meetforever/GRPO_CL huggingface dataset:
user: Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan’s pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair $(a, b)$ of positive integers and tells it to Nathan. Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most $N$ rounds.
Answer: 9
user: A triangle $ABC$ with $AC = 20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance from $A$ to $t$ is $25$ and the distance from $C$ to $t$ is $16$. If $S$ denotes the area of triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$.
Answer: 10
user: Let $A$, $B$, $C$, $D$, $E$, and $F$ be 6 points around a circle, listed in clockwise order. We have $AB = 3\sqrt{2}$, $BC = 3\sqrt{3}$, $CD = 6\sqrt{6}$, $DE = 4\sqrt{2}$, and $EF = 5\sqrt{2}$. Given that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent, determine the square of $AF$.
Answer: 225
user: Find the last three digits of the sum:
\[ 2005^{11} + 2005^{12} + \ldots + 2005^{2006} \]
Answer: 500
user: Find the largest positive integer $N$ satisfying the following properties:

- $N$ is divisible by $7$.
- Swapping the $i^{\text{th}}$ and $j^{\text{th}}$ digits of $N$ (for any $i$ and $j$ with $i \neq j$) gives an integer which is \textit{not} divisible by $7$.
Answer: 987546
user: In the plane, there are $10$ lines in general position, meaning that no two are parallel and no three are concurrent. At each intersection of two lines, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?
Answer: 4050
user: There is an interval $[a, b]$ that is the solution to the inequality $|3x - 80| \le |2x - 105|$. Find $a + b$.
Answer: 12
user: Given $\triangle ABC$, where $A$ is at $(0,0)$, $B$ is at $(20,0)$, and $C$ is on the positive $y$-axis. Cone $M$ is formed when $\triangle ABC$ is rotated about the $x$-axis, and cone $N$ is formed when $\triangle ABC$ is rotated about the $y$-axis. If the volume of cone $M$ minus the volume of cone $N$ is $140\pi$, find the length of $\overline{BC}$.
Answer: 29
user: Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183$ and $a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac{9}{5} (a + b)$.
Answer: 73
user: Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that:
- $P$ passes through $A$ and $B$,
- the focus $F$ of $P$ lies on $E$,
- the orthocenter $H$ of $\triangle FAB$ lies on the directrix of $P$.

If the major and minor axes of $E$ have lengths $50$ and $14$, respectively, compute $AH^2 + BH^2$.
Answer: 2402
user: Let $ABCDEF$ be a regular hexagon with a side length of $3$. Points $X$, $Y$, and $Z$ are located on segments $AB$, $CD$, and $EF$ respectively, such that $AX = CY = EZ = 1$. The area of triangle $XYZ$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $\gcd(a, c) = 1$. Find $100a + 10b + c$.
Answer: 2134
user: Consider \(13\) marbles that are labeled with positive integers such that the product of all \(13\) integers is \(360\). Moor randomly picks up \(5\) marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
Answer: 24
user: The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle?
Answer: 150
user: Find the largest integer $n \ge 3$ for which there exists an $n$-digit number $\overline{a_1a_2a_3\ldots a_n}$ with non-zero digits $a_1$, $a_2$, and $a_n$, such that the number is divisible by $\overline{a_2a_3\ldots a_n}$.
Answer: 6
user: Let $ABCD$ be a square with $AB = 6$. A point $P$ inside the square is 2 units away from side $BC$ and 3 units away from side $CD$. What is the distance from $P$ to $A$?
Answer: 5
user: Let $p$ be the largest prime less than $2013$ for which \[ N = 20 + p^{p^{p+1}-13} \] is also prime. Find the remainder when $N$ is divided by $10^4$.
Answer: 1781
user: Consider the sequence $$1, 7, 8, 49, 50, 56, 57, 343, \ldots$$ which consists of sums of distinct powers of $7$, that is, $7^0$, $7^1$, $7^0 + 7^1$, $7^2$, $\ldots$ in increasing order. At what position will $16856$ occur in this sequence?
Answer: 36
user: Given that $a_n a_{n-2} - a_{n-1}^2 + a_n - n a_{n-2} = -n^2 + 3n - 1$ and initial conditions $a_0 = 1$, $a_1 = 3$, find $a_{20}$.
Answer: 1048596
user: For each set of five integers $S= \{a_1, a_2, a_3, a_4, a_5\}$, let $P_S$ be the product of all differences between two of the elements, namely:

$$P_S = (a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)$$

Determine the greatest integer $n$ such that given any set $S$ of five integers, $n$ divides $P_S$.
Answer: 288
user: John and Mary each select a natural number and tell them to Bill. Bill writes down their sum and product on two separate pieces of paper, hides one, and shows the other to John and Mary. John looks at the number, which is $2002$, and declares he cannot determine Mary's number. Knowing this, Mary also says she cannot determine John's number. What was Mary's number?
Answer: 1001
user: A set $A$ contains exactly $n$ integers, each greater than $1$, where every prime factor of these integers is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer.
Answer: 17
user: Evaluate the sum: \( \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_{20}} \), where \( a_n = \sqrt{1 + \left(1 - \frac{1}{n}\right)^2} + \sqrt{1 + \left(1 + \frac{1}{n}\right)^2} \) for \( n \ge 1 \).
Answer: 7
user: A rectangular storage bin measures $10$ feet by $12$ feet, is $3$ feet tall, and sits on a flat plane. A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground $4$ feet away from the base of the storage bin as shown. The number of cubic feet of dirt needed to form the pile can be written as $m + n \pi$ where $m$ and $n$ are positive integers. Find $m + n.$
Answer: 280
user: How many ordered triplets $(a, b, c)$ of positive integers satisfy the inequality $30a + 50b + 70c \leq 343$?
Answer: 30
user: Let $p = 101$ and let $S$ be the set of $p$-tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that:
- $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$.
- $f(a) = f(b)$ whenever all components of $a-b$ are divisible by $p$.

Compute the number of positive integer divisors of $N$. (Here addition and subtraction in $\mathbb{Z}^p$ are done component-wise.)
Answer: 5152
user: Let $x$ be the largest root of $x^4 - 2009x + 1$. Find the nearest integer to $\frac{1}{x^3-2009}$.
Answer: -13
user: The diagram below shows a $18 \times 35$ rectangle with eight points marked that divide each side into three equal parts. Four triangles are removed from each of the corners of the rectangle, leaving the shaded region. Find the area of this shaded region.
Answer: 490
user: Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying:
- $f(1) = 1$,
- For $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$.

Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019.
Answer: 1889
user: Five teams participated in a commercial football tournament. Each team was supposed to play exactly one match against every other team. However, due to financial difficulties, some games were canceled. 

By the end of the tournament, all teams had different scores, and no team had zero points. What is the minimum number of games that could have been played in this tournament, given that:
- A win awards 3 points,
- A draw awards 1 point,
- A defeat awards 0 points?
Answer: 6
user: Emma's calculator has ten buttons: one for each digit $1, 2, \ldots, 9$, and one marked "clear". When Emma presses one of the buttons marked with a digit, that digit is appended to the right of the display. When she presses the "clear" button, the display is completely erased. If Emma starts with an empty display and presses five (not necessarily distinct) buttons at random, where all ten buttons have equal probability of being chosen, the expected value of the number produced is $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$. (Take an empty display to represent the number 0.)
Answer: 6642902
user: Let $a_1, a_2, \ldots, a_{100}$ be non-negative integers such that:

1. There exists a positive integer $k \leq 100$ such that $a_1 \leq a_2 \leq \cdots \leq a_k$ and $a_i = 0$ for $i > k$.
2. $a_1 + a_2 + a_3 + \cdots + a_{100} = 100$.
3. $a_1 + 2a_2 + 3a_3 + \cdots + 100a_{100} = 2022$.

Find the minimum value of $a_1 + 2^2a_2 + 3^2a_3 + \cdots + 100^2a_{100}$.
Answer: 40940
user: Evaluate $ \int_{\sqrt{2}-1}^{\sqrt{2}+1} \frac{x^4+x^2+2}{(x^2+1)^2}\, dx.$
Answer: 2
user: Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.
Answer: 1997
user: What is the smallest possible value of \( x^2 + y^2 - x - y - xy \)?
Answer: -1
user: Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave smudges that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. 

For each number of distinct digits that could be used in the access code, calculate the number of possible combinations when the digits are known but their order and frequency are not known. 

For example, if there are smudges on $3$ and $9$, two possible codes are $393939$ and $993999$. Which number of distinct digits in the access code offers the most combinations?
Answer: 5
user: Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1 \le x_2 \le \cdots \le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000.
Answer: 601
user: How many seven-digit positive integers do not either start or end with \(7\)?
Answer: 7200000
user: Let $S$ be a subset of $\{1, 2, 3, \cdots, 1989 \}$ in which no two members differ by exactly $4$ or by exactly $7$. What is the largest number of elements $S$ can have?
Answer: 905
user: Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$.
Answer: 804
user: Let $ABCD$ be a quadrilateral with $\angle ADC = 70^{\circ}$, $\angle ACD = 70^{\circ}$, $\angle ACB = 10^{\circ}$, and $\angle BAD = 110^{\circ}$. Determine the measure of $\angle CAB$ (in degrees).
Answer: 70
user: There exist two triples of real numbers \((a,b,c)\) such that \(a-\frac{1}{b}, b-\frac{1}{c}, c-\frac{1}{a}\) are the roots to the cubic equation \(x^3-5x^2-15x+3\) listed in increasing order. Denote those \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\). If \(a_1, b_1,\) and \(c_1\) are the roots to the monic cubic polynomial \(f\) and \(a_2, b_2,\) and \(c_2\) are the roots to the monic cubic polynomial \(g\), find \(f(0)^3+g(0)^3\).
Answer: -14
user: Find the largest positive integer $n$ such that for each prime $p$ with $2 < p < n$, the difference $n - p$ is also a prime number.
Answer: 10
user: A $70$ foot pole stands vertically in a plane and is supported by three $490$ foot wires. All wires are attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. Determine the distance, in feet, between any two of those anchor points.
Answer: 840
user: At a party, there are $2011$ people, each with a glass of fruit juice, sitting around a circular table. Every second, they clink glasses according to the following two rules:
1. They do not clink glasses crosswise.
2. At each point in time, everyone can clink glasses with at most one other person.

How many seconds are needed, at a minimum, for everyone to have clinked glasses with everyone else?
Answer: 2011
user: A circle with area $40$ is tangent to a circle with area $10$. Let $R$ be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$.
Answer: 240
user: Suppose $a$, $b$, and $c$ are positive integers with the property that $ab$, $bc$, and $ac$ are pairwise distinct perfect squares. What is the smallest possible value of $a + b + c$?
Answer: 14
user: Find all prime numbers $p$ such that $p^2 - p + 1$ is a perfect cube.
Answer: 19
user: Let $N = 1890 \times 1930 \times 1970$. Find the number of divisors of $N$ which are not divisors of $45$.
Answer: 506
user: Calculate $ \lfloor \log_3 5 +\log_5 7 +\log_7 3 \rfloor $.
Answer: 3
user: Compute the remainder when $1^{2547} + 2^{2547} + \ldots + 2547^{2547}$ is divided by $25$.
Answer: 4
user: Gary has $2$ children. We know one is a boy born on a Friday. Assume birthing boys and girls are equally likely, being born on any day of the week is equally likely, and that these properties are independent of each other, as well as independent from child to child. The probability that both of Gary's children are boys is $\frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Find $a + b.$
Answer: 40
user: In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}$ and $\overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, and $PF = 2$.
Answer: 84
user: Determine the number of $8$-tuples of nonnegative integers $(a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $0 \le a_k \le k$, for each $k = 1,2,3,4$, and $a_1 + a_2 + a_3 + a_4 + 2b_1 + 3b_2 + 4b_3 + 5b_4 = 19$.
Answer: 1540
user: Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.
Answer: 15
user: The diagonals of convex quadrilateral $BSCT$ meet at the midpoint $M$ of $\overline{ST}$. Lines $BT$ and $SC$ meet at $A$, and $AB = 91$, $BC = 98$, $CA = 105$. Given that $\overline{AM} \perp \overline{BC}$, find the positive difference between the areas of $\triangle SMC$ and $\triangle BMT$.
Answer: 336
user: 12. What is the sum of all possible $\binom{i}{j}$ subject to the restrictions that $i \geq 10$, $j \geq 0$, and $i+j \leq 20$? Count different $i, j$ that yield the same value separately - for example, count both $\binom{10}{1}$ and $\binom{10}{9}$.
Answer: 27633
user: A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m$.
Answer: 1333
user: For how many ordered pairs of positive integers \((x, y)\) is the least common multiple of \(x\) and \(y\) equal to \(1,003,003,001\)?
Answer: 343
user: For a positive integer $n$, let $\sigma(n)$ be the sum of the divisors of $n$ (for example, $\sigma(10) = 1 + 2 + 5 + 10 = 18$). For how many $n \in \{1, 2, \ldots, 100\}$ do we have $\sigma(n) < n + \sqrt{n}$?
Answer: 26
user: For some integers $a$ and $b$, the function $f(x) = ax + b$ has the properties that $f(f(0)) = 0$ and $f(f(f(4))) = 9$. Find $f(f(f(f(10))))$.
Answer: 10
user: Given $n=2020$, sort the 6 values: $$n^{n^2},\, 2^{2^{2^n}},\, n^{2^n},\, 2^{2^{n^2}},\, 2^{n^n},\, \text{and}\, 2^{n^{2^2}}$$ from least to greatest. Provide your answer as a 6-digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.
Answer: 163542
user: Suppose you have a $6$-sided die with $3$ faces colored red, $2$ faces colored blue, and $1$ face colored green. You roll this die $20$ times and record the color that shows up on top. What is the expected value of the product of the number of red faces, blue faces, and green faces?
Answer: 190
user: Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the large circle. Calculate the area of quadrilateral $ABCD$.
Answer: 24
user: An integer $n$ satisfies the following conditions:

1. $\Bigl\lfloor \frac{n}{9} \Bigr\rfloor$ is a three-digit number with equal digits.
2. $\Bigl\lfloor \frac{n-172}{4} \Bigr\rfloor$ is a four-digit number with the digits $2, 0, 2, 4$ in some order.

Determine the remainder when $n$ is divided by $100$.
Answer: 91
user: Let $x, y$ be distinct positive real numbers satisfying:
\[
\frac{1}{\sqrt{x + y} - \sqrt{x - y}} + \frac{1}{\sqrt{x + y} + \sqrt{x - y}} = \frac{x}{\sqrt{y^3}}.
\]
If \( \frac{x}{y} = \frac{a + \sqrt{b}}{c} \) for positive integers \( a, b, c \) with \( \gcd(a, c) = 1 \), find \( a + b + c \).
Answer: 8
user: Find the smallest positive three-digit integer $n$ such that $3^n + 4^n$ is divisible by $5$.
Answer: 102
user: Determine the smallest natural number that can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.
Answer: 10010
user: Let $k$ be the largest integer such that $2^k$ divides $$\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}\right)^2\right)\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}^2\right)\right).$$ Find $k$.
Answer: 707
user: Find the number of triples of nonnegative integers $(x, y, z)$ such that 
\[x^2 + 2xy + y^2 - z^2 = 9.\]
Answer: 10
user: A rectangular piece of paper is divided into polygonal areas in the following way: at each step, one of the existing surfaces is cut by a straight line, resulting in two new areas. What is the minimum number of cuts needed such that among the obtained polygons, there are $251$ polygons with $11$ sides?
Answer: 2007
user: The area of a circle (in square inches) is numerically larger than its circumference (in inches). What is the smallest possible integral area of the circle, in square inches?
Answer: 13
user: Find an odd natural number not exceeding $1000$ such that the sum of the last digits of all its divisors (including $1$ and the number itself) is $33$.
Answer: 729
user: Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with their product; when it is Sara's turn, she can pick any two integers on the board and replace them with their sum. Play continues until exactly one integer remains on the board. Paul wins if that integer is odd, and Sara wins if it is even.

Initially, there are $2021$ integers on the board, each one sampled uniformly at random from the set $\{0, 1, 2, 3, \ldots, 2021\}$. Assuming both players play optimally, the probability that Paul wins is $\frac{m}{n}$, where $m, n$ are positive integers and $\gcd(m, n) = 1$. Find the remainder when $m + n$ is divided by $1000$.
Answer: 3
user: A circle with radius $r$ has an area of $505$. Compute the area of a circle with a diameter of $2r$.
Answer: 2020
user: Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is worth $n$ points. Assume that entering answers and moving between or skipping problems takes no time.
Answer: 70
user: On triangle $ABC$, let $D$ be the point on $AB$ such that $CD$ is an altitude of the triangle, and $E$ be the point on $BC$ such that $AE$ bisects angle $BAC$. Let $G$ be the intersection of $AE$ and $CD$, and let point $F$ be the intersection of side $AC$ and the ray $BG$. If $AB$ has length $28$, $AC$ has length $14$, and $CD$ has length $10$, then the length of $CF$ can be written as $\frac{k-m\sqrt{p}}{n}$ where $k$, $m$, $n$, and $p$ are positive integers, $k$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $k - m + n + p$.
Answer: 623
user: In the coordinate plane, the line passing through the points $(2023, 0)$ and $(-2021, 2024)$ also passes through the point $(1, c)$ for a constant $c$. Find the value of $c$.
Answer: 1012
user: Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$. Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.
Answer: -1002
user: Leo the fox has a $5 \times 5$ checkerboard grid with alternating red and black squares. He fills in the grid with the numbers $1, 2, 3, \dots, 25$ such that any two consecutive numbers are in adjacent squares (sharing a side) and each number is used exactly once. He then computes the sum of the numbers in the $13$ squares that are the same color as the center square. Compute the maximum possible sum Leo can obtain.
Answer: 169
user: Consider all pairs of points $(a, b, c)$ and $(d, e, f)$ in the $3$-D coordinate system such that $ad + be + cf = -2023$. What is the least positive integer that can be the distance between such a pair of points?
Answer: 90
user: A book is published in three volumes, with the pages numbered consecutively starting from $1$. The numbering continues from the first volume to the second and then to the third. The second volume contains $50$ more pages than the first volume, and the third volume contains one and a half times as many pages as the second volume. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?
Answer: 17
user: Find the number of pairs of integers $(x, y)$ such that $x^2 + 2y^2 < 25$.
Answer: 55
user: For real numbers $B$, $M$, and $T$, we have the following conditions:  
1. $B^2 + M^2 + T^2 = 2022$  
2. $B + M + T = 72$

Compute the sum of the minimum and maximum possible values of $T$.
Answer: 48
user: Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
- She rolled a $6$ exactly three times.
- The product of her first three rolls is the same as the product of her last three rolls.

How many distinct sequences of six rolls could Kara have rolled?
Answer: 72
user: How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?
Answer: 28
user: The product $55 \cdot 60 \cdot 65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number among these 5 numbers.
Answer: 20
user: $N$ is a 50-digit number in decimal representation. All digits except the 26th digit (from the left) are 1. If $N$ is divisible by 13, find its 26th digit.
Answer: 3
user: A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?
Answer: 13