square, writing $1$, $2$, $3$, $\ldots$, $10$ in the first column (from top to bottom), then $11$, $12$, $\ldots$, $20$ in the second column, and so forth. 

When this process is finished, how many squares will have the property that their two numbers sum to $101$?
Answer: 10
user: Compute the number of nonempty subsets $S \subseteq \{-10, -9, -8, \ldots, 8, 9, 10\}$ that satisfy $|S| + \min(S) \cdot \max(S) = 0$.
Answer: 335
user: There are $1988$ birds in $994$ cages, two in each cage. Every day we change the arrangement of the birds so that no cage contains the same two birds as ever before. What is the greatest possible number of days we can keep doing so?
Answer: 1987
user: Let $N$ be the number of ordered pairs of integers $(x, y)$ such that \[ 4x^2 + 9y^2 \le 1000000000. \] Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$. What is the value of $10a + b$?
Answer: 52
user: On a 30-question test, Question 1 is worth 1 point, Question 2 is worth 2 points, and so on up to Question 30. David takes the test and afterward finds out he answered nine of the questions incorrectly. However, he was not told which nine were incorrect. What is the highest possible score he could have attained?
Answer: 420
user: Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, with $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
Answer: 84
user: Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$. Let $D$ be the point such that $AB \parallel CD$ and $CA \parallel BD$. Let $R$ be a point within triangle $BCD$. Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively. Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively, and $m$ meets $CA$ and $BC$ at $Q$ and $Q^\prime$ respectively. If $S$ denotes the largest possible sum of the areas of triangle $BPP^\prime$, $RP^\prime Q^\prime$, and $CQQ^\prime$, determine the value of $S^2$.
Answer: 180
user: Given the integers $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are equal to the same integer. Additionally, the following conditions are satisfied:
- $b \leq d$
- $c \geq a$
- $a \leq e$
- $b \geq e$
- $d \neq 5$

Determine the value of $a^b + c^d + e$.
Answer: 628
user: A village is surrounded by a circular wall with four gates oriented towards the north, south, east, and west. A tree is located outside the village, precisely $16 \, \mathrm{m}$ north of the north gate. This tree is visible on the horizon from a point $48 \, \mathrm{m}$ east of the south gate. Determine the diameter, in meters, of the wall that encircles the village.
Answer: 48
user: A store offers packages of $12$ pens for $\$10$ and packages of $20$ pens for $\$15$. Using only these two types of packages of pens, find the greatest number of pens $\$173$ can buy at this store.
Answer: 224
user: Let $\zeta = e^{2\pi i/99}$ and $\omega = e^{2\pi i/101}$. The polynomial \[ x^{9999} + a_{9998}x^{9998} + \cdots + a_1x + a_0 \] has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$.
Answer: 0
user: Equilateral triangle $\triangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\overline{BC}$ such that $BD = CE$ and $B$, $D$, $E$, $C$ are collinear in that order. Points $F$ and $G$ lie on $\overline{AB}$ such that $\overline{FD} \perp \overline{BC}$, and $GF = GA$. If the minimum possible value of the sum of the areas of $\triangle BFD$ and $\triangle DGE$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $\gcd(a, c) = 1$ and $b$ squarefree, find $a + b + c$.
Answer: 14
user: Find the number of positive integers $k \leq 2018$ for which there exist integers $m$ and $n$ such that $k = 2^m + 2^n$. For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$.
Answer: 65
user: Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx - 3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x) = 0$ and $g(x) = 0$ have a common root?
Answer: 0
user: Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence.
Answer: -2
user: Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
Answer: 90720
user: Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital \textit{every} month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs.

Hannah comes up with a plan that the committee likes. Beginning in October, the committee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that \textit{must} go by (including October when the first set of five teachers is selected, but not September) such that some five-member committee \textit{must have} taught together twice (all five members are the same during two different months)?
Answer: 1261
user: In the Bank of Shower, a bored customer lays $n$ coins in a row. Each second, the customer performs "The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up.

Define the function $f$ as follows: If there exists some initial arrangement of the coins so that the customer never stops, then $f(n) = 0$. Otherwise, $f(n)$ is the average number of seconds until the customer stops over all initial configurations. It is given that whenever $n = 2^k-1$ for some positive integer $k$, $f(n) > 0$.

Let $N$ be the smallest positive integer so that 
\[
M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right)
\]
is a positive integer. If $M = \overline{b_kb_{k-1}\cdots b_0}$ in base two, compute $N + b_0 + b_1 + \cdots + b_k$.
Answer: 1040
user: Let $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.
Answer: 880
user: In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ inches. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac{2}{3}$. Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches.
Answer: 8
user: Recall that the Fibonacci numbers $F(n)$ are defined by:
- $F(0) = 0$
- $F(1) = 1$
- $F(n) = F(n - 1) + F(n - 2)$ for $n \geq 2$.

Determine the last digit of $F(2006)$.
Answer: 3
user: During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour, he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $\frac{n^2}{2}$ miles on the $n^{\text{th}}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\text{th}}$ day?
Answer: 580
user: Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$. Find the number of elements in $S$ that do not divide \( \frac{L}{2016} \).
Answer: 44
user: Determine the number of real solutions $a$ to the equation:
\[ \left\lfloor \frac{1}{2}a \right\rfloor + \left\lfloor \frac{1}{3}a \right\rfloor + \left\lfloor \frac{1}{5}a \right\rfloor = a. \]
Here, if $x$ is a real number, then $\left\lfloor x \right\rfloor$ denotes the greatest integer that is less than or equal to $x$.
Answer: 30
user: Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$.
Answer: 48
user: One thousand students participate in the $2011$ Canadian Closed Mathematics Challenge. Each student is assigned a unique three-digit identification number $abc$, where each of $a$, $b$, and $c$ is a digit between $0$ and $9$, inclusive. Later, when the contests are marked, a number of markers will be hired. Each of the markers will be given a unique two-digit identification number $xy$, with each of $x$ and $y$ a digit between $0$ and $9$, inclusive. Marker $xy$ will be able to mark any contest with an identification number of the form $xyA$, $xAy$, or $Axy$, for any digit $A$. What is the minimum possible number of markers to be hired to ensure that all contests will be marked?
Answer: 50
user: Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i). $$
Answer: 673
user: Five persons of different heights stand next to each other on numbered booths to take a picture. In how many ways can they be arranged such that the people in positions $1$ and $3$ are both taller than the person in position $2$?
Answer: 40
user: Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid?
Answer: 122
user: Let $ABC$ and $DEF$ be two triangles such that $AB = DE = 20$, $BC = EF = 13$, and $\angle A = \angle D$. If $AC - DF = 10$, determine the area of $\triangle ABC$.
Answer: 126
user: In $\triangle ABC$, let point $D$ be on $\overline{BC}$ such that the perimeters of $\triangle ADB$ and $\triangle ADC$ are equal. Let point $E$ be on $\overline{AC}$ such that the perimeters of $\triangle BEA$ and $\triangle BEC$ are equal. Let point $F$ be the intersection of $\overline{AB}$ with the line that passes through $C$ and the intersection of $\overline{AD}$ and $\overline{BE}$. Given that $BD = 10$, $CD = 2$, and $BF/FA = 3$, what is the perimeter of $\triangle ABC$?
Answer: 36
user: Find the remainder when $20^{20} + 21^{21} - 21^{20} - 20^{21}$ is divided by 100.
Answer: 20
user: On a blackboard, a stranger writes the values of $s_7(n)^2$ for $n = 0, 1, \ldots, 7^{20} - 1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.
Answer: 3680
user: For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \leq 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$.
Answer: 632
user: If $a, b, c$ are the roots of the polynomial $x^3 + 20x^2 + 1x + 5$, compute $(a^2 + 1)(b^2 + 1)(c^2 + 1)$.
Answer: 225
user: Rachel starts with the number $1000$ in her hands. Each minute, she flips a fair coin. If it lands heads, she adds 1 to the number by placing it in her left pocket, transforming it to $x+1$. If it lands tails, she inverts the number by placing it in her right pocket, transforming it to $x^{-1}$. After eight minutes, the expected value of the number in Rachel's hands is $E$. Compute $\left\lfloor\frac{E}{10}\right\rfloor$.
Answer: 13
user: Suppose that $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all possible values of $x^2 + y^2$.
Answer: -90
user: We color each number in the set $S = \{1, 2, \ldots, 61\}$ with one of $25$ given colors. It is not necessary for every color to be used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$?
Answer: 119
user: An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. The bands are colored alternately red and blue, with the smallest band colored red. The difference between the total area in red and the total area in blue is $20\, \text{cm}^2$. 

What is the area of triangle $ABC$?
Answer: 200
user: Katherine makes Benj play a game called \(50\) Cent. Benj starts with \(\$0.50\), and every century thereafter has a \(50\%\) chance of doubling his money and a \(50\%\) chance of having his money reset to \(\$0.50\). What is the expected value of the amount of money Benj will have, in dollars, after \(50\) centuries?
Answer: 13
user: The roots of the equation $x^3 - x + 1 = 0$ are $a$, $b$, and $c$. Find $a^8 + b^8 + c^8$.
Answer: 10
user: Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.
Answer: 40
user: Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EFGH$ be another regular tetrahedron such that the volume of $EFGH$ is $\frac{1}{8}$-th the volume of $ABCD$. The height of $EFGH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are positive coprime integers. What is $a + b$?
Answer: 7
user: Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small.

Nair builds triangular figures by following these rules:
- Figure $1$ is made up of $4$ small pieces.
- Figure $2$ is made up of $2$ large pieces and $8$ small pieces.
- Figure $3$ is made up of $6$ large pieces and $12$ small pieces, and so on.
- The central column must be made up exclusively of small parts.
- Outside the central column, only large pieces can be placed.

Following the pattern, how many pieces will Nair use to build Figure $20$?
Answer: 460
user: An equilateral triangle $T$ with side length $2022$ is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown in the image below, which consist of $4$ equilateral unit triangles and can be rotated by any angle $k \cdot 60^{\circ}$ for $k \in \{1,2,3,4,5\}$.

The covering satisfies the following conditions:

1) It is possible not to use figures of some type and it is possible to use several figures of the same type. The unit triangles in the figures correspond to the unit triangles in the grid.
2) Every unit triangle in the grid is covered, no two figures overlap, and every figure is fully contained in $T$.

Determine the smallest possible number of figures of type $1$ that can be used in such a covering.
Answer: 1011
user: Let $\triangle ABC$ be an equilateral triangle with side length $1$ and $M$ be the midpoint of side $AC$. Let $N$ be the foot of the perpendicular from $M$ to side $AB$. The angle bisector of angle $\angle BAC$ intersects $MB$ and $MN$ at $X$ and $Y$, respectively. If the area of triangle $\triangle MXY$ is $\mathcal{A}$, and $\mathcal{A}^2$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.
Answer: 769
user: The straight line $y = ax + 16$ intersects the graph of $y = x^3$ at 2 distinct points. What is the value of $a$?
Answer: 12
user: Let $a_n$ be a sequence with $a_0 = 1$ and defined recursively by
\[
a_{n+1} = \begin{cases}
a_n + 2 & \text{if } n \text{ is even},\\
2a_n & \text{if } n \text{ is odd.}
\end{cases}
\]
What are the last two digits of $a_{2015}$?
Answer: 38
user: Let $AB$ be a diameter of a circle and let $C$ be a point on the segment $AB$ such that $AC : CB = 6 : 7$. Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$. Let $DE$ be the diameter through $D$. If $[XYZ]$ denotes the area of the triangle $XYZ$, find $\frac{[ABD]}{[CDE]}$ to the nearest integer.
Answer: 13
user: The real numbers $x$ and $y$ satisfy the following system of equations:
\[
\begin{cases} 
\sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\
\sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt{2} 
\end{cases}
\]
Find the numerical value of the ratio $\frac{y}{x}$.
Answer: 6
user: Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x) = (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$.
Answer: 545
user: Consider all ordered pairs \((a, b)\) of positive integers such that \(\frac{a^2 + b^2 + 2}{ab}\) is an integer and \(a \leq b\). We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus \(P_1 = (1, 1)\), \(P_2 = (1, 3)\), and so on. If \(P_{2020} = (x, y)\), then compute the remainder when \(x + y\) is divided by \(2017\).
Answer: 52
user: Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address but forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 5
user: Determine the natural number $a = \frac{p+q}{r} + \frac{q+r}{p} + \frac{r+p}{q}$ where $p$, $q$, and $r$ are prime positive numbers.
Answer: 6
user: Let $x$ and $y$ be integers satisfying both $x^2 - 16x + 3y = 20$ and $y^2 + 4y - x = -12$. Find $x + y$.
Answer: 18
user: Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane such that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What is the largest possible number of these angles that can be equal to $90^\circ$?
Answer: 2042220
user: For a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\frac{S(n)}{S(16n)}$.
Answer: 13
user: How many (possibly empty) sets of lattice points $\{P_1, P_2, \ldots, P_M\}$, where each point $P_i = (x_i, y_i)$ for $x_i, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$? An infinite slope, e.g., $P_i$ is vertically above $P_j$, does not count as positive.
Answer: 3432
user: Let \( f(x) = x^4 - 6x^3 + 26x^2 - 46x + 65 \).
Let the roots of \( f(x) \) be \( a_k + ib_k \) for \( k = 1, 2, 3, 4 \).
Given that the \( a_k \), \( b_k \) are all integers, find \( |b_1| + |b_2| + |b_3| + |b_4| \).
Answer: 10
user: For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. For example, the number $r = 0.123123123\ldots$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r = \overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$.
Answer: 49995000
user: How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x\]?
Answer: 5
user: There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$.
Answer: 5
user: Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX = 3$ and $XQ = 1$. Compute the largest possible area of $ABCD$.
Answer: 32
user: Find all natural numbers $n$ for which $2^8 + 2^{11} + 2^n$ is a perfect square.
Answer: 12
user: How many non-empty subsets of $\{1,2,\dots,11\}$ are there such that the product of its elements is the cube of an integer?
Answer: 19
user: A company of \( n \) soldiers is such that:
1. \( n \) is a palindrome number (reads the same forwards and backwards).
2. When the soldiers arrange in rows of 3, 4, or 5 soldiers, the last row contains 2, 3, and 5 soldiers, respectively.

Find the smallest \( n \) satisfying these conditions and prove that there are infinitely many such numbers \( n \).
Answer: 515
user: How many factors of $$(20^{12})^2$$ less than $$20^{12}$$ are not factors of $$20^{12}$$?
Answer: 288
user: Let $S=\{1,2,3,\ldots,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product?
Answer: 31
user: For positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1$, $d(i,1) = 1$ for all $i$ and $j$, and for $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$. Compute the remainder when $d(3,2016)$ is divided by $1000$.
Answer: 481
user: The sequence $a_n$ satisfies the equation $a_{m+n} + a_{m-n} = \frac{1}{2}(a_{2m} + a_{2n})$ for all $m \geq n \geq 0$. Given that $a_1 = 1$, find $a_{1995}$.
Answer: 3980025
user: Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a + b$.
Answer: 132
user: Let $N$ be the number of distinct rearrangements of the 34 letters in "SUPERCALIFRAGILISTICEXPIALIDOCIOUS". How many positive factors does $N$ have?
Answer: 3225600
user: Starting with a list of three numbers, the "Make-My-Day" procedure creates a new list by replacing each number with the sum of the other two. For example, from \( \{1, 3, 8\} \) "Make-My-Day" gives \( \{11, 9, 4\} \) and a new "Make-My-Day" leads to \( \{13, 15, 20\} \). If we begin with \( \{20, 1, 8\} \), what is the maximum difference between two numbers on the list after 2018 consecutive "Make-My-Day"s?
Answer: 19
user: Alice starts with the number $0$. She can apply $100$ operations on her number. In each operation, she can either add $1$ to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain?
Answer: 94
user: The integers $1, 2, \dots, 20$ are written on the blackboard. Consider the following operation as one step: choose two integers $a$ and $b$ such that $a - b \ge 2$ and replace them with $a - 1$ and $b + 1$. Determine the maximum number of steps that can be done.
Answer: 330
user: A \textit{divisibility chain} is a sequence of positive integers $(a_1, a_2, \ldots, a_n)$ such that $a_k$ divides $a_{k+1}$ for all $1 \le k < n$. Compute the number of divisibility chains of the form $(a, b, a^2, c, a^3, 360^9)$.
Answer: 1617000
user: The diagram below shows a large square with each of its sides divided into four equal segments. The shaded square, whose sides are diagonals drawn to these division points, has an area of $13$. Find the area of the large square.
Answer: 104