meetforever/GRPO_CL huggingface dataset:
user: 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
user: Calculate the result of the expression $1990-1980+1970-1960+\cdots -20+10$.
Answer: 1000
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: What is the largest quotient that can be formed using two numbers chosen from the set $\{ -24, -3, -2, 1, 2, 8 \}$?
Answer: 12
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Calculate the value of $(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193)$.
Answer: 167400
user: When $10^{93}-93$ is expressed as a single whole number, what is the sum of the digits?
Answer: 826
user: 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
user: 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
user: For how many three-digit whole numbers does the sum of the digits equal $25$? Provide your answer as an integer.
Answer: 6
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.  
Rule $1$: If the integer is less than $10$, multiply it by $9$.  
Rule $2$: If the integer is even and greater than 9, divide it by $2$.   
Rule $3$: If the integer is odd and greater than $9$, subtract $5$ from it.   
$A$ sample sequence: $23, 18, 9, 81, 76, \cdots$.   
Find the value of the $98^\text{th}$ term of the sequence that begins $98, 49, \cdots$.
Answer: 27
user: If the pattern in the diagram continues, what fraction of the eighth triangle would be shaded? Provide your answer in the form \frac{k}{m}, where the fraction is in simplest form. Then, calculate the value of k + m.
Answer: 23
user: A rectangular board of $8$ columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered $1$ through $8$, row two is $9$ through $16$, and so on. A student shades square $1$, then skips one square and shades square $3$, skips two squares and shades square $6$, skips $3$ squares and shades square $10$, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
Answer: 120
user: Three generous friends, each with some money, redistribute the money as follows:  
Amy gives enough money to Jan and Toy to double each amount they have.  
Jan then gives enough to Amy and Toy to double their amounts.  
Finally, Toy gives enough to Amy and Jan to double their amounts.  
If Toy had $36$ dollars at the beginning and $36$ dollars at the end, what is the total amount that all three friends have?
Answer: 252
user: To make the statement $(6\ ?\ 3) + 4 - (2 - 1) = 5$ true, replace the question mark with the correct arithmetic operation. Convert the correct arithmetic operation to its ASCII code value and provide this value as your answer.
Answer: 47
user: Three flower beds overlap as shown. Bed $A$ has $500$ plants, bed $B$ has $450$ plants, and bed $C$ has $350$ plants. Beds $A$ and $B$ share $50$ plants, while beds $A$ and $C$ share $100$. What is the total number of plants?
Answer: 1150
user: Each of the five numbers $1, 4, 7, 10$, and $13$ is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. What is the largest possible value for the horizontal or vertical sum?
Answer: 24
user: Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\}$, the second from $\{A,I,O\}$, and the third from $\{D,M,N,T\}$.    
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of additional license plates that can be made by adding two letters?
Answer: 40
user: In a far-off land, three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? Provide your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers, and give the value of \(k + m\).
Answer: 11
user: Square $ABCD$ has sides of length $3$. Segments $CM$ and $CN$ divide the square's area into three equal parts. If the length of segment $CM$ is \( \sqrt{x} \), what is the value of \( x \)?
Answer: 13
user: Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then find the total area of the shaded triangles rounded to the nearest integer.
Answer: 6
user: In the year $2001$, the United States will host the International Mathematical Olympiad. Let $I, M,$ and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. What is the largest possible value of the sum $I + M + O$?
Answer: 671
user: The Fibonacci sequence $1,1,2,3,5,8,13,21,\cdots$ starts with two $1s$, and each term afterwards is the sum of its two predecessors. Find the last of the ten digits to appear in the units position of a number in the Fibonacci sequence.
Answer: 6
user: Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, find how many of the following four quantities change:  
  
$(a)$ the length of the segment $MN$  
  
$(b)$ the perimeter of $\triangle PAB$  
  
$(c)$ the area of $\triangle PAB$  
  
$(d)$ the area of trapezoid $ABNM$
Answer: 1
user: If $|x - 2| = p$, where $x < 2$, then find the value of $x - p$. The original answer is in the form $a - bp$, please provide the value of $a + b$.
Answer: 4
user: In rectangle \(ABCD\), \(AD=1\), \(P\) is on \(\overline{AB}\), and \(\overline{DB}\) and \(\overline{DP}\) trisect \(\angle ADC\). The perimeter of \(\triangle BDP\) is in the form \(k+\frac{m\sqrt{n}}{3}\), where \(k\), \(m\), and \(n\) are integers. Find the value of \(k + m + n\).
Answer: 9
user: What is the minimum possible product of three different numbers from the set \(\{-8,-6,-4,0,3,5,7\}\)? The original answer is in the form of -k, please give the value of k.
Answer: 280
user: Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? Provide your answer as an integer.
Answer: 20201
user: Three dice with faces numbered $1$ through $6$ are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). Find the total number of dots NOT visible in this view.
Answer: 41
user: Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walters entered?
Answer: 80
user: The point $P = (1,2,3)$ is reflected in the $xy$-plane, then its image $Q$ is rotated by $180^\circ$ about the $x$-axis to produce $R$, and finally, $R$ is translated by 5 units in the positive-$y$ direction to produce $S$. What are the coordinates of $S$? Provide your answer in the form $(x, y, z)$, and calculate the value of x + y + z.
Answer: 7
user: The sides of a triangle with positive area have lengths $4$, $6$, and $x$. The sides of a second triangle with positive area have lengths $4$, $6$, and $y$. Find the smallest positive number that is not a possible value of $|x-y|$.
Answer: 8
user: Two non-zero real numbers, $a$ and $b,$ satisfy $ab = a - b$. Calculate the value of the expression $\frac {a}{b} + \frac {b}{a} - ab$ and provide the result as an integer.
Answer: 2
user: The number $64$ has the property that it is divisible by its unit digit. Find how many whole numbers between 10 and 50 have this property.
Answer: 17
user: Let A, M, and C be nonnegative integers such that A + M + C = 12. What is the maximum value of \( A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C \)?
Answer: 112
user: A block wall $100$ feet long and $7$ feet high will be constructed using blocks that are $1$ foot high and either $2$ feet long or $1$ foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?
Answer: 353
user: Segments $AD=10$, $BE=6$, $CF=24$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle, find the value of $x$. The original answer is in \frac{x}{y} format, please give the value of x + y.
Answer: 43
user: There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg to be placed on a triangular peg board. Find the number of ways the pegs can be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color.
Answer: 1
user: One morning each member of Angela's family drank an $8$-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
Answer: 5
user: 在三角形 $CAT$ 中，$\angle ACT =\angle ATC$ 且 $\angle CAT = 36^\circ$。如果 $\overline{TR}$ 平分 $\angle ATC$，求 $\angle CRT$ 的度数。如果答案是 $x^\circ$，求 $x$ 的值。
Answer: 72
user: Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. The original answer is in the form -\frac{k}{m}, where the fraction is in simplest terms. Please provide the value of k + m.
Answer: 10
user: Triangles $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB = 4$, find the perimeter of figure $ABCDEFG$.
Answer: 15
user: A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$, the second row $18,19,\ldots,34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$, the second column $14,15,\ldots,26$ and so on across the board, find the sum of the numbers in the squares that have the same numbers in both numbering systems.
Answer: 555