ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: The number of positive integers $k$ for which the equation $kx - 12 = 3k$ has an integer solution for $x$ is ___.
Answer: 6
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: The value of the expression $\frac{(304)^5}{(29.7)(399)^4}$ is closest to what integer?
Answer: 3
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: In how many ways can $47$ be written as the sum of two primes?
Answer: 0
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). If the postage for a letter weighing $4.5$ ounces is x cents, what is the value of x?
Answer: 118
Problem: Calculate the result of the expression $1990-1980+1970-1960+\cdots -20+10$.
Answer: 1000
Problem: Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?
Answer: 36
Problem: The annual incomes of $1,000$ families range from $8,200$ dollars to $98,000$ dollars. Due to an error, the largest income was entered on the computer as $980,000$ dollars. Calculate the absolute difference between the mean of the incorrect data and the mean of the actual data, and provide your answer as an integer in dollars.
Answer: 882
Problem: A list of \(8\) numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Given that the last three numbers are \(16\), \(64\), and \(1024\), find the first number in the list. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).
Answer: 5
Problem: If an integer of two digits is \(k\) times the sum of its digits, find the expression that represents the number formed by interchanging the digits as the sum of the digits multiplied by \(x\). The original answer is in the form \(m - k\). Please find the value of \(m + n\).
Answer: 11
Problem: We write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$s can be written into the unit squares?
Answer: 198
Problem: Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then how many students are at the table?
Answer: 11
Problem: How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different. Find the number of such different patterns.
Answer: 8
Problem: Which number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down, rows go right and left.)  
$$\begin{matrix}10&6&4&3&2\\11&7&14&10&8\\8&3&4&5&9\\13&4&15&12&1\\8&2&5&9&3\\\end{matrix}$$  
Fill in the blank with the correct number.
Answer: 7
Problem: What is the largest quotient that can be formed using two numbers chosen from the set $\{ -24, -3, -2, 1, 2, 8 \}$?
Answer: 12
Problem: Several students are competing in a series of three races. A student earns 5 points for winning a race, 3 points for finishing second, and 1 point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?
Answer: 13
Problem: An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, find the maximum number of students that can be seated for an exam.
Answer: 200
Problem: The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. Find the number of males in the band who are $NOT$ in the orchestra.
Answer: 10
Problem: A cube of edge $3 \text{ cm}$ is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, find the value of $N$.
Answer: 20
Problem: The digit-sum of $998$ is $9+9+8=26$. Find the number of $3$-digit whole numbers, whose digit-sum is $26$, that are even.
Answer: 1
Problem: The distance between the $5^\text{th}$ and $26^\text{th}$ exits on an interstate highway is $118$ miles. If any two consecutive exits are at least $5$ miles apart, find the largest number of miles there can be between two consecutive exits that are between the $5^\text{th}$ and $26^\text{th}$ exits.
Answer: 18
Problem: Eight $1\times 1$ square tiles are arranged as shown so their outside edges form a polygon with a perimeter of $14$ units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. What could be the perimeter of the new figure?
Answer: 18
Problem: Calculate the value of \(1000 \times 1993 \times 0.1993 \times 10\) and express your answer in the form of \(x^2\). What is the value of \(x\)?
Answer: 1993
Problem: To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, how many months would the supply of medicine last approximately if the answer is given in months?
Answer: 8
Problem: Consider the operation $*$ defined by the following table:
$\begin{array}{c|cccc} * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 3 & 1 & 4 & 2 \\ 4 & 4 & 3 & 2 & 1 \end{array}$
For example, $3*2=1$. Find the value of $(2*4)*(1*3)$.
Answer: 4
Problem: The nine squares in the table shown are to be filled so that every row and every column contains each of the numbers $1, 2, 3$. Find the value of $A + B$.
$$\begin{array}{|c|c|c|}\hline1&&\\\hline&2&\text{A}\\\hline&&\text{B}\\\hline\end{array}$$
Answer: 4
Problem: The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. Find the area of quadrilateral $ABDF$.
Answer: 320
Problem: Calculate the value of $(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193)$.
Answer: 167400
Problem: When $10^{93}-93$ is expressed as a single whole number, what is the sum of the digits?
Answer: 826
Problem: What number is directly above $142$ in this array of numbers?  
$$\begin{array}{cccccc}& & & 1 & &\\ & & 2 & 3 & 4 &\\ & 5 & 6 & 7 & 8 & 9\\ 10 & 11 & 12 &\cdots & &\\ \end{array}$$  
Find the number directly above $142$.
Answer: 120
Problem: Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\text{7:25 A.M.}$ and takes her lunch break at noon, at what time will her working day end? Please provide the answer in the format of AA:BB PM, where AA is the hour and BB is the minutes. Calculate the sum of AA and BB.
Answer: 14
Problem: For how many three-digit whole numbers does the sum of the digits equal $25$? Provide your answer as an integer.
Answer: 6
Problem: Two children at a time can play pairball. For $90$ minutes, with only two children playing at a time, five children take turns so that each one plays the same amount of time. How many minutes does each child play?
Answer: 36
Problem: Let \(W, X, Y,\) and \(Z\) be four different digits selected from the set  

\(\{ 1,2,3,4,5,6,7,8,9\}.\)  

Find the smallest possible value of the sum \(\frac{W}{X} + \frac{Y}{Z}\). The original answer is in the format \(\frac{m}{n}\). Please provide the value of \(m + n\).
Answer: 97
Problem: A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares. Provide your answer as an integer.
Answer: 6
Problem: Find the sum of the digits in the answer to  
  
$\underbrace{9999\cdots 99}_{94\mathrm{ nines}} \times \underbrace{4444\cdots 44}_{94\mathrm{ fours}}$  
  
where a string of $94$ nines is multiplied by a string of $94$ fours. What is the sum of the digits in the result?
Answer: 846
Problem: Which operation has the same effect on a number as multiplying by \(\dfrac{3}{4}\) and then dividing by \(\dfrac{3}{5}\)? Provide your answer in the form of a fraction \(\frac{k}{m}\), and find the value of \(k + m\).
Answer: 9
Problem: Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. What is the perimeter of $III$?
Answer: 36
Problem: At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home? Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and give the value of \(k + m\).
Answer: 23
Problem: Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $4$ and passes through points $P$ and $R$. Find the area of the rectangle.
Answer: 32
Problem: A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: The product of the month times the day equals the last two digits of the year. For example, $1956$ is a lucky year because it has the date $7/8/56$ and $7\times 8 = 56$. Find the year from the following list that is NOT a lucky year: 1990, 1991, 1992, 1993, 1994.
Answer: 1994
Problem: The table below gives the percent of students in each grade at Annville and Cleona elementary schools:  
$$  
\begin{array}{rccccccc} 
& \underline{\mathbf{K}} & \underline{\mathbf{1}} & \underline{\mathbf{2}} & \underline{\mathbf{3}} & \underline{\mathbf{4}} & \underline{\mathbf{5}} & \underline{\mathbf{6}} \\
\mathrm { Annville: } & 16 \% & 15 \% & 15 \% & 14 \% & 13 \% & 16 \% & 11 \% \\
\mathrm { Cleona: } & 12 \% & 15 \% & 14 \% & 13 \% & 15 \% & 14 \% & 17 \%
\end{array}
$$  
Annville has $100$ students and Cleona has $200$ students. In the two schools combined, what is the value of x if x\% of the students are in grade $6$?
Answer: 15
Problem: Diana and Apollo each roll a standard die obtaining a number at random from \(1\) to \(6\). What is the probability that Diana's number is larger than Apollo's number? Provide your answer as a fraction in simplest form \(\frac{k}{m}\), and then calculate the value of \(k + m\).
Answer: 17
Problem: How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different? Provide your answer as a whole number.
Answer: 1400
Problem: What is the smallest result that can be obtained from the following process?  
Choose three different numbers from the set $\{3,5,7,11,13,17\}$.  
Add two of these numbers.  
Multiply their sum by the third number.  
Find the smallest possible result.
Answer: 36
Problem: Six different digits from the set  
$\{ 1,2,3,4,5,6,7,8,9\}$ 
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is $23$ and the sum of the entries in the horizontal row is $12$.  
What is the sum of the six digits used?
Answer: 29
Problem: How many subsets containing three different numbers can be selected from the set  
$\{ 89,95,99,132,166,173 \}$  
so that the sum of the three numbers is even?
Answer: 12
Problem: Ahn chooses a two-digit integer, subtracts it from $200$, and doubles the result. What is the largest number Ahn can get? Provide your answer as an integer.
Answer: 380
Problem: Walter gets up at $6:30$ a.m., catches the school bus at $7:30$ a.m., has $6$ classes that last $50$ minutes each, has $30$ minutes for lunch, and has $2$ hours additional time at school. He takes the bus home and arrives at $4:00$ p.m. How many minutes has he spent on the bus?
Answer: 60
Problem: Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. Find the ratio of the area of the inscribed square to the area of the large square. The original answer is in \(\dfrac{k}{m}\) format, please give the value of k + m.
Answer: 14
Problem: A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. Find the total number of diagonals in a cube.
Answer: 16
Problem: If the product $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9$, find the sum of $a$ and $b$.
Answer: 35
Problem: A pair of $8$-sided dice have sides numbered $1$ through $8$. Each side has the same probability (chance) of landing face up. Find the probability that the product of the two numbers that land face-up exceeds $36$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 37
Problem: There are positive integers that have these properties:  
  
the sum of the squares of their digits is $50$, and  
each digit is larger than the one to its left.  
Find the product of the digits of the largest integer with both properties.
Answer: 36
Problem: All of the even numbers from $2$ to $98$ inclusive, excluding those ending in $0,$ are multiplied together. What is the rightmost digit (the units digit) of the product?
Answer: 6
Problem: Calculate the value of \(100 \times 19.98 \times 1.998 \times 1000\) and express your answer in the form of \(x^2\). What is the value of \(x\)?
Answer: 1998
Problem: Each of the letters $W$, $X$, $Y$, and $Z$ represents a different integer in the set $\{ 1,2,3,4\}$, but not necessarily in that order. If $\dfrac{W}{X} - \dfrac{Y}{Z}=1$, then what is the sum of $W$ and $Y$?
Answer: 7
Problem: Harry has $3$ sisters and $5$ brothers. His sister Harriet has $S$ sisters and $B$ brothers. What is the product of $S$ and $B$?
Answer: 12
Problem: An Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are $15$ students in the Science Club. How many students are in the Math Club?
Answer: 40
Problem: Tamika从集合\{8,9,10\}中随机选择两个不同的数字并将它们相加。Carlos从集合\{3,5,6\}中随机选择两个不同的数字并将它们相乘。求Tamika的结果大于Carlos的结果的概率。答案以\frac{k}{m}的形式表示，请给出k+m的值。
Answer: 13