ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Find the number of quadruples $(a,b,c,d)$ of integers which satisfy both \begin{align*}\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} &= \frac{1}{2}\qquad\text{and}\\\\2(a+b+c+d) &= ab + cd + (a+b)(c+d) + 1.\end{align*}
Answer: 48
Problem: A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise?
Answer: 77
Problem: The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $165$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
Answer: 30
Problem: Functions $f$ and $g$ are defined so that $f(1) = 4$, $g(1) = 9$, and for each integer $n \ge 1$, $f(n+1) = 2f(n) + 3g(n) + 2n $ and $g(n+1) = 2g(n) + 3 f(n) + 5$. Find $f(2005) - g(2005)$.
Answer: 1999
Problem: Let $T$ be a $30-60-90$ triangle with hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.
Answer: 50
Problem: We have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers.
Answer: 120
Problem: At the beginning of each hour from $1$ o’clock AM to $12$ NOON and from $1$ o’clock PM to $12$ MIDNIGHT a coo-coo clock’s coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock’s coo-coo bird coo-coos a single time at $30$ minutes past each hour. How many times does the coo-coo bird coo-coo from $12:42$ PM on Monday until $3:42$ AM on Wednesday?
Answer: 279
Problem: Find the sum of all the prime numbers less than $100$ which are one more than a multiple of six.
Answer: 527
Problem: $12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed?
Answer: 138600
Problem: Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$. Find the sum of all the possible values of $n$ in these ordered pairs.
Answer: 237
Problem: In triangle $ABC$, $AB = 52$, $BC = 56$, $CA = 60$. Let $D$ be the foot of the altitude from $A$ and $E$ be the intersection of the internal angle bisector of $\angle BAC$ with $BC$. Find $DE$.
Answer: 6
Problem: The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. This continued through the day when they caught more squirrels than rabbits. Up through that day how many rabbits did they catch in all?
Answer: 5491
Problem: At a movie theater tickets for adults cost $4$ dollars more than tickets for children. One afternoon the theater sold $100$ more child tickets than adult tickets for a total sales amount of $1475$ dollars. How much money would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?
Answer: 1875
Problem: Consider the polynomials \begin{align*}P(x) &= (x + \sqrt{2})(x^2 - 2x + 2)\\Q(x) &= (x - \sqrt{2})(x^2 + 2x + 2)\\R(x) &= (x^2 + 2)(x^8 + 16).\end{align*} Find the coefficient of $x^4$ in $P(x)\cdot Q(x)\cdot R(x)$.
Answer: 0
Problem: We draw a radius of a circle. We draw a second radius $23$ degrees clockwise from the first radius. We draw a third radius $23$ degrees clockwise from the second. This continues until we have drawn $40$ radii each $23$ degrees clockwise from the one before it. What is the measure in degrees of the smallest angle between any two of these $40$ radii?
Answer: 7
Problem: Michael is celebrating his fifteenth birthday today. How many Sundays have there been in his lifetime?
Answer: 783
Problem: In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ satisfying $x < y < z$ ?
Answer: 784
Problem: Find the sum of all positive integers less than $2006$ which are both multiples of six and one more than a multiple of seven.
Answer: 47094
Problem: We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference of the circle rotated $75$ degrees clockwise from the first triangle so that it shares a side with the first triangle. We draw a third, fourth, and fifth such triangle each rotated $75$ degrees clockwise from the previous triangle. The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by the intersection?
Answer: 120
Problem: There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?
Answer: 225
Problem: Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\{ 2, 3, 5, 7 \}$.
Answer: 1932
Problem: Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$.
Answer: 168
Problem: Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$.
Answer: 9
Problem: Heather and Kyle need to mow a lawn and paint a room. If Heather does both jobs by herself, it will take her a total of nine hours. If Heather mows the lawn and, after she finishes, Kyle paints the room, it will take them a total of eight hours. If Kyle mows the lawn and, after he finishes, Heather paints the room, it will take them a total of seven hours. If Kyle does both jobs by himself, it will take him a total of six hours. It takes Kyle twice as long to paint the room as it does for him to mow the lawn. The number of hours it would take the two of them to complete the two tasks if they worked together to mow the lawn and then worked together to paint the room is a fraction $\tfrac{m}{n}$where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 41
Problem: At a movie theater tickets for adults cost $4$ dollars more than tickets for children. One afternoon the theater sold $100$ more child tickets than adult tickets for a total sales amount of $1475$ dollars. How many dollars would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?
Answer: 1875
Problem: Let $F_0 = 0, F_{1} = 1$, and for $n \ge 1, F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$ . Then there are rational numbers $A$ and $B$ such that $\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}$. Find $A + B$.
Answer: 34
Problem: Square $ABCD$ has side length $36$.  Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$.  Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$.
Answer: 144
Problem: Tom can run to Beth's house in $63$ minutes.  Beth can run to Tom's house in $84$ minutes.  At noon Tom starts running from his house toward Beth's house while at the same time Beth starts running from her house toward Tom's house.  When they meet, they both run at Beth's speed back to Beth's house.  At how many minutes after noon will they arrive at Beth's house?
Answer: 72
Problem: If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list.
Answer: 240
Problem: 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
Problem: A dart board looks like three concentric circles with radii of 4, 6, and 8.  Three darts are thrown at the board so that they stick at three random locations on then board.  The probability that one dart sticks in each of the three regions of the dart board is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers.   Find $m + n$.
Answer: 617
Problem: Six chairs sit in a row.  Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left.  There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right.  The probability that this will [b]not[/b] happen is given by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 1106
Problem: Let $S$ be the graph of $y=x^3$, and $T$ be the graph of $y=\sqrt[3]{y}$. Let $S^*$ be $S$ rotated around the origin $15$ degrees clockwise, and $T^*$ be T rotated around the origin 45 degrees counterclockwise. $S^*$ and $T^*$ will intersect at a point in the first quadrant a distance $M+\sqrt{N}$ from the origin where $M$ and $N$ are positive integers. Find $M+N$.
Answer: 16
Problem: For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers.  Find the largest value for $a+b$.
Answer: 125
Problem: A bowl contained $10\%$ blue candies and $25\%$ red candies.  A bag containing three quarters red candies and one quarter blue candies was added to the bowl.  Now the bowl is $16\%$ blue candies.  What percentage of the candies in the bowl are now red?
Answer: 45
Problem: Penelope plays a game where she adds $25$ points to her score each time she wins a game and deducts $13$ points from her score each time she loses a game.  Starting with a score of zero, Penelope plays $m$ games and has a total score of $2007$ points.  What is the smallest possible value for $m$?
Answer: 97
Problem: How many numbers $n$ have the property that both $\frac{n}{2}$ and $2n$ are four digits whole numbers?
Answer: 1500
Problem: Two circles with radius $2$ and radius $4$ have a common center at P.  Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle.  Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$.
Answer: 192
Problem: Let $x$ be a positive integer less than $200$, and let $y$ be obtained by writing the base 10 digits of $x$ in reverse order. Given that $x$ and $y$ satisfy $11x^2+363y=7xy+6571$, find $x$.
Answer: 133
Problem: $F(0)=3$ and $F(n)=F(n-1)+4$ when $n$ is positive.  Find $F(F(F(5)))$.
Answer: 383
Problem: The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$.  We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$.  If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$.  We continually apply this permutation to each $T_m$ to get $T_{m+1}$.  Find the smallest positive integer $n$ so that $T_n=T_0$.
Answer: 252
Problem: We have some identical paper squares which are black on one side of the sheet and white on the other side.  We can join nine squares together to make a $3$ by $3$ sheet of squares by placing each of the nine squares either white side up or black side up.  Two of these $3$ by $3$ sheets are distinguishable if neither can be made to look like the other by rotating the sheet or by turning it over.  How many distinguishable $3$ by $3$ squares can we form?
Answer: 70
Problem: Find the maximum possible value of $8\cdot 27^{\log_6 x}+27\cdot 8^{\log_6 x}-x^3$ as $x$ varies over the positive real numbers.
Answer: 216
Problem: You know that the Jones family has five children, and the Smith family has three children.  Of the eight children you know that there are five girls and three boys.  Let $\dfrac{m}{n}$  be the probability that at least one of the families has only girls for children.  Given that $m$ and $n$ are relatively prime positive integers,  find $m+ n$.
Answer: 67
Problem: What is the greatest positive integer $m$ such that $ n^2(1+n^2-n^4)\equiv 1\pmod{2^m} $ for all odd integers $n$?
Answer: 7
Problem: The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference.  Find their sum.
Answer: 770
Problem: Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class.  Each year at the college one third of the freshman class drops out of school, $40$ students in the sophomore class drop out of school, and one tenth of the junior class drops out of school.  Given that the college only admits new freshman students, and that it wants to begin each school year with $3400$ students enrolled, how many students does it need to admit into the freshman class each year?
Answer: 1185
Problem: Evaluate the sum \[1^2+2^2-3^2-4^2+5^2+6^2-7^2-8^2+\cdots-1000^2+1001^2\]
Answer: 1001
Problem: Starting with a sequence of $n 1's$, you can insert plus signs to get various sums. For example, when $n = 10$,  you can get the sum $1 + 1 + 1 + 11 + 11 + 111 = 136$, and the sum $1 + 1 + 11 + 111 + 111 = 235$.  Find the number of values of $n$ so that the sum of $1111$ is possible.
Answer: 121
Problem: Three congruent ellipses are mutually tangent. Their major axes are parallel.  Two of the ellipses are tangent at the end points of their minor axes as shown. The distance between the centers of these two ellipses is $4$. The distances from those two centers to the center of the third ellipse are both $14$.  There are positive integers m and n so that the area between these three ellipses is $\sqrt{n}-m \pi$. Find $m+n$.
Answer: 776
Problem: Last Sunday at noon the date on the calendar was 15 (April 15, 2007).  What will be the date on the calendar one million minutes after that time?
Answer: 9
Problem: The four sets A, B, C, and D each have $400$ elements.  The intersection of any two of the sets has $115$ elements.  The intersection of any three of the sets has $53$ elements.  The intersection of all four sets has $28$ elements.  How many elements are there in the union of the four sets?
Answer: 1094
Problem: Let $a=3^{1/223}+1$ and for all $n \ge 3$ let \[f(n)= \dbinom{n}{0} a^{n-1} - \dbinom{n}{1} a^{n-2} + \dbinom{n}{2} a^{n-3}- ... +(-1)^{n-1} \dbinom{n}{n-1} a^0.\] Find  $f(2007)+f(2008).$
Answer: 19683
Problem: A city is laid out with a rectangular grid of roads with 10 streets numbered from 1 to 10 running east-west and 16 avenues numbered from 1 to 16 running northsouth.  All streets end at First and Sixteenth Avenues, and all avenues end at First and Tenth Streets.  A rectangular city park is bounded on the north and south by Sixth Street and Eighth Street, and bounded on the east and west by Fourth Avenue and Twelfth Avenue. Although there are no breaks in the roads that bound the park, no road goes through the park.  The city paints a crosswalk from every street corner across any adjacent road.  Thus, where two roads cross such as at Second Street and Second Avenue, there are four crosswalks painted, while at corners such as First Street and First Avenue, there are only two crosswalks painted.  
How many crosswalks are there painted on the roads of this city?
Answer: 544
Problem: Each of the distinct letters in the following subtraction problem represents a different digit. Find the number represented by the word [b]TEAM[/b]

[size=150][b]
  PURPLE
- COMET
________
[color=#FFFFFF].....[/color]TEAM [/b][/size]
Answer: 6852
Problem: Find the sum of all the integers  $N  >  1$  with the properties that the each prime factor of  $N $ is either  $2, 3,$ or $5,$ and  $N$  is not divisible by any perfect cube greater than $1.$
Answer: 2820
Problem: If you roll six fair dice, let $\mathsf{ p}$ be the probability that exactly five different numbers appear on the upper faces of the six dice.  If  $\mathsf{p} = \frac{m}{n}$ where $ m $ and  $n$ are relatively prime positive integers, find $m+n.$
Answer: 133
Problem: Find the least integer  $n$ greater than $345$ such that $\frac{3n+4}{5},  \frac{4n+5}{3},$ and $\frac{5n+3}{4}$ are all integers.
Answer: 397
Problem: Find the sum of all the digits in the decimal representations of all the positive integers less than $1000.$
Answer: 13500
Problem: When Tim was Jim’s age, Kim was twice as old as Jim.  When Kim was Tim’s age, Jim was 30.  When Jim becomes Kim’s age, Tim will be 88.  When Jim becomes Tim’s age, what will be the sum of the ages of Tim, Jim, and Kim?
Answer: 221
Problem: Let $S$ S be a point chosen at random from the interior of the square $ABCD$, which has side $AB$ and diagonal $AC$.  Let $P$ be the probability that the segments $AS$, $SB$, and $AC$ are congruent to the sides of a triangle.  Then $P$ can be written as $\dfrac{a-\pi\sqrt{b}-\sqrt{c}}{d}$ where $a,b,c,$ and $d$ are all positive integers and $d$ is as small as possible.  Find $ab+cd$.
Answer: 160
Problem: A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon $5$ centimeters of rain falls in the open field partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.
Answer: 60
Problem: A canister contains two and a half cups of flour.  Greg and Sally have a brownie recipe which calls for one and one third cups of flour.  Greg and Sally want to make one and a half recipes of brownies.  To the nearest whole percent, what percent of the flour in the canister would they use?
Answer: 80
Problem: A 16 ×16 square sheet of paper is folded once in half horizontally and once in half vertically to make an 8 × 8 square.  This square is again folded in half twice to make a 4 × 4 square.  This square is folded in half twice to make a 2 × 2 square. This square is folded in half twice to make a 1 × 1 square.  Finally, a scissor is used to make cuts through both diagonals of all the layers of the 1 × 1 square.  How many pieces of paper result?
Answer: 544
Problem: We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday.  On the first Monday the number we record is ten and a half.  On every Tuesday and every Thursday the number we record is one third of what it was on the previous workday.  On every Monday, Wednesday, and Friday the number we record is double what it was on the previous workday.  This will go on forever. What is the sum of all of the numbers we will record?
Answer: 252
Problem: Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits.
Answer: 313
Problem: Starting on April 15, 2008, you can go one day backward and one day forwards to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer k so that if you go k days backward and k days forward you get two calendar dates that are the same.
Answer: 76
Problem: Let $@$ be a binary operation on the natural numbers satisfying the properties that, for all  $a, b,$ and $c$, $(a + b)@c  = (a@c) + (b@c)$ and $a@(b + c) = (a@b)@c. $  Given that  $5@5=160$,   find the value of  $7@7$
Answer: 896
Problem: Ralph is standing along a road which heads straight east.  If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike.  At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour.  Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point.   Let $M$ and $N$ be relatively prime positive integers such that $\frac{M}{N}$ is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.
Answer: 29
Problem: For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$. 
Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.
Answer: 2000000000000000000000
Problem: The area of the quadrilateral with vertices at the four points in three dimensional space $(0,0,0)$, $(2,6,1)$, $(-3,0,3)$ and $(-4,2,5)$ is the number $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers.  Find $m+n$.
Answer: 53
Problem: Three friends who live in the same apartment building leave the building at the same time to go rock climbing on a cliff just outside of town.  Susan walks to the cliff and climbs to the top of the cliff.  Fred runs to the cliff twice as fast as Susan walks and climbs the top of the cliff at a rate that is only two-thirds as fast as Susan climbs.  Ralph bikes to the cliff at a speed twice as fast as Fred runs and takes two hours longer to climb to the top of the cliff than Susan does.  If all three friends reach the top of the cliff at the same time, how many minutes after they left home is that?
Answer: 320
Problem: Let $A_1A_2A_3...A_{12}$ be a regular dodecagon. Find the number of right triangles whose vertices are in the set ${A_1A_2A_3...A_{12}}$
Answer: 60
Problem: What is the least positive integer with the property that the product of its digits is $9! ?$
Answer: 2578899
Problem: There are three men and eleven women taking a dance class.  In how many different ways can each man be paired with a woman partner and then have the eight remaining women be paired into four pairs of two?
Answer: 103950
Problem: At Mallard High School there are three intermural sports leagues: football, basketball, and baseball.  There are 427 students participating in these sports: 128 play on football teams, 291 play on basketball teams, and 318 play on baseball teams.  If exactly 36 students participate in all three of the sports, how many students participate in exactly two of the sports?
Answer: 238