meetforever/GRPO_CL huggingface dataset:
user: If $x$ is a number satisfying the equation $\sqrt[3]{x+9} - \sqrt[3]{x-9} = 3$, find the value of $x^2$ and determine which interval it falls into among the following: $[55, 65]$, $[65, 75]$, $[75, 85]$, $[85, 95]$, $[95, 105]$. The original answer is in the form $[a, b]$, please give the value of $a + b$.
Answer: 160
user: Flights are arranged between 13 countries. For $k \ge 2$, the sequence $A_1, A_2, \ldots, A_k$ is said to be a cycle if there exists a flight from $A_1$ to $A_2$, from $A_2$ to $A_3$, $\ldots$, from $A_{k-1}$ to $A_k$, and from $A_k$ to $A_1$. Find the smallest possible number of flights such that, regardless of how the flights are arranged, there exists a cycle.
Answer: 79
user: A cone-shaped mountain has its base on the ocean floor and has a height of $8000$ feet. The top $\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain, in feet?
Answer: 4000
user: A wooden cube has edges of length $3$ meters. Square holes, with a side of $1$ meter, are centered in each face and cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. What is the total surface area, including the inside, in square meters?
Answer: 72
user: Let \(a\), \(b\), and \(c\) be the roots of the polynomial \(x^3 + 4x^2 - 7x - 1\). Find the polynomial that has roots \(ab\), \(bc\), and \(ac\). The original answer is in the form \(mx^3 + nx^2 + px + q\). Please give the value of \(m + n + p + q\).
Answer: 3
user: In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, find the perimeter of $\triangle AMN$.
Answer: 30
user: Let $ABCD$ be a convex quadrilateral with $\angle C = \angle D = 90^\circ$. The circle $K$ passing through $A$ and $B$ is tangent to $CD$ at $C$. Let $E$ be the intersection of $K$ and $[AD]$. If $|BC| = 20$, $|AD| = 16$, then find the length of $|CE|$. The original answer is in the form $k\sqrt{m}$, please give the value of $k + m$.
Answer: 9
user: Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?
Answer: 15
user: [BD] is a median of \( \triangle ABC \). \( m(\widehat{ABD}) = 90^\circ \), \(|AB| = 2\), and \(|AC| = 6\). Find the length of \(|BC|\). The original answer is in the form of \(k\sqrt{m}\). Please give the value of \(k + m\).
Answer: 8
user: Find the largest positive value attained by the function
\[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \ x\ \text{a real number} \]
If the answer is in the form $a\sqrt{b}$, where $a$ and $b$ are integers, find the value of $a$.
Answer: 2
user: For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer? Provide the number of such positive integers.
Answer: 7
user: On circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. Find the ratio of the area of the smaller sector $COD$ to the area of the circle. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 31
user: Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P \in [AC]$, $Q \in [BC]$, $R \in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K \in [BC]$, $M \in [AB]$, and $L \in [AC]$. Find $|AB|+|AC|$.
Answer: 12
user: Let \(D\) be a point on side \([BC]\) of \(\triangle ABC\) such that \(|AB|=3\), \(|CD|=1\), and \(|AC|=|BD|=\sqrt{5}\). If the \(B\)-altitude of \(\triangle ABC\) meets \(AD\) at \(E\), find the length of \(|CE|\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
user: If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ is on $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, find the ratio of the length of $PQ$ to the length of $AQ$. The original answer is in \(\frac{k\sqrt{3}}{m}\) format, please give the value of k + m.
Answer: 4
user: When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value of $S$?
Answer: 337
user: The chord $[CD]$ of the circle with diameter $[AB]$ is perpendicular to $[AB]$. Let $M$ and $N$ be the midpoints of $[BC]$ and $[AD]$, respectively. If $|BC|=6$ and $|AD|=2\sqrt{3}$, find the length of $|MN|$. The original answer is in the form \(\sqrt{m}\). Please find the value of \(m\).
Answer: 21
user: Medians \(AD\) and \(CE\) of triangle \(ABC\) intersect at point \(M\). The midpoint of \(AE\) is \(N\). The area of triangle \(MNE\) is \(k\) times the area of triangle \(ABC\). The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Answer: 13
user: Find the sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$. The original answer is in the form an(b + c), where a, b, and c are constants. Please give the value of a + b + c.
Answer: 4
user: Given three squares as in the figure, where the largest square has area 1, and the area $A$ is known. Find the area $B$ of the smallest square in terms of $A$. The original answer is in the form \frac{(k - A)^m}{n}, where k, m, and n are integers. Please calculate the value of k + m + n.
Answer: 7
user: If $a$ and $b$ are the roots of the equation $x^2 - 2cx - 5d = 0$, and $c$ and $d$ are the roots of the equation $x^2 - 2ax - 5b = 0$, where $a, b, c, d$ are distinct real numbers, find the value of $a + b + c + d$.
Answer: 30
user: If \(x \neq 0\) or \(4\) and \(y \neq 0\) or \(6\), then \(\frac{2}{x}+\frac{3}{y}=\frac{1}{2}\) is equivalent to the equation \(\frac{ay}{y-b}=x\). Please determine the sum of a and b.
Answer: 10
user: Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
Answer: 1680
user: Two medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \sqrt{15}$ square inches. Find the length of the third median in inches. The original answer is in the form $k \sqrt{m}$, please give the value of $k + m$.
Answer: 9
user: Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? Express your answer as a fraction in simplest form. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 10
user: For every $n \ge 2$, $a_n = \frac{\sqrt[3]{n^3 + n^2 - n - 1}}{n}$. Find the least value of positive integer $k$ satisfying $a_2a_3\cdots a_k > 3$.
Answer: 106
user: In the overlapping triangles $\triangle{ABC}$ and $\triangle{ABE}$ sharing common side $AB$, $\angle{EAB}$ and $\angle{ABC}$ are right angles, $AB = 4$, $BC = 6$, $AE = 8$, and $\overline{AC}$ and $\overline{BE}$ intersect at $D$. Find the difference between the areas of $\triangle{ADE}$ and $\triangle{BDC}$.
Answer: 4
user: The pattern of figures $\triangle$ $\bullet$ $\square$ $\blacktriangle$ $\circ$ is repeated in the sequence:
\[
\triangle, \bullet, \square, \blacktriangle, \circ, \triangle, \bullet, \square, \blacktriangle, \circ, \ldots
\]
Determine the 214th figure in the sequence. Write the position of the figure in the pattern as a number from 1 to 5.
Answer: 4
user: The equation \(z(z+i)(z+3i)=2002i\) has a zero of the form \(a+bi\), where \(a\) and \(b\) are positive real numbers. The original answer is in the form \(\sqrt{k}\). Please find the value of \(k\).
Answer: 118
user: How many integer triples $(x,y,z)$ are there satisfying the equation $x^3+y^3=x^2yz+xy^2z+2$?
Answer: 4
user: What is the sum of distinct real numbers \(x\) such that \((2x^2+5x+9)^2=56(x^3+1)\)? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 11
user: Find the shortest distance from an integral point to the line $y=\frac{5}{3}x+\frac{4}{5}$. The original answer is in the form \frac{\sqrt{k}}{m}, where k and m are integers. Please find the value of k + m.
Answer: 119
user: The hour part of a defective digital watch displays only the numbers from $1$ to $12$. After one minute from $n: 59$, although it must display $(n + 1): 00$, it displays $2n: 00$ (considering $\mod\, 12$). For example, after $7: 59$, it displays $2: 00$ instead of $8: 00$. If we set the watch to an arbitrary time, find the probability that the hour part displays $4$ after exactly one day. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 4
user: How many ordered triples of integers $(a, b, c)$ satisfy the following equations?
\[ |a + b| + c = 19 \]
\[ ab + |c| = 97 \]
Provide the number of such ordered triples.
Answer: 12
user: A man buys a house for $10000$ dollars and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep, pays $325$ dollars a year in taxes, and realizes $5\frac{1}{2}\%$ on his investment. If the monthly rent is $x$ dollars, what is the value of $100x$?
Answer: 8333
user: For all non-zero numbers $x$ and $y$ such that $x = \frac{1}{y}$, find the value of \[\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)\]. The original answer is in the form $ax^2 - by^2$. Please find the value of $a + b$.
Answer: 2
user: There are integers \( a \), \( b \), and \( c \), each greater than 1, such that \( \sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}} \) for all \( N > 1 \). Find the value of \( b \).
Answer: 3
user: If $a=\sqrt[3]{9}-\sqrt[3]{3}+1$, find the value of $\left(\frac{4-a}{a}\right)^6$.
Answer: 9
user: What is the sum of the real roots of the equation $(x\sqrt{x})^x = x^{x\sqrt{x}}$? Provide your answer in the form of a simplified fraction \frac{k}{m}, and then give the value of k + m.
Answer: 17
user: Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have an area of $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? Provide your answer as an integer.
Answer: 20
user: Emily has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any previous boxes. How many balls in total are in the boxes as a result of Emily's $2017$th step?
Answer: 9
user: Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream? The answer should be in the form \frac{k}{m}, where k and m are coprime. Please provide the value of k + m.
Answer: 7
user: For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which 
\[n(A) + n(B) + n(C) = n(A \cup B \cup C)\] 
and 
\[|A| = |B| = 100,\] 
then find the minimum possible value of $|A \cap B \cap C|$.
Answer: 97
user: Let $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\sqrt{6}$, $m(\widehat{ABD})=60^\circ$, $m(\widehat{BDC})=45^\circ$, and $|BD|=13+3\sqrt{3}$. Find the length of $|AC|$.
Answer: 16
user: We have a square grid of $4 \times 4$ points. Find the number of triangles that can be formed with vertices on these points, given that the three vertices must not lie on a straight line.
Answer: 516
user: The zeroes of the function $f(x) = x^2 - ax + 2a$ are integers. Find the sum of all possible values of $a$.
Answer: 16
user: What is the remainder when the sum
\[\binom{2013}{1} + 2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013}\]
is divided by $41$?
Answer: 20
user: Five points on a circle are numbered 1, 2, 3, 4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle. If it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps, on which point will it be?
Answer: 4
user: How many natural numbers (i.e., 1, 2, 3,...) are equal to three times the sum of their digits? Find the number of such natural numbers.
Answer: 1
user: Let $ABC$ be an isosceles triangle with $|AB| = |AC| = 10$ and $|BC| = 12$. Points $P$ and $R$ are on $[BC]$ such that $|BP| = |RC| = 3$. Points $S$ and $T$ are the midpoints of $[AB]$ and $[AC]$, respectively. If $M$ and $N$ are the feet of the perpendiculars from $S$ and $R$ to $PT$, find the length of $|MN|$. The original answer is in the form \frac{k\sqrt{m}}{n}, please provide the value of k + m + n.
Answer: 36
user: Let $f(x) = x - \frac{1}{x}$. How many different solutions are there to the equation $f(f(f(x))) = 1$? Provide your answer as an integer.
Answer: 8
user: The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1$? The original answer is in the form of $k \pi \sqrt{m}$, please give the value of $k + m$.

[asy]
unitsize(20);
filldraw(circle((0,0),2*\sqrt{3}+1),rgb(0.5,0.5,0.5));
filldraw(circle((-2,0),1),white);
filldraw(circle((0,0),1),white);
filldraw(circle((2,0),1),white);
filldraw(circle((1,\sqrt{3}),1),white);
filldraw(circle((3,\sqrt{3}),1),white);
filldraw(circle((-1,\sqrt{3}),1),white);
filldraw(circle((-3,\sqrt{3}),1),white);
filldraw(circle((1,-1*\sqrt{3}),1),white);
filldraw(circle((3,-1*\sqrt{3}),1),white);
filldraw(circle((-1,-1*\sqrt{3}),1),white);
filldraw(circle((-3,-1*\sqrt{3}),1),white);
filldraw(circle((0,2*\sqrt{3}),1),white);
filldraw(circle((0,-2*\sqrt{3}),1),white);
[/asy]
Answer: 7
user: A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0)$, $(2020, 0)$, $(2020, 2020)$, and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) If the answer is $d$ to the nearest tenth, what is the value of $10d$?
Answer: 4
user: Two equal parallel chords are drawn $8$ inches apart in a circle of radius $8$ inches. Find the area of that part of the circle that lies between the chords in the form of $k\sqrt{3} + \frac{m}{n}\pi$, where $k$, $m$, and $n$ are integers, and $m/n$ is in simplest form. Please provide the value of $k + m + n$.
Answer: 99
user: Lines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1)$. Find the equation of one of these lines. The equation is in the form mx + ny + p = 0. Please give the value of m + n + p.
Answer: 10
user: A sequence of three real numbers forms an arithmetic progression with a first term of $9$. If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?
Answer: 1
user: Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}$, $AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$?
Answer: 360
user: The number of positive integers $k$ for which the equation $kx - 12 = 3k$ has an integer solution for $x$ is ___.
Answer: 6
user: Let \[x = .123456789101112\ldots998999,\] where the digits are obtained by writing the integers 1 through 999 in order. What is the 1983rd digit to the right of the decimal point?
Answer: 7
user: Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
Answer: 137
user: What is the largest area of a regular hexagon that can be drawn inside an equilateral triangle with a side length of \(3\)? Provide your answer in the form of \(\frac{a\sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are integers. Please find the value of a + b + c.
Answer: 8
user: Let $P(x)=x^3+ax^2+bx+c$ where $a,b,c$ are positive real numbers. If $P(1)\geq 2$ and $P(3)\leq 31$, find how many integers $P(4)$ can take.
Answer: 4
user: In a certain base \(b\) (different from \(10\)), \(57_b^2 = 2721_b\). What is \(17_b^2\) in this base? Provide your answer in the form of \(k_{b}m_{b}n_{b}\), where \(k\), \(m\), and \(n\) are digits in base \(b\). Calculate the value of \(k + m + n\).
Answer: 9
user: A circle is inscribed in an equilateral triangle. Three nested sequences of circles are then constructed as follows: each circle touches the previous circle and has two edges of the triangle as tangents.

Find the ratio of the area of the largest circle to the combined area of all the other circles. The original answer is in the format \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please calculate the value of \(k + m\).
Answer: 11
user: Calculate the value of \(\dfrac{3\times 5}{9\times 11}\times \dfrac{7\times 9\times 11}{3\times 5\times 7}\).
Answer: 1
user: A "stair-step" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. How many black squares are in the $37$th row?
Answer: 36
user: How many whole numbers between $100$ and $400$ contain the digit $2$? Fill in the blank with the number of such whole numbers.
Answer: 138
user: Nine copies of a certain pamphlet cost less than $＄$ $10.00$ while ten copies of the same pamphlet (at the same price) cost more than $＄$ $11.00$. If the answer is $＄x$, what is the value of $x \times 100$?
Answer: 111
user: In a magic triangle, each of the six whole numbers $10-15$ is placed in one of the circles so that the sum, $S$, of the three numbers on each side of the triangle is the same. Find the largest possible value for $S$.
Answer: 39
user: In July $1861, 366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? Provide your answer as a simplified fraction of the form \(\frac{m}{n}\), and calculate the value of \(m + n\).
Answer: 185
user: A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $25\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter? If the answer is $x$ million, what is the value of $x \times 10$?
Answer: 25
user: A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m?
Answer: 12
user: At the beginning of a trip, the mileage odometer read $56,200$ miles. The driver filled the gas tank with $6$ gallons of gasoline. During the trip, the driver filled his tank again with $12$ gallons of gasoline when the odometer read $56,560$. At the end of the trip, the driver filled his tank again with $20$ gallons of gasoline. The odometer read $57,060$. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip? If the answer is $x$, what is the value of $10x$?
Answer: 269
user: The value of the expression $\frac{(304)^5}{(29.7)(399)^4}$ is closest to what integer?
Answer: 3
user: The $600$ students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. Find the probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 10
user: The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. Find the total number of smaller cubes that must have at least one face shaded.
Answer: 20
user: If $A$ and $B$ are nonzero digits, then find the number of digits (not necessarily different) in the sum of the three whole numbers:
$$\begin{array}{cccc}9&8&7&6\\&{A}&3&2\\&{B}&1&\\\hline\end{array}$$
Answer: 5
user: A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48$. What is the maximum number of questions he could have answered correctly?
Answer: 12
user: Calculate the product \(8 \times 0.25 \times 2 \times 0.125\). Provide your answer in the form of \(\frac{k}{m}\), where the fraction is in its simplest form. Find the value of \(k + m\).
Answer: 3
user: If rose bushes are spaced about 1 foot apart, approximately how many bushes are needed to surround a circular patio whose radius is 12 feet? Provide your answer as an integer.
Answer: 75
user: A fifth number, $n$, is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median. Find the number of possible values of $n$.
Answer: 3
user: Maria buys computer disks at a price of $4$ for $5 and sells them at a price of $3$ for $5. How many computer disks must she sell in order to make a profit of $100?
Answer: 240
user: When placing each of the digits $2,4,5,6,9$ in exactly one of the boxes of this subtraction problem, find the smallest possible difference.  
  
$$\begin{array}{rrrr}&&\square&\square&\square\\&-&&\square&\square\\\hline\end{array}$$
Answer: 149
user: In how many ways can $47$ be written as the sum of two primes?
Answer: 0
user: The letters $A$, $J$, $H$, $S$, $M$, $E$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list:
$$\begin{aligned}&&&\mathrm{AJHSME}\quad1989\\&1.&&\mathrm{JHSMEA}\quad9891\\&2.&&\mathrm{HSMEAJ}\quad8919\\&3.&&\mathrm{SMEAJH}\quad9198\\&&&.........\end{aligned}$$
What is the number of the line on which $AJHSME 1989$ will appear for the first time?
Answer: 12
user: Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed—a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 11
user: Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even? Express your answer as a fraction in simplest form. The original answer is in the format \frac{k}{m}, please give the value of k + m.
Answer: 3
user: What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?
Answer: 1047
user: In the decimal number $0.12345$, which digit, when changed to $9$, gives the largest possible number? Indicate the original position of this digit.
Answer: 1
user: There are twenty-four $4$-digit numbers that use each of the four digits $2$, $4$, $5$, and $7$ exactly once. Listed in numerical order from smallest to largest, what is the number in the $17$th position in the list?
Answer: 5724