data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Cut a 12cm long thin iron wire into three segments with lengths a, b, and c, \n(1) Find the maximum volume of the rectangular solid with lengths a, b, and c as its dimensions; \n(2) If these three segments each form an equilateral triangle, find the minimum sum of the areas o... | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {3}}{3}"
} | {
"index": 7300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cut a 3-meter-long rope into 7 equal segments. Each segment accounts for \\_\\_\\_\\_\\_\\_ of the total length, and each segment is \\_\\_\\_\\_\\_\\_ meters long."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"index": 7301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cut a cube into two cuboids. If the ratio of their surface areas is 1:2, what is the ratio of their volumes?"
}
] | math | {
"style": "rule",
"ground_truth": "1:5"
} | {
"index": 7302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cutting equilateral triangle $BFC$ out of square $ABCD$ and translating it to the left of the square creates the shaded figure $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. What is the perimeter, in inches, of $ABFCDE$?\n\n[asy]\nfilldraw((0,0)--(sqrt(3),1)--(sqrt(3)+... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 7303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cybil and Ronda are sisters. The 10 letters from their names are placed on identical cards so that each of 10 cards contains one letter. Without replacement, two cards are selected at random from the 10 cards. What is the probability that one letter is from each sister's name? ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 7304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cyclic pentagon $ABCDE$ has a right angle $\\angle ABC=90^{\\circ}$ and side lengths $AB=15$ and $BC=20$. Supposing that $AB=DE=EA$, find $CD$."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 7305,
"solution": "By Pythagoras, $AC=25$. Since $\\overline{AC}$ is a diameter, angles $\\angle ADC$ and $\\angle AEC$ are also right, so that $CE=20$ and $AD^{2}+CD^{2}=AC^{2}$ as well. Beginning with Ptolemy's theorem, $$\\begin{aligned} & (AE \\cdot CD+AC \\cdot DE)^{2}=AD^{2} \\cdot EC^{2}=\\left(AC... |
DeepScaleR | [
{
"role": "user",
"content": "Cyclic pentagon $ABCDE$ has side lengths $AB=BC=5, CD=DE=12$, and $AE=14$. Determine the radius of its circumcircle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{225\\sqrt{11}}{88}"
} | {
"index": 7306,
"solution": "Let $C^{\\prime}$ be the point on minor arc $BCD$ such that $BC^{\\prime}=12$ and $C^{\\prime}D=5$, and write $AC^{\\prime}=BD=C^{\\prime}E=x, AD=y$, and $BD=z$. Ptolemy applied to quadrilaterals $ABC^{\\prime}D, BC^{\\prime}DE$, and $ABDE$ gives $$\\begin{aligned} & x^{2}=12y+5^{2} \\... |
DeepScaleR | [
{
"role": "user",
"content": "Cyclic pentagon \\( A B C D E \\) has a right angle \\( \\angle A B C = 90^\\circ \\) and side lengths \\( A B = 15 \\) and \\( B C = 20 \\). Supposing that \\( A B = D E = E A \\), find \\( C D \\)."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cyclic quadrilateral $ABCD$ satisfies $\\angle ABD = 70^\\circ$ , $\\angle ADB=50^\\circ$ , and $BC=CD$ . Suppose $AB$ intersects $CD$ at point $P$ , while $AD$ intersects $BC$ at point $Q$ . Compute $\\angle APQ-\\angle AQP$ ."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cyclic quadrilateral $A B C D$ has side lengths $A B=1, B C=2, C D=3$ and $D A=4$. Points $P$ and $Q$ are the midpoints of $\\overline{B C}$ and $\\overline{D A}$. Compute $P Q^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{116}{35}"
} | {
"index": 7309,
"solution": "Construct $\\overline{A C}, \\overline{A Q}, \\overline{B Q}, \\overline{B D}$, and let $R$ denote the intersection of $\\overline{A C}$ and $\\overline{B D}$. Because $A B C D$ is cyclic, we have that $\\triangle A B R \\sim \\triangle D C R$ and $\\triangle A D R \\sim \\triangle B C... |
DeepScaleR | [
{
"role": "user",
"content": "Cyclic quadrilateral \\(ABCD\\) has side lengths \\(AB = 1\\), \\(BC = 2\\), \\(CD = 3\\), and \\(AD = 4\\). Determine \\(\\frac{AC}{BD}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "5/7"
} | {
"index": 7310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cylinder $B$'s height is equal to the radius of cylinder $A$ and cylinder $B$'s radius is equal to the height $h$ of cylinder $A$. If the volume of cylinder $A$ is twice the volume of cylinder $B$, the volume of cylinder $A$ can be written as $N \\pi h^3$ cubic units. What is t... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cylinder $C$'s height is equal to the diameter of cylinder $D$ and cylinder $C$'s diameter is equal to the height $h$ of cylinder $D$. If the volume of cylinder $D$ is three times the volume of cylinder $C$, the volume of cylinder $D$ can be written as $M \\pi h^3$ cubic units.... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"index": 7312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cynthia and Lynnelle are collaborating on a problem set. Over a $24$ -hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$ -hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during... | math | {
"style": "rule",
"ground_truth": "4/9"
} | {
"index": 7313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she m... | math | {
"style": "rule",
"ground_truth": "469"
} | {
"index": 7314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cátia leaves school every day at the same time and returns home by bicycle. When she pedals at $20 \\mathrm{~km/h}$, she arrives home at $4:30$ PM. If she pedals at $10 \\mathrm{~km/h}$, she arrives home at $5:15$ PM. At what speed should she pedal to arrive home at $5:00$ PM?"... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \\mathrm{~km} / \\mathrm{h}$, she arrives home at $16 \\mathrm{~h} 30 \\mathrm{~m}$. If she pedals at $10 \\mathrm{~km} / \\mathrm{h}$, she arrives home at $17 \\mathrm{~h} 15 \\ma... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \\mathrm{~km} / \\mathrm{h}$, she arrives home at $4:30$ PM. If she pedals at $10 \\mathrm{~km} / \\mathrm{h}$, she arrives home at $5:15$ PM. At what speed should she pedal to arr... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Célia wants to trade with Guilherme stickers from an album about Brazilian animals. Célia wants to trade four butterfly stickers, five shark stickers, three snake stickers, six parakeet stickers, and six monkey stickers. All of Guilherme's stickers are of spiders. They know tha... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"index": 7318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dad is $a$ years old this year, which is 4 times plus 3 years more than Xiao Hong's age this year. Xiao Hong's age expressed in an algebraic expression is ____. If Xiao Hong is 7 years old this year, then Dad's age is ____ years old."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 7319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dad says he is exactly 35 years old, not counting weekends. How old is he really?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 7320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dan is walking down the left side of a street in New York City and must cross to the right side at one of 10 crosswalks he will pass. Each time he arrives at a crosswalk, however, he must wait $t$ seconds, where $t$ is selected uniformly at random from the real interval $[0,60]... | math | {
"style": "rule",
"ground_truth": "60\\left(1-\\left(\\frac{1}{10}\\right)^{\\frac{1}{9}}\\right)"
} | {
"index": 7321,
"solution": "With probability $\\left(1-\\frac{k}{60}\\right)^{9}$, Dan reaches the last crosswalk without crossing at any previous site, in which case the expected value of his wait time is 30 seconds. Otherwise, with probability $1-\\left(1-\\frac{k}{60}\\right)^{9}$, Dan crosses at an earlier cr... |
DeepScaleR | [
{
"role": "user",
"content": "Dana goes first and Carl's coin lands heads with probability $\\frac{2}{7}$, and Dana's coin lands heads with probability $\\frac{3}{8}$. Find the probability that Carl wins the game."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{31}"
} | {
"index": 7322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dani wrote the integers from 1 to \\( N \\). She used the digit 1 fifteen times. She used the digit 2 fourteen times.\nWhat is \\( N \\) ?"
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 7323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM ,each mango costs $4$ RM , and each papaya costs $5$ RM . He bought at least one of each fruit, and paid exactly $50$ RM. What is the maximum number of fruits that he could hav... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a $3$-digit number with $a\\ge1$ and $a+b+c\\le7$. At the end of the trip, the odometer showed ... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 7325,
"solution": "1. **Set up the equation based on the problem statement:**\n Danica drove for $x$ hours at a speed of $55$ miles per hour. The odometer reading at the start was $abc$ and at the end was $cba$. In terms of digits, this translates to:\n \\[\n 100a + 10b + c \\text{ at the start and... |
DeepScaleR | [
{
"role": "user",
"content": "Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a \\geq 1$ and $a+b+c \\leq 7$. At the end of the trip, the odometer showe... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 7326,
"solution": "1. **Understanding the Problem:**\n Danica drove her car for a whole number of hours at an average speed of 55 miles per hour. The odometer readings at the start and end of the trip are given as $abc$ and $cba$, respectively, where $abc$ and $cba$ are three-digit numbers formed by di... |
DeepScaleR | [
{
"role": "user",
"content": "Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the greatest number of additional cars she must buy in order to be able to arrange all her cars this way?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 7327,
"solution": "1. **Identify the Requirement**: Danica wants to arrange her model cars in rows with exactly 6 cars in each row. This means that the total number of cars must be a multiple of 6.\n\n2. **Current Number of Cars**: She currently has 23 model cars.\n\n3. **Finding the Next Multiple of 6**... |
DeepScaleR | [
{
"role": "user",
"content": "Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a $60 \\%$ chance of winning each point, what is the probability that he will win the game?... | math | {
"style": "rule",
"ground_truth": "9 / 13"
} | {
"index": 7328,
"solution": "Consider the situation after two points. Daniel has a $9 / 25$ chance of winning, Scott, $4 / 25$, and there is a $12 / 25$ chance that the players will be tied. In the latter case, we revert to the original situation. In particular, after every two points, either the game returns to t... |
DeepScaleR | [
{
"role": "user",
"content": "Daniel worked for 50 hours per week for 10 weeks during the summer, earning \\$6000. If he wishes to earn an additional \\$8000 during the school year which lasts for 40 weeks, how many fewer hours per week must he work compared to the summer if he receives the same hourly wage... | math | {
"style": "rule",
"ground_truth": "33.33"
} | {
"index": 7329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Daniel works at an electronics store, and he claims that the popularity of a toaster (measured in number of sales) is inversely proportional to its cost. If 12 customers buy a toaster that costs $\\$500$, according to Daniel's theory, how many customers would buy a toaster that... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 7330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's li... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Daniel wrote all the positive integers from 1 to $n$ inclusive on a piece of paper. After careful observation, he realized that the sum of all the digits that he wrote was exactly 10,000. Find $n$."
}
] | math | {
"style": "rule",
"ground_truth": "799"
} | {
"index": 7332,
"solution": "Let $S(n)$ denote the sum of the digits of $n$, and let $f(x)=\\sum_{n=0}^{x} S(n)$. (We may add $n=0$ because $S(0)=0$.) Observe that: $$f(99)=\\sum_{a=0}^{9}\\left(\\sum_{b=0}^{9}(a+b)\\right)=10 \\sum_{b=0}^{9} b+10 \\sum_{a=0}^{9} a=900$$ If $a$ is an integer between 1 and 9 inclus... |
DeepScaleR | [
{
"role": "user",
"content": "Danielle Bellatrix Robinson is organizing a poker tournament with 9 people. The tournament will have 4 rounds, and in each round the 9 players are split into 3 groups of 3. During the tournament, each player plays every other player exactly once. How many different ways can Dan... | math | {
"style": "rule",
"ground_truth": "20160"
} | {
"index": 7333,
"solution": "We first split the 9 people up arbitrarily into groups of 3. There are $\\frac{\\binom{9}{3}\\binom{6}{3}\\binom{3}{3}}{3!}=280$ ways of doing this. Without loss of generality, label the people 1 through 9 so that the first round groups are $\\{1,2,3\\},\\{4,5,6\\}$, and $\\{7,8,9\\}$.... |
DeepScaleR | [
{
"role": "user",
"content": "Danielle picks a positive integer $1 \\leq n \\leq 2016$ uniformly at random. What is the probability that \\operatorname{gcd}(n, 2015)=1?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1441}{2016}"
} | {
"index": 7334,
"solution": "We split the interval $[1,2016]$ into $[1,2015]$ and 2016. The number of integers in $[1,2015]$ that are relatively prime to 2015 is $\\phi(2015)=\\frac{4}{5} \\cdot \\frac{12}{13} \\cdot \\frac{30}{31} \\cdot 2015=1440$. Also, 2016 is relatively prime to 2015, so there are a total of ... |
DeepScaleR | [
{
"role": "user",
"content": "Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 7335,
"solution": "1. **Identify the Periodicity**: \n Alice, Beatrix, and Claire visit every 3, 4, and 5 days respectively. The least common multiple (LCM) of 3, 4, and 5 is:\n \\[\n \\text{LCM}(3, 4, 5) = 60\n \\]\n This means every 60 days, all three friends visit Daphne together.\n\n2. **Di... |
DeepScaleR | [
{
"role": "user",
"content": "Darren has borrowed $100$ clams from Ethan at a $10\\%$ simple daily interest. Meanwhile, Fergie has borrowed $150$ clams from Gertie at a $5\\%$ simple daily interest. In how many days will Darren and Fergie owe the same amounts, assuming that they will not make any repayments... | math | {
"style": "rule",
"ground_truth": "20\\text{ days}"
} | {
"index": 7336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Darryl has a six-sided die with faces $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face\ncomes up with probability $1/2$ and the other five faces have equal probability of coming up. He\nunfortunately does not know which side is weighted, but he knows each f... | math | {
"style": "rule",
"ground_truth": "3/26"
} | {
"index": 7337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dasha added 158 numbers and obtained a sum of 1580. Then Sergey tripled the largest of these numbers and decreased another number by 20. The resulting sum remained unchanged. Find the smallest of the original numbers."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at ran... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 7339,
"solution": "There are $12 \\cdot 11 = 132$ possible situations ($12$ choices for the initially assigned gate, and $11$ choices for which gate Dave's flight was changed to). We are to count the situations in which the two gates are at most $400$ feet apart.\n If we number the gates $1$ through $12$... |
DeepScaleR | [
{
"role": "user",
"content": "Dave rode 30 miles at 10 miles per hour and 10 miles at 30 miles per hour. What was his average speed, in miles per hour, for the entire ride?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\\dfrac mn$ is the probability that the number of tim... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 7341,
"solution": "Let $p$ be the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. (We will call this event \"a win\", and the opposite event will be \"a loss\".)\n Let both players roll their first die.\n With probability $\\fr... |
DeepScaleR | [
{
"role": "user",
"content": "Dave walks to school and averages 85 steps per minute, with each step being 80 cm long. It now takes him 15 minutes to get to school. Jack, walking the same route to school, takes steps that are 72 cm long and averages 104 steps per minute. Find the time it takes Jack to reach ... | math | {
"style": "rule",
"ground_truth": "13.62"
} | {
"index": 7342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dave's sister Amy baked $4$ dozen pies. Among these:\n- $5/8$ of them contained chocolate.\n- $3/4$ of them contained marshmallows.\n- $2/3$ of them contained cayenne.\n- $1/4$ of them contained salted soy nuts.\nAdditionally, all pies with salted soy nuts also contained marshm... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 7343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dave's sister baked $3$ dozen pies of which half contained chocolate, two thirds contained marshmallows, three-fourths contained cayenne, and one-sixths contained salted soy nuts. What is the largest possible number of pies that had none of these ingredients?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 7344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "David and Evan are playing a game. Evan thinks of a positive integer $N$ between 1 and 59, inclusive, and David tries to guess it. Each time David makes a guess, Evan will tell him whether the guess is greater than, equal to, or less than $N$. David wants to devise a strategy t... | math | {
"style": "rule",
"ground_truth": "36440"
} | {
"index": 7345,
"solution": "We can represent each strategy as a binary tree labeled with the integers from 1 to 59, where David starts at the root and moves to the right child if he is too low and to the left child if he is too high. Our tree must have at most 6 layers as David must guess at most 5 times. Once Da... |
DeepScaleR | [
{
"role": "user",
"content": "David and Evan each repeatedly flip a fair coin. David will stop when he flips a tail, and Evan will stop once he flips 2 consecutive tails. Find the probability that David flips more total heads than Evan."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 7346,
"solution": "Solution 1: We can find the values of the functions $D(h)$ and $E(h)$, the probabilities that David and Evan, respectively, flip exactly $h$ heads. It is easy to see that $D(h)=2^{-h-1}$. In order to find $E(h)$, we note that each sequence must end with the flips HTT (unless Evan flips... |
DeepScaleR | [
{
"role": "user",
"content": "David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ m... | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 7347,
"solution": "Let's denote the total distance from David's home to the airport as $d$ miles. According to the problem, David drives the first 35 miles in one hour. If he continues at this speed for the entire journey, he would be 1 hour late. This means that the total time required to travel at 35 m... |
DeepScaleR | [
{
"role": "user",
"content": "David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\\text{ }B,\\text{ }C$, which can each be inscribed in a circle with radius $1$. Let $\\varphi_A$ denote the measure of the acute angle made by the diagona... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 7348,
"solution": "Let the sides of the quadrilaterals be $a,b,c,$ and $d$ in some order such that $A$ has $a$ opposite of $c$, $B$ has $a$ opposite of $b$, and $C$ has $a$ opposite of $d$. Then, let the diagonals of $A$ be $e$ and $f$. Similarly to solution $2$, we get that $\\tfrac{2}{3}(ac+bd)=\\tfrac... |
DeepScaleR | [
{
"role": "user",
"content": "David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?\n\n*Ray Li*"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 7349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "David has a unit triangular array of 10 points, 4 on each side. A looping path is a sequence $A_{1}, A_{2}, \\ldots, A_{10}$ containing each of the 10 points exactly once, such that $A_{i}$ and $A_{i+1}$ are adjacent (exactly 1 unit apart) for $i=1,2, \\ldots, 10$. (Here $A_{11... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7350,
"solution": "There are $10 \\cdot 2$ times as many loop sequences as loops. To count the number of loops, first focus on the three corners of the array: their edges are uniquely determined. It's now easy to see there are 3 loops (they form \" $V$-shapes\"), so the answer is $10 \\cdot 2 \\cdot 3=60... |
DeepScaleR | [
{
"role": "user",
"content": "David has a very unique calculator that performs only two operations, the usual addition $(+)$ and another operation denoted by $*$, which satisfies:\n\n(i) $a * a = a$,\n\n(ii) $a * 0 = 2a$ and \n\n(iii) $(a * b) + (c * d) = (a + c) * (b + d)$,\n\nfor any integers $a$ and $b$.... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 7351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that ex... | math | {
"style": "rule",
"ground_truth": "9/14"
} | {
"index": 7352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "David, Delong, and Justin each showed up to a problem writing session at a random time during the session. If David arrived before Delong, what is the probability that he also arrived before Justin?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 7353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "David, Ellie, Natasha, and Lucy are tutors in their school science lab. Their working schedule is as follows: David works every fourth school day, Ellie works every fifth school day, Natasha works every sixth school day, and Lucy works every eighth school day. Today, they all h... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 7354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dean is playing a game with calculators. The 42 participants (including Dean) sit in a circle, and Dean holds 3 calculators. One calculator reads 1, another 0, and the last one -1. Dean starts by pressing the cube button on the calculator that shows 1, pressing the square butto... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 7355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean's scoring average. How many points, in total, did Ruth score?"
}
] | math | {
"style": "rule",
"ground_truth": "171"
} | {
"index": 7356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"index": 7357,
"solution": "1. **Identify the Required Sequence**: \n Debra must get two heads in a row, but she must see a second tail before seeing the second head. This means the sequence of flips must start with \"THT\" (Tail, Head, Tail).\n\n2. **Calculate the Probability of Starting with THT**:\n - The ... |
DeepScaleR | [
{
"role": "user",
"content": "Decide whether there is an integer \\( n > 1 \\) with the following properties:\n\n(a) \\( n \\) is not a prime number.\n\n(b) For all integers \\( a \\), \\( a^{n} - a \\) is divisible by \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "561"
} | {
"index": 7358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Decompose $\\frac{1}{4}$ into unit fractions."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}+\\frac{1}{12}+\\frac{1}{24}"
} | {
"index": 7359,
"solution": "$\\frac{1}{8}+\\frac{1}{12}+\\frac{1}{24}$"
} |
DeepScaleR | [
{
"role": "user",
"content": "Deepen and Expand: Suppose set $A$ contains 4 elements, and set $B$ contains 3 elements. Now, establish a mapping $f: A \\rightarrow B$, and make every element in $B$ have a pre-image in $A$. Then, the number of such mappings is ____."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define\n\\[\nd_k = k + \\cfrac{1}{3k + \\cfrac{1}{3k + \\cfrac{1}{3k + \\dotsb}}}.\n\\]\nCalculate $\\sum_{k = 1}^{10} (d_k^2 + 2)$."
}
] | math | {
"style": "rule",
"ground_truth": "405"
} | {
"index": 7361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define\n\\[ A' = \\frac{1}{1^2} + \\frac{1}{7^2} - \\frac{1}{11^2} - \\frac{1}{13^2} + \\frac{1}{19^2} + \\frac{1}{23^2} - \\dotsb, \\]\nwhich omits all terms of the form $\\frac{1}{n^2}$ where $n$ is an odd multiple of 5, and\n\\[ B' = \\frac{1}{5^2} - \\frac{1}{25^2} + \\frac... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 7362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define\n\\[A = \\frac{1}{1^2} + \\frac{1}{5^2} - \\frac{1}{7^2} - \\frac{1}{11^2} + \\frac{1}{13^2} + \\frac{1}{17^2} - \\dotsb,\\]which omits all terms of the form $\\frac{1}{n^2}$ where $n$ is an odd multiple of 3, and\n\\[B = \\frac{1}{3^2} - \\frac{1}{9^2} + \\frac{1}{15^2}... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define\n\\[c_k = k + \\cfrac{1}{2k + \\cfrac{1}{2k + \\cfrac{1}{2k + \\dotsb}}}.\\]Calculate $\\sum_{k = 1}^{11} c_k^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "517"
} | {
"index": 7364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $ a \\circledast b = a + b-2ab $ . Calculate the value of $$ A=\\left( ...\\left(\\left(\\frac{1}{2014}\\circledast \\frac{2}{2014}\\right)\\circledast\\frac{3}{2014}\\right)...\\right)\\circledast\\frac{2013}{2014} $$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 7365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $H_n = 1+\\frac{1}{2}+\\cdots+\\frac{1}{n}$ . Let the sum of all $H_n$ that are terminating in base 10 be $S$ . If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$ .\n\n*Proposed by Lewis Chen*"
}
] | math | {
"style": "rule",
"ground_truth": "9920"
} | {
"index": 7366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\displaystyle{f(x) = x + \\sqrt{x + \\sqrt{x + \\sqrt{x + \\sqrt{x + \\ldots}}}}}$ . Find the smallest integer $x$ such that $f(x)\\ge50\\sqrt{x}$ .\n\n(Edit: The official question asked for the \"smallest integer\"; the intended question was the \"smallest positi... | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"index": 7367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\lfloor x \\rfloor$ as the largest integer less than or equal to $x$ . Define $\\{x \\} = x - \\lfloor x \\rfloor$ . For example, $\\{ 3 \\} = 3-3 = 0$ , $\\{ \\pi \\} = \\pi - 3$ , and $\\{ - \\pi \\} = 4-\\pi$ . If $\\{n\\} + \\{ 3n\\} = 1.4$ , then find the ... | math | {
"style": "rule",
"ground_truth": "145"
} | {
"index": 7368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\phi_n(x)$ to be the number of integers $y$ less than or equal to $n$ such that $\\gcd(x,y)=1$ . Also, define $m=\\text{lcm}(2016,6102)$ . Compute $$ \\frac{\\phi_{m^m}(2016)}{\\phi_{m^m}(6102)}. $$"
}
] | math | {
"style": "rule",
"ground_truth": "339/392"
} | {
"index": 7369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $f\\left(n\\right)=\\textrm{LCM}\\left(1,2,\\ldots,n\\right)$ . Determine the smallest positive integer $a$ such that $f\\left(a\\right)=f\\left(a+2\\right)$ .\n\n*2017 CCA Math Bonanza Lightning Round #2.4*"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $m(n)$ to be the greatest proper natural divisor of $n\\in \\mathbb{N}$ . Find all $n \\in \\mathbb{N} $ such that $n+m(n) $ is a power of $10$ .\n\n*N. Agakhanov*"
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 7371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $p(n)$ to be th product of all non-zero digits of $n$ . For instance $p(5)=5$ , $p(27)=14$ , $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:\n\\[p(1)+p(2)+p(3)+...+p(999).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 7372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $A\\star B$ as $A\\star B = \\frac{(A+B)}{3}$. What is the value of $(2\\star 10) \\star 5$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 7373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $E(a,b,c) = a \\cdot b^2 + c$. What value of $a$ is the solution to the equation $E(a,4,5) = E(a,6,7)$?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{10}"
} | {
"index": 7374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $E(a,b,c) = a \\times b^c$. What positive value of $r$ is the solution to the equation $E(r,r,3) = 625$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 7375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $E(n)$ as the sum of the even digits of $n$ and $O(n)$ as the sum of the odd digits of $n$. Find the value of $E(1) + O(1) + E(2) + O(2) + \\dots + E(150) + O(150)$.\nA) 1200\nB) 1300\nC) 1350\nD) 1400\nE) 1450"
}
] | math | {
"style": "rule",
"ground_truth": "1350"
} | {
"index": 7376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $F(x, y, z) = x \\times y^z$. What positive value of $s$ is the solution to the equation $F(s, s, 2) = 1024$?"
}
] | math | {
"style": "rule",
"ground_truth": "8 \\cdot \\sqrt[3]{2}"
} | {
"index": 7377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $L(x) = x - \\frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by\n\\[\n a_n = L \\Bigl( L \\Bigl( L \\Bigl( \\cdots L \\Bigl( \\frac{17}{n} \\Bigr) \\cdots \\Bigr) \\Bigr) \\Bigr),\n\\]where there are $n$ iterations of $L$. For exam... | math | {
"style": "rule",
"ground_truth": "\\frac{34}{19}"
} | {
"index": 7378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $P=\\{\\mathrm{S}, \\mathrm{T}\\}$ and let $\\mathcal{P}$ be the set of all proper subsets of $P$. (A proper subset is a subset that is not the set itself.) How many ordered pairs $(\\mathcal{S}, \\mathcal{T})$ of proper subsets of $\\mathcal{P}$ are there such that (a) ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 7379,
"solution": "For ease of notation, we let $0=\\varnothing, 1=\\{\\mathrm{S}\\}, 2=\\{\\mathrm{T}\\}$. Then both $\\mathcal{S}$ and $\\mathcal{T}$ are proper subsets of $\\{0,1,2\\}$. We consider the following cases: Case 1. If $\\mathcal{S}=\\varnothing$, then $\\mathcal{S}$ is a proper subset of a... |
DeepScaleR | [
{
"role": "user",
"content": "Define $[a,b,c]$ to mean $\\frac {a+b}c$, where $c \\neq 0$. What is the value of $\\left[[60,30,90],[2,1,3],[10,5,15]\\right]?$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 7380,
"solution": "1. **Understanding the operation $[a,b,c]$:** \n The operation $[a,b,c]$ is defined as $\\frac{a+b}{c}$, where $c \\neq 0$.\n\n2. **Simplifying each component:** \n - For $[60,30,90]$, we calculate:\n \\[\n [60,30,90] = \\frac{60+30}{90} = \\frac{90}{90} = 1\n \\]\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\#N$ by the formula $\\#N = .5(N) + 1$. Calculate $\\#(\\#(\\#50))$."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 7381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\#N$ by the formula $\\#N = .5(N) + 1$. Calculate $\\#(\\#(\\#58))$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 7382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\mathbf{A} = \\begin{pmatrix} 0 & 1 \\\\ 3 & 0 \\end{pmatrix}.$ Find the vector $\\mathbf{v}$ such that\n\\[(\\mathbf{A}^8 + \\mathbf{A}^6 + \\mathbf{A}^4 + \\mathbf{A}^2 + \\mathbf{I}) \\mathbf{v} = \\begin{pmatrix} 0 \\\\ 11 \\end{pmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 \\\\ 1/11 \\end{pmatrix}"
} | {
"index": 7383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\operatorname{gcd}(a, b)$ as the greatest common divisor of integers $a$ and $b$. Given that $n$ is the smallest positive integer greater than 1000 that satisfies:\n$$\n\\begin{array}{l}\n\\operatorname{gcd}(63, n+120) = 21, \\\\\n\\operatorname{gcd}(n+63, 120) = 60\n\... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 7384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the number of integers $2 \\leq n \\leq 50$ such that $n$ divides $\\phi^{!}(n)+1$."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 7385,
"solution": "Note that, if $k$ is relatively prime to $n$, there exists a unique $0<k^{-1}<n$ such that $k k^{-1} \\equiv 1(\\bmod n)$. Hence, if $k^{2} \\not \\equiv 1(\\bmod n)$, we can pair $k$ with its inverse to get a product of 1. If $k^{2} \\equiv 1(\\bmod n)$, then $(n-k)^{2} \\equiv 1(\\bm... |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when $$ \\sum_{\\substack{2 \\leq n \\leq 50 \\\\ \\operatorname{gcd}(n, 50)=1}} \\phi^{!}(n) $$ is divided by 50 ."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7386,
"solution": "First, $\\phi^{!}(n)$ is even for all odd $n$, so it vanishes modulo 2 . To compute the remainder modulo 25 , we first evaluate $\\phi^{!}(3)+\\phi^{!}(7)+\\phi^{!}(9) \\equiv 2+5 \\cdot 4+5 \\cdot 3 \\equiv 12$ $(\\bmod 25)$. Now, for $n \\geq 11$ the contribution modulo 25 vanishes a... |
DeepScaleR | [
{
"role": "user",
"content": "Define $\\varphi^{k}(n)$ as the number of positive integers that are less than or equal to $n / k$ and relatively prime to $n$. Find $\\phi^{2001}\\left(2002^{2}-1\\right)$. (Hint: $\\phi(2003)=2002$.)"
}
] | math | {
"style": "rule",
"ground_truth": "1233"
} | {
"index": 7387,
"solution": "$\\varphi^{2001}\\left(2002^{2}-1\\right)=\\varphi^{2001}(2001 \\cdot 2003)=$ the number of $m$ that are relatively prime to both 2001 and 2003, where $m \\leq 2003$. Since $\\phi(n)=n-1$ implies that $n$ is prime, we must only check for those $m$ relatively prime to 2001, except for 2... |
DeepScaleR | [
{
"role": "user",
"content": "Define $a * b$ as $2a - b^2$. If $a * 5 = 9$, what is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 7388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $a \\Delta b = a^2 -b $. What is the value of $ (2^{4 \\Delta13})\\Delta(3^{3\\Delta5})$"
}
] | math | {
"style": "rule",
"ground_truth": "-17"
} | {
"index": 7389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $a$ $\\$$ $b$ to be $a(b + 1) + ab$. What is the value of $(-2)$ $\\$$ $3$?"
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"index": 7390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $a$ ? $=(a-1) /(a+1)$ for $a \\neq-1$. Determine all real values $N$ for which $(N ?)$ ?=\\tan 15."
}
] | math | {
"style": "rule",
"ground_truth": "-2-\\sqrt{3}"
} | {
"index": 7391,
"solution": "Let $x=N$ ?. Then $(x-1) \\cos 15=(x+1) \\sin 15$. Squaring and rearranging terms, and using the fact that $\\cos ^{2} 15-\\sin ^{2} 15=\\cos 30=\\frac{\\sqrt{3}}{2}$, we have $3 x^{2}-4 \\sqrt{3} x+3=0$. Solving, we find that $x=\\sqrt{3}$ or \\frac{\\sqrt{3}}{3}$. However, we may rej... |
DeepScaleR | [
{
"role": "user",
"content": "Define $a@b = ab - b^{2}$ and $a\\#b = a + b - ab^{2}$. What is $\\frac {6@2}{6\\#2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 7392,
"solution": "1. **Calculate $6@2$:**\n Using the operation defined as $a@b = ab - b^2$, we substitute $a = 6$ and $b = 2$:\n \\[\n 6@2 = 6 \\cdot 2 - 2^2 = 12 - 4 = 8.\n \\]\n\n2. **Calculate $6\\#2$:**\n Using the operation defined as $a\\#b = a + b - ab^2$, we substitute $a = 6$ and $b ... |
DeepScaleR | [
{
"role": "user",
"content": "Define $f(x) = \\frac{3}{27^x + 3}.$ Calculate the sum\n\\[ f\\left(\\frac{1}{2001}\\right) + f\\left(\\frac{2}{2001}\\right) + f\\left(\\frac{3}{2001}\\right) + \\dots + f\\left(\\frac{2000}{2001}\\right). \\]"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 7393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $f(x)=3x-8$. If $f^{-1}$ is the inverse of $f$, find the value(s) of $x$ for which $f(x)=f^{-1}(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $g$ by $g(x) = 3x + 2$. If $g(x) = f^{-1}(x) - 5$ and $f^{-1}(x)$ is the inverse of the function $f(x) = cx + d$, find $7c + 3d$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{14}{3}"
} | {
"index": 7395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $g$ by $g(x)=5x-4$. If $g(x)=f^{-1}(x)-3$ and $f^{-1}(x)$ is the inverse of the function $f(x)=ax+b$, find $5a+5b$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 7396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $n!!$ as in the original problem. Evaluate $\\sum_{i=1}^{5} \\frac{(2i-1)!!}{(2i)!!}$, and express the result as a fraction in lowest terms."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{437}{256}"
} | {
"index": 7397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\dfrac{ab}{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 7398,
"solution": "Using the steps of the previous solution we get $c = \\sum_{i=1}^{2009} {2i\\choose i} \\cdot 2^{2\\cdot 2009 - 2i}$ and if you do the small cases(like $1, 2, 3, 4, 5, 6$) you realize that you can \"thin-slice\" the problem and simply look at the cases where $i=2009, 2008$(they're near... |
DeepScaleR | [
{
"role": "user",
"content": "Define $n_a!$ for $n$ and $a$ positive to be\n$n_a ! = n (n-a)(n-2a)(n-3a)...(n-ka)$\nwhere $k$ is the greatest integer for which $n>ka$. Then the quotient $72_8!/18_2!$ is equal to"
}
] | math | {
"style": "rule",
"ground_truth": "4^9"
} | {
"index": 7399,
"solution": "1. **Understanding $n_a!$:** \n The factorial-like function $n_a!$ is defined as:\n \\[ n_a! = n(n-a)(n-2a)(n-3a)\\ldots \\]\n where the product continues until $n-ka$ where $k$ is the largest integer such that $n > ka$.\n\n2. **Calculating $72_8!$:** \n We need to find the p... |
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