data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A classroom consists of a $5 \\times 5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are... | math | {
"style": "rule",
"ground_truth": "962"
} | {
"split": "train",
"index": 19349,
"solution": "The set of empty desks must be of the form (non-full rows) $\\times$ (non-full columns): each empty desk is in a non-full column and a non-full row, and the given condition implies that each desk in such a position is empty. So if there are fewer than 25 students, ... |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\( m \\) and \\( n \\) are such that \\( m > n \\), \\( m \\) is not divisible by \\( n \\), and \\( m \\) has the same remainder when divided by \\( n \\) as \\( m + n \\) has when divided by \\( m - n \\).\n\nFind the ratio \\( m : n \\). Let's think step by ... | math | {
"style": "rule",
"ground_truth": "5/2"
} | {
"split": "train",
"index": 19350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the weekend, Lou advertises the sale: \"Ten percent off the listed price. Sale starts Monday.\" How much does a pair of shoes co... | math | {
"style": "rule",
"ground_truth": "39.60"
} | {
"split": "train",
"index": 19351,
"solution": "1. **Calculate the price increase on Friday:**\n The price of the shoes on Thursday is $40$ dollars. On Friday, Lou increases the prices by $10\\%$. Therefore, the price on Friday can be calculated as follows:\n \\[\n \\text{Price on Friday} = \\text{Price on... |
deepscaler | [
{
"role": "user",
"content": "Given the right focus $F$ and the right directrix $l$ of the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, with eccentricity $e = \\frac{\\sqrt{5}}{5}$. Draw $AM \\perp l$ through the vertex $A(0, b)$, with $M$ as the foot of the perpendicular. Find the slope o... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 19352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are positive integers $x$ and $y$ that satisfy the system of equations \\begin{align*} \\log_{10} x + 2 \\log_{10} (\\text{gcd}(x,y)) &= 60\\\\ \\log_{10} y + 2 \\log_{10} (\\text{lcm}(x,y)) &= 570. \\end{align*} Let $m$ be the number of (not necessarily distinct) prime f... | math | {
"style": "rule",
"ground_truth": "880"
} | {
"split": "train",
"index": 19353,
"solution": "Let $x=10^a$ and $y=10^b$ and $a<b$. Then the given equations become $3a=60$ and $3b=570$. Therefore, $x=10^{20}=2^{20}\\cdot5^{20}$ and $y=10^{190}=2^{190}\\cdot5^{190}$. Our answer is $3(20+20)+2(190+190)=\\boxed{880}$."
} |
deepscaler | [
{
"role": "user",
"content": "Given that six coal freight trains are organized into two groups of three trains, with trains 'A' and 'B' in the same group, determine the total number of different possible departure sequences for the six trains. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 19354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x=2018$, what is the value of the expression $x^{2}+2x-x(x+1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 19355,
"solution": "For any value of $x$, we have $x^{2}+2x-x(x+1)=x^{2}+2x-x^{2}-x=x$. When $x=2018$, the value of this expression is thus 2018."
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \\sqrt{5}c$ and $\\cos C = \\frac{3}{5}$. \n$(Ⅰ)$ Find the value of $\\sin A$. \n$(Ⅱ)$ If $b = 11$, find the area of $\\triangle ABC$. Let's... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 19356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ in the equation $16^{16}+16^{16}+16^{16}+16^{16}=2^x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 19357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jeff wants to calculate the product $0.52 \\times 7.35$ using a calculator. However, he mistakenly inputs the numbers as $52 \\times 735$ without the decimal points. The calculator then shows a product of $38220$. What would be the correct product if Jeff had correctly entered ... | math | {
"style": "rule",
"ground_truth": "3.822"
} | {
"split": "train",
"index": 19358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Karl was attempting to calculate economic figures. He found the following equation to be true:\\[fp-w=10000\\]If $f=5$ and $w=5+125i$, what is $p$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2001+25i"
} | {
"split": "train",
"index": 19359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets $A=\\{-1,1,2\\}$ and $B=\\{-2,1,2\\}$, a number $k$ is randomly selected from set $A$ and a number $b$ is randomly selected from set $B$. The probability that the line $y=kx+b$ does not pass through the third quadrant is $\\_\\_\\_\\_\\_\\_$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "P = \\frac{2}{9}"
} | {
"split": "train",
"index": 19360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A television station is broadcasting 5 advertisements in a row, which include 3 different commercial advertisements and 2 different National Games promotional advertisements. The requirements are that the last advertisement must be one of the National Games promotional advertis... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 19361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the scores $93$, $89$, $92$, $95$, $93$, $94$, $93$ of seven referees, determine the average value of the remaining data after removing the highest and lowest scores. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 19362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many triangles can be formed using the vertices of a regular hexadecagon (a 16-sided polygon). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 19363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the square \\(ABCD\\), points \\(K\\) and \\(M\\) are marked (point \\(M\\) is inside triangle \\(ABD\\), point \\(K\\) is inside \\(BMC\\)) such that triangles \\(BAM\\) and \\(DKM\\) are congruent \\((AM = KM, BM = MD, AB = KD)\\). Find \\(\\angle KCM\\) if \\(\\angle ... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 19364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^3+z^3|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 19365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 5-letter words with at least one consonant can be constructed from the letters $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $I$? Each letter can be used more than once, and $B$, $C$, $D$, $F$, $G$ are consonants. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "32525"
} | {
"split": "train",
"index": 19366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola \\( C: \\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 \\) with \\( a > 0 \\) and \\( b > 0 \\), the eccentricity is \\( \\frac{\\sqrt{17}}{3} \\). Let \\( F \\) be the right focus, and points \\( A \\) and \\( B \\) lie on the right branch of the hyperbola.... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 19367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters with no restrictions? (The triplets are treated as distinguishable.) Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"split": "train",
"index": 19368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles p... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 19369,
"solution": "1. **Calculate the time to reach the shore**: \n Steve rows at a speed of 4 miles per hour. Since they are 1 mile from shore, the time \\( t \\) it takes to reach the shore is given by:\n \\[\n t = \\frac{\\text{distance}}{\\text{speed}} = \\frac{1 \\text{ mi... |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable and neither are the boxes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to poin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 19371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particular coin has a $\\frac{1}{3}$ chance of landing on heads (H), $\\frac{1}{3}$ chance of landing on tails (T), and $\\frac{1}{3}$ chance of landing vertically in the middle (M). When continuously flipping this coin, what is the probability of observing the continuous seq... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 19372,
"solution": "For a string of coin flips $S$, let $P_{S}$ denote the probability of flipping $H M M T$ before $H M T$ if $S$ is the starting sequence of flips. We know that the desired probability, $p$, is $\\frac{1}{3} P_{H}+\\frac{1}{3} P_{M}+\\frac{1}{3} P_{T}$. Now, using co... |
deepscaler | [
{
"role": "user",
"content": "Given a right prism $ABC-A_{1}B_{1}C_{1}$, where $AB=3$, $AC=4$, and $AB \\perp AC$, $AA_{1}=2$, find the sum of the surface areas of the inscribed sphere and the circumscribed sphere of the prism. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33\\pi"
} | {
"split": "train",
"index": 19373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\\sin(C+\\frac{π}{3})-c\\sin B=0$. \n$(1)$ Find the value of angle $C$. \n$(2)$ If the area of $\\triangle ABC$ is $10\\sqrt{3}$ and $D$ is the ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 19374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, satisfying $2c+b-2a\\cos B=0$. \n$(1)$ Find angle $A$; \n$(2)$ If $a=2\\sqrt{3}$, $\\overrightarrow{BA}\\cdot \\overrightarrow{AC}=\\frac{3}{2}$, and $AD$ is the median of $\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 19375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 19376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct positive integers $a$ and $b$ are factors of 48. If $a\\cdot b$ is not a factor of 48, what is the smallest possible value of $a\\cdot b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 19377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Zhang Hua has to go through four traffic posts A, B, C, and D on his way to school. The probability of encountering a red light at posts A and B is $\\frac{1}{2}$ each, and at posts C and D, it is $\\frac{1}{3}$ each. Assuming that the events of encountering red lights at the f... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 19378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is either a 10 or a Jack? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{442}"
} | {
"split": "train",
"index": 19379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 19380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$1.15"
} | {
"split": "train",
"index": 19381,
"solution": "1. Let $n$ represent the number of nickels Patty has, and $d$ represent the number of dimes. Since Patty has a total of 20 coins, we can express the number of dimes in terms of nickels:\n \\[\n d = 20 - n\n \\]\n\n2. Calculate the total value of the coins whe... |
deepscaler | [
{
"role": "user",
"content": "Antônio needs to find a code with 3 different digits \\( A, B, C \\). He knows that \\( B \\) is greater than \\( A \\), \\( A \\) is less than \\( C \\), and also:\n\\[\n\\begin{array}{cccc}\n & B & B \\\\\n+ & A & A \\\\\n\\hline\n& C & C \\\\\n\\end{array} = 242\n\\]\n\nWhat... | math | {
"style": "rule",
"ground_truth": "232"
} | {
"split": "train",
"index": 19382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $5\\,41G\\,507\\,2H6$ is divisible by $72.$ If $G$ and $H$ each represent a single digit, what is the sum of all distinct possible values of the product $GH?$ (Count each possible value of $GH$ only once, even if it results from multiple $G,$ $H$ pairs.) Let's think ... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 19383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two similar triangles $\\triangle ABC\\sim\\triangle FGH$, where $BC = 24 \\text{ cm}$ and $FG = 15 \\text{ cm}$. If the length of $AC$ is $18 \\text{ cm}$, find the length of $GH$. Express your answer as a decimal to the nearest tenth. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "11.3"
} | {
"split": "train",
"index": 19384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$, $y$, $z$ are positive real numbers, find the maximum value of $\\dfrac{xy+yz}{x^{2}+y^{2}+z^{2}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 19385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$ . $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\\angle ZWX$ is a right angle. Given that $WX = WZ = \\sqrt{13}$ and ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{65}"
} | {
"split": "train",
"index": 19386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six green balls and four red balls are in a bag. A ball is taken from the bag, its color recorded, then placed back in the bag. A second ball is taken and its color recorded. What is the probability the two balls are the same color? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{13}{25}"
} | {
"split": "train",
"index": 19387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all of the solutions of the equation $\\frac{4x}{20}=\\frac{5}{x}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 19388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of $$\\frac {1}{x}$$ in the expansion of $$(1-x^2)^4\\left(\\frac {x+1}{x}\\right)^5$$ is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-29"
} | {
"split": "train",
"index": 19389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(1+x)^3+(1+x)^4+\\ldots+(1+x)^{19}$, the coefficient of the $x^2$ term is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1139"
} | {
"split": "train",
"index": 19390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$ whose side edge length is equal to the base edge length, find the sine value of the angle formed by $AB_{1}$ and the side face $ACC_{1}A_{1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{4}"
} | {
"split": "train",
"index": 19391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest possible real value of $x^2 + 8x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 19392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Mr. Abraham's class, $10$ of the $15$ students received an $A$ on the latest exam. If the same ratio of students received an $A$ on Mrs. Berkeley's latest exam, and if Mrs. Berkeley has $24$ students total, how many students in Mrs. Berkeley's class received an $A$? Let's th... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 19393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a point $P$ on the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with foci $F_{1}$, $F_{2}$, find the eccentricity of the ellipse given that $\\overrightarrow{PF_{1}} \\cdot \\overrightarrow{PF_{2}}=0$ and $\\tan \\angle PF_{1}F_{2}= \\frac{1}{2}$. Le... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 19394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kai has begun to list, in ascending order, the positive integers which are not factors of 240. What is the sixth number on Kai's list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 19395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the exact value of the series\n\\[\n\\frac{1}{3 + 1} + \\frac{2}{3^2 + 1} + \\frac{4}{3^4 + 1} + \\frac{8}{3^8 + 1} + \\frac{16}{3^{16} + 1} + \\dotsb.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 19396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $x$ and $y$, define $x \\spadesuit y = (x+y)(x-y)$. What is $3 \\spadesuit (4 \\spadesuit 5)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-72"
} | {
"split": "train",
"index": 19397,
"solution": "1. **Define the operation $\\spadesuit$:**\n Given $x \\spadesuit y = (x+y)(x-y)$, we need to evaluate $3 \\spadesuit (4 \\spadesuit 5)$.\n\n2. **Evaluate $4 \\spadesuit 5$:**\n \\[\n 4 \\spadesuit 5 = (4+5)(4-5) = 9 \\times (-1) = -9\n \\]\n\n3. **Substitu... |
deepscaler | [
{
"role": "user",
"content": "Let \\( A B C D \\) be an isosceles trapezoid with \\( [A B] \\) as the larger base. It is given that the diagonals intersect at a point \\( O \\) such that \\(\\frac{O A}{O C}=2\\). Given that the area of triangle \\( B O C \\) is 10, what is the area of the trapezoid \\( A B ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 19398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the vertices of a regular 100-sided polygon \\( A_{1}, A_{2}, A_{3}, \\ldots, A_{100} \\), in how many ways can three vertices be selected such that they form an obtuse triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "117600"
} | {
"split": "train",
"index": 19399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least possible value of \\((x+1)(x+2)(x+3)(x+4)+2023\\) where \\(x\\) is a real number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 19400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the area and the circumference of a circle with the center at the point \\( R(2, -1) \\) and passing through the point \\( S(7, 4) \\). Express your answer in terms of \\( \\pi \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\pi \\sqrt{2}"
} | {
"split": "train",
"index": 19401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On three faces of a cube, diagonals are drawn such that a triangle is formed. Find the angles of this triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 19402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Form a five-digit number with no repeated digits using the numbers 0, 1, 2, 3, and 4, where exactly one even number is sandwiched between two odd numbers. The total number of such five-digit numbers is Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 19403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $G$ and $N$ are chosen on the interiors of sides $E D$ and $D O$ of unit square $D O M E$, so that pentagon GNOME has only two distinct side lengths. The sum of all possible areas of quadrilateral $N O M E$ can be expressed as $\\frac{a-b \\sqrt{c}}{d}$, where $a, b, c, ... | math | {
"style": "rule",
"ground_truth": "10324"
} | {
"split": "train",
"index": 19404,
"solution": "Since $M O=M E=1$, but $O N$ and $G E$ are both less than 1, we must have either $O N=N G=G E=x$ (call this case 1) or $O N=G E=x, N G=1$ (call this case 2). Either way, the area of $N O M E$ (a trapezoid) is $\\frac{1+x}{2}$, and triangle $N G T$ is a $45-45-90$ t... |
deepscaler | [
{
"role": "user",
"content": "During the process of choosing trial points using the 0.618 method, if the trial interval is $[3, 6]$ and the first trial point is better than the second, then the third trial point should be at _____. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "5.292"
} | {
"split": "train",
"index": 19405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of distinct roots of \\prod_{k=1}^{2012}\\left(x^{k}-1\\right)$. Give lower and upper bounds $L$ and $U$ on $N$. If $0<L \\leq N \\leq U$, then your score will be \\left[\\frac{23}{(U / L)^{1.7}}\\right\\rfloor$. Otherwise, your score will be 0 . Let's thi... | math | {
"style": "rule",
"ground_truth": "1231288"
} | {
"split": "train",
"index": 19406,
"solution": "For $x$ to be such a number is equivalent to $x$ being an $k^{\\text {th }}$ root of unity for some $k$ up to 2012. For each $k$, there are \\varphi(k)$ primitive $k^{\\text {th }}$ roots of unity, so the total number of roots is \\sum_{k=1}^{2012} \\varphi(k)$. We... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number \\( N \\) that is divisible by \\( p \\), ends with \\( p \\), and has a digit sum equal to \\( p \\), given that \\( p \\) is a prime number and \\( 2p+1 \\) is a cube of a natural number. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "11713"
} | {
"split": "train",
"index": 19407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After folding the long rope in half 6 times, calculate the number of segments the rope will be cut into. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 19408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nPoints \\( A \\) and \\( B \\) are located on a straight highway running from west to east. Point \\( B \\) is 9 km east of \\( A \\). A car leaves point \\( A \\) heading east at a speed of 40 km/h. At the same time, a motorcycle leaves point \\( B \\) in the same direction ... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 19409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the power function $y=(m^2-5m-5)x^{2m+1}$ is a decreasing function on $(0, +\\infty)$, then the real number $m=$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 19410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a s_k + b s_{k-1} + c s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$? Let's ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19411,
"solution": "1. **Identify the polynomial and its roots**: Let $p, q, r$ be the roots of the polynomial $x^3 - 5x^2 + 8x - 13 = 0$. \n\n2. **Express the polynomial equations for each root**:\n \\[\n p^3 - 5p^2 + 8p - 13 = 0, \\quad q^3 - 5q^2 + 8q - 13 = 0, \\quad r^3 - 5r^... |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin(\\omega x+\\varphi)$, where $(\\omega>0, |\\varphi|<\\frac{\\pi}{2})$, passes through the point $(0,-\\frac{1}{2})$. Find the minimum value of $\\omega$ if the graph of this function is shifted to the right by $\\frac{\\pi}{3}$ units and be... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 19412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $150\\times 324\\times 375$ rectangular solid is made by gluing together $1\\times 1\\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\\times 1\\times 1$ cubes? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 19413,
"solution": "Consider a point travelling across the internal diagonal, and let the internal diagonal have a length of $d$. The point enters a new unit cube in the $x,y,z$ dimensions at multiples of $\\frac{d}{150}, \\frac{d}{324}, \\frac{d}{375}$ respectively. We proceed by usi... |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common divisor of 128, 144 and 480? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 19414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the given $5 \\times 5$ grid, there are 6 letters. Divide the grid along the lines to form 6 small rectangles (including squares) of different areas, so that each rectangle contains exactly one letter, and each letter is located in a corner square of its respective rectangle... | math | {
"style": "rule",
"ground_truth": "34216"
} | {
"split": "train",
"index": 19415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log_{10}{m}= b-\\log_{10}{n}$, then $m=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10^{b}}{n}"
} | {
"split": "train",
"index": 19416,
"solution": "1. Start by expressing $b$ in terms of logarithm base 10:\n \\[\n b = \\log_{10}{10^b}\n \\]\n This follows from the property of logarithms that $\\log_b{b^x} = x$.\n\n2. Substitute this expression for $b$ into the given equation:\n \\[\n \\log_{10}{m} ... |
deepscaler | [
{
"role": "user",
"content": "In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off... | math | {
"style": "rule",
"ground_truth": "766"
} | {
"split": "train",
"index": 19417,
"solution": "Suppose $1000$ workers can complete one quarter of the job in one day. After the first day, there were $900$ workers remaining so the second quarter was completed in $\\frac{10}{9}$ days. Now there are only $800$ workers remaining so the third quarter can be comple... |
deepscaler | [
{
"role": "user",
"content": "If Ravi shortens the length of one side of a $5 \\times 7$ index card by $1$ inch, the card would have an area of $24$ square inches. What is the area of the card in square inches if instead he shortens the length of the other side by $1$ inch? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 19418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\\Delta ABC$ at $A$ and the line perpendicular to $GO$ at $G$. Let $Y$ be the intersection of lines $XG$ and $BC$. Given that t... | math | {
"style": "rule",
"ground_truth": "592"
} | {
"split": "train",
"index": 19419,
"solution": "Notice that $\\triangle ABC$ looks isosceles, so we assume it's isosceles. Then, let $\\angle BAC = \\angle ABC = 13x$ and $\\angle BCA = 2x.$ Taking the sum of the angles in the triangle gives $28x=180,$ so $13x = \\frac{13}{28} \\cdot 180 = \\frac{585}{7}$ so the... |
deepscaler | [
{
"role": "user",
"content": "How many such nine-digit positive integers can Ricardo make if he wants to arrange three 1s, three 2s, two 3s, and one 4 with the properties that there is at least one 1 before the first 2, at least one 2 before the first 3, and at least one 3 before the 4, and no digit 2 can b... | math | {
"style": "rule",
"ground_truth": "254"
} | {
"split": "train",
"index": 19420,
"solution": "Case 1: $N$ begins 12. There are 10 possible pairs of positions for the 2s. There are 10 pairs of positions for the 1s. There are 2 orders for the 3s and 4. In this case, there are $10 \\times 10 \\times 2=200$ possible integers $N$. Case 2: $N$ begins 112. There a... |
deepscaler | [
{
"role": "user",
"content": "Find the 150th term of the sequence that consists of all those positive integers which are either powers of 3 or sums of distinct powers of 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2280"
} | {
"split": "train",
"index": 19421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lattice shown is continued for $9$ rows. What will be the third number in the $9$th row? \\begin{tabular}{rccccc}\nRow 1: & 1 & 2 & 3 & 4 & 5 \\\\\nRow 2: & 6 & 7 & 8 & 9 & 10 \\\\\nRow 3: & 11 & 12 & 13 & 14 & 15 \\\\\nRow 4: & 16 & 17 & 18 & 19 & 20\n\\end{tabular} Let's ... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 19422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10-7\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 19423,
"solution": "1. **Identify the shape of the faces after slicing**: After slicing off the corners of the unit cube, each face becomes a regular octagon. This implies that each edge of the cube is divided into three segments: two segments forming the slanted edges of the octagon ... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the maximum of \\( |x^2 - 2xy| \\) over \\( 0 \\leq x \\leq 1 \\) for \\( y \\) in \\( \\mathbb{R} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"split": "train",
"index": 19424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\\sin 2C=c\\sin B$.\n\n1. Find angle $C$.\n2. If $\\sin \\left(B-\\frac{\\pi }{3}\\right)=\\frac{3}{5}$, find the value of $\\sin A$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}-3}{10}"
} | {
"split": "train",
"index": 19425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of all the divisors of $36$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10077696"
} | {
"split": "train",
"index": 19426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A \\(10 \\times 1\\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct... | math | {
"style": "rule",
"ground_truth": "1022"
} | {
"split": "train",
"index": 19427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the product: $500 \\times 2019 \\times 0.02019 \\times 5.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.25 \\times 2019^2"
} | {
"split": "train",
"index": 19428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\sin ^{2}A-\\sin ^{2}C=(\\sin A-\\sin B)\\sin B$, then angle $C$ equals to $\\dfrac {\\pi}{6}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\pi}{3}"
} | {
"split": "train",
"index": 19429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the hyperbola $C: \\frac{x^2}{a^2} - y^2 = 1$ ($a > 0$) intersect with the line $l: x + y = 1$ at two distinct points $A$ and $B$.\n\n(Ⅰ) Find the range of values for the eccentricity $e$ of the hyperbola $C$.\n\n(Ⅱ) Let the intersection of line $l$ with the y-axis be $P$, ... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{13}"
} | {
"split": "train",
"index": 19430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two friends, Alice and Bob, start cycling towards a park 80 miles away. Alice cycles 3 miles per hour slower than Bob. Upon reaching the park, Bob immediately turns around and starts cycling back, meeting Alice 15 miles away from the park. Find the speed of Alice. Let's think s... | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"split": "train",
"index": 19431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations\r\n\\[ \\begin{cases}x^3 \\equal{} 3x\\minus{}12y\\plus{}50, \\\\ y^3 \\equal{} 12y\\plus{}3z\\minus{}2, \\\\ z^3 \\equal{} 27z \\plus{} 27x. \\end{cases}\\]\r\n\r\n[i]Razvan Gelca.[/i] Let's think... | math | {
"style": "rule",
"ground_truth": "(2, 4, 6)"
} | {
"split": "train",
"index": 19432,
"solution": "\nWe are given the system of equations:\n\\[\n\\begin{cases}\nx^3 = 3x - 12y + 50, \\\\\ny^3 = 12y + 3z - 2, \\\\\nz^3 = 27z + 27x.\n\\end{cases}\n\\]\n\nTo find all triples \\((x, y, z)\\) of real numbers that satisfy these equations, we analyze the behavior of th... |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is a root of the equation $x^{2}+x-6=0$, simplify $\\frac{x-1}{\\frac{2}{{x-1}}-1}$ and find its value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 19433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a $16 \\times 16$ table of integers, each row and column contains at most 4 distinct integers. What is the maximum number of distinct integers that there can be in the whole table? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 19434,
"solution": "First, we show that 50 is too big. Assume for sake of contradiction that a labeling with at least 50 distinct integers exists. By the Pigeonhole Principle, there must be at least one row, say the first row, with at least 4 distinct integers in it; in this case, tha... |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 19435,
"solution": "1. **Identify the first digit**: The first digit of $N$ must be the largest possible digit that is a factor of $120$ and also maximizes the remaining product for the other four digits. We start by checking from the largest possible digit:\n - $9$ is not a factor ... |
deepscaler | [
{
"role": "user",
"content": "If $x, y$, and $z$ are real numbers such that $2 x^{2}+y^{2}+z^{2}=2 x-4 y+2 x z-5$, find the maximum possible value of $x-y+z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 19436,
"solution": "The equation rearranges as $(x-1)^{2}+(y+2)^{2}+(x-z)^{2}=0$, so we must have $x=1$, $y=-2, z=1$, giving us 4 ."
} |
deepscaler | [
{
"role": "user",
"content": "Billy Bones has two coins - one gold and one silver. One of them is symmetrical, and the other is not. It is unknown which coin is asymmetrical, but it is known that the asymmetrical coin lands heads with a probability of $p=0.6$.\n\nBilly Bones tossed the gold coin, and it lan... | math | {
"style": "rule",
"ground_truth": "5/9"
} | {
"split": "train",
"index": 19437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a revised game of Deal or No Deal, participants choose a box at random from a set of $30$, each containing one of the following values:\n\\[\n\\begin{array}{|c|c|}\n\\hline\n\\$0.50 & \\$50,000 \\\\\n\\hline\n\\$5 & \\$100,000 \\\\\n\\hline\n\\$20 & \\$150,000 \\\\\n\\hline\... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number \\( 1 \\leq n \\leq 221 \\) is called lucky if, when dividing 221 by \\( n \\), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 19439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation \\[-x^2 = \\frac{3x+1}{x+3}.\\]Enter all solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 19440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\oplus(\\oplus(2,4,5), \\oplus(3,5,4), \\oplus(4,2,5))$, where $\\oplus(x,y,z) = \\frac{x+y-z}{y-z}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{13}"
} | {
"split": "train",
"index": 19441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "63 or 12"
} | {
"split": "train",
"index": 19442,
"solution": "1. **Define Variables:**\n Let $d$ be the distance between the two towns, $p$ be the speed of the plane in still air, and $w$ be the speed of the wind.\n\n2. **Equation for the trip against the wind:**\n The time taken to fly against the wind is given as 84 min... |
deepscaler | [
{
"role": "user",
"content": "There are 99 positive integers, and their sum is 101101. The greatest possible value of the greatest common divisor of these 99 positive integers is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 19443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possi... | math | {
"style": "rule",
"ground_truth": "33,\\!800,\\!000"
} | {
"split": "train",
"index": 19444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\phi = \\tfrac{1+\\sqrt 5}2$ be the positive root of $x^2=x+1$ . Define a function $f:\\mathbb N\\to\\mathbb N$ by\n\\begin{align*}\nf(0) &= 1\nf(2x) &= \\lfloor\\phi f(x)\\rfloor\nf(2x+1) &= f(2x) + f(x).\n\\end{align*}\nFind the remainder when $f(2007)$ is divid... | math | {
"style": "rule",
"ground_truth": "2007"
} | {
"split": "train",
"index": 19445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$ be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest possible value of $\\frac{a b+c}{a+b+c}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 19446,
"solution": "We have $$\\frac{a b+c}{a+b+c}=\\frac{a b-a-b}{a+b+c}+1$$ We note that $\\frac{a b-a-b}{a+b+c}<0 \\Leftrightarrow(a-1)(b-1)<1$, which only occurs when either $a=1$ or $b=1$. Without loss of generality, let $a=1$. Then, we have a value of $$\\frac{-1}{b+c+a}+1$$ We ... |
deepscaler | [
{
"role": "user",
"content": "An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"split": "train",
"index": 19447,
"solution": "This problem can be solved via recursion since we are \"building a string\" of numbers with some condition. We want to create a new string by adding a new digit at the front so we avoid complications($0$ can't be at the front and no digit is less than $9$). There a... |
deepscaler | [
{
"role": "user",
"content": "The third and fourth terms of a geometric sequence are 12 and 16, respectively. What is the first term of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{4}"
} | {
"split": "train",
"index": 19448,
"solution": ""
} |
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