Datasets:

1231czx
/

9b_gemma2pt_iter1_random_pair

gt stringlengths 1 39	rej sequencelengths 1 1	chosen listlengths 4 12	rejected listlengths 4 12
\frac{1}{4}	[ "-2.6666666666666665" ]	[ { "content": "What is the sum of the following infinite geometric series: $\\frac{1}{3} - \\frac{1}{9} + \\frac{1}{27} - \\frac{1}{81} + \\frac{1}{243} - \\frac{1}{729} + \\frac{1}{2187} - \\cdots$?", "role": "user" }, { "content": "To sum the given infinite geometric series: $\\frac{1}{3} - \\frac{...	[ { "content": "What is the sum of the following infinite geometric series: $\\frac{1}{3} - \\frac{1}{9} + \\frac{1}{27} - \\frac{1}{81} + \\frac{1}{243} - \\frac{1}{729} + \\frac{1}{2187} - \\cdots$?", "role": "user" }, { "content": "Let's define the series using sympy.\n```python\nfrom sympy import ...
4	[ "20" ]	[ { "content": "A line segment $\\overline{AB}$ has its midpoint at $M(4,4)$. If the coordinates of point A are known to be $(8,4)$, calculate the total of the x and y coordinates for the other endpoint, point $B$.", "role": "user" }, { "content": "Let's denote unknown $B$ coordinates as $(x, y)$.\nTh...	[ { "content": "A line segment $\\overline{AB}$ has its midpoint at $M(4,4)$. If the coordinates of point A are known to be $(8,4)$, calculate the total of the x and y coordinates for the other endpoint, point $B$.", "role": "user" }, { "content": "To find the missing endpoint $B$, we can use the foll...
187	[ "62" ]	[ { "content": "What is the remainder when $2x^3 - x^2 - 17x + 47$ is divided by $x-5$?", "role": "user" }, { "content": "According to the remainder theorem, if $p(x)$ is a polynomial of degree $n \\geq 1$ and $a$ is a real number, then $r = p(a)$ is the remainder when $p(x)$ is divided by $(x - a)$.\...	[ { "content": "What is the remainder when $2x^3 - x^2 - 17x + 47$ is divided by $x-5$?", "role": "user" }, { "content": "To find the remainder of a polynomial when divided by $x-5$, we can use the remainder theorem.\nAccording to the theorem, if we substitute $x=5$ into the given polynomial, we will ...
1	[ "\\frac{7}{19}" ]	[ { "content": "If $a>0$ and $b>0,$ a new operation $\\nabla$ is defined as follows: $$a \\nabla b = \\frac{a + b}{1 + ab}.$$For example, $$3 \\nabla 6 = \\frac{3 + 6}{1 + 3 \\times 6} = \\frac{9}{19}.$$Calculate $(1 \\nabla 2) \\nabla 3.$", "role": "user" }, { "content": "First, we calculate $1 \\na...	[ { "content": "If $a>0$ and $b>0,$ a new operation $\\nabla$ is defined as follows: $$a \\nabla b = \\frac{a + b}{1 + ab}.$$For example, $$3 \\nabla 6 = \\frac{3 + 6}{1 + 3 \\times 6} = \\frac{9}{19}.$$Calculate $(1 \\nabla 2) \\nabla 3.$", "role": "user" }, { "content": "Let's name $a = 1$, $b = 2$...
8	[ "0" ]	[ { "content": "If Suzy is currently 20 years old and in four years she will be twice Mary's age at that time, what is Mary's current age?", "role": "user" }, { "content": "Let's solve this problem using Python's sympy library.\n```python\nimport sympy as sp\n\n# Let's denote Mary's current age as y\n...	[ { "content": "If Suzy is currently 20 years old and in four years she will be twice Mary's age at that time, what is Mary's current age?", "role": "user" }, { "content": "Let's solve this problem using Python's sympy library.\n```python\nimport sympy as sp\n\n# both will be in years\ncurrent_suzy_ag...
2	[ "2.25" ]	[ { "content": "Alexia designed a logo X inches wide and 1.5 inches tall to be used on her school's notebooks. The school wants the logo on the notebook to be 8 inches wide. How tall, in inches, It will be 6 if it is enlarged proportionately. What is the value of unknown variable X?", "role": "user" }, { ...	[ { "content": "Alexia designed a logo X inches wide and 1.5 inches tall to be used on her school's notebooks. The school wants the logo on the notebook to be 8 inches wide. How tall, in inches, It will be 6 if it is enlarged proportionately. What is the value of unknown variable X?", "role": "user" }, { ...
234	[ "3.9" ]	[ { "content": "If Flynn watches 30 minutes of TV every night on weekdays and an additional 2 hours in total on weekends, how many hours of TV does she watch in 52 weeks?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\n# flynn watches 30 minutes every weekday\...	[ { "content": "If Flynn watches 30 minutes of TV every night on weekdays and an additional 2 hours in total on weekends, how many hours of TV does she watch in 52 weeks?", "role": "user" }, { "content": "First let's figure out how much time Flynn watches TV on weekdays. Since we have to find how much...
4	[ "28" ]	[ { "content": "Bugs are thriving in a garden of 30 plants. The bugs are very hungry today and completely ate 20 plants. The next day, they were still full and only ate half of the remaining plants. After that, they ate only 1 of the plants that were left. How many plants remain?", "role": "user" }, { ...	[ { "content": "Bugs are thriving in a garden of 30 plants. The bugs are very hungry today and completely ate 20 plants. The next day, they were still full and only ate half of the remaining plants. After that, they ate only 1 of the plants that were left. How many plants remain?", "role": "user" }, { ...
21	[ "17996.658227740118" ]	[ { "content": "Let $a \\oslash b = (\\sqrt{2a+b})^3$. If $u \\oslash 4 = 512$, and $v \\oslash 4 = 64$, find the value of $(u+2v) \\div 2$.", "role": "user" }, { "content": "Using sympy,\n```python\nfrom sympy import symbols, Pow, Eq, solve\n\n# define the variables\nu, v = symbols('u v')\n\n# given ...	[ { "content": "Let $a \\oslash b = (\\sqrt{2a+b})^3$. If $u \\oslash 4 = 512$, and $v \\oslash 4 = 64$, find the value of $(u+2v) \\div 2$.", "role": "user" }, { "content": "Since $u \\oslash 4 = 512$ and $v \\oslash 4 = 64$, $u \\oslash 4 = 8^3$ and $v \\oslash 4 = 4^3$.\nUsing these values, we can ...
33	[ "0.3333" ]	[ { "content": "If the price of a stamp is X cents, what is the maximum number of stamps that could be purchased with $\\$32$?\nIf we know the answer to the above question is 96, what is the value of unknown variable X?", "role": "user" }, { "content": "This problem requires a method of solving Dioph...	[ { "content": "If the price of a stamp is X cents, what is the maximum number of stamps that could be purchased with $\\$32$?\nIf we know the answer to the above question is 96, what is the value of unknown variable X?", "role": "user" }, { "content": "Let's define our cost constraint in sympy.\n```...
90	[ "144" ]	[ { "content": "In a right triangle, the ratio of the measures of the acute angles is 8:1. What is the measure, in degrees, of the largest angle in the triangle?", "role": "user" }, { "content": "We know that the sum of the three angles in a triangle is 180 degrees.\nLet's represent the ratio of the a...	[ { "content": "In a right triangle, the ratio of the measures of the acute angles is 8:1. What is the measure, in degrees, of the largest angle in the triangle?", "role": "user" }, { "content": "Let's denote the smaller, larger, and largest angles by $A$, $B$, and $C$ respectively.\nLet's also denote...
0	[ "-\\frac{1}{6}" ]	[ { "content": "If $z = 2y$ and $y = \\displaystyle\\frac{1}{3x+1}$, what is the value of $x$ when $z = 2$?", "role": "user" }, { "content": "To solve this problem, we first substitute $y = \\frac{1}{3x+1}$ into the first equation and solve for $z$ in terms of $x$:\n```python\nfrom sympy import symbol...	[ { "content": "If $z = 2y$ and $y = \\displaystyle\\frac{1}{3x+1}$, what is the value of $x$ when $z = 2$?", "role": "user" }, { "content": "Given $z = 2y$ and $y = \\frac{1}{3x+1}$, we have $z = 2 \\cdot \\frac{1}{3x+1}$.\nFrom this equation, we can solve for $x$ when $z=2$.\nLet's rewrite the equat...
2	[ "343" ]	[ { "content": "How many natural-number factors does $N$ have if $N = 2^4 \\cdot 3^3 \\cdot 5^2 \\cdot X$? The answer is 180. What is the value of unknown variable X?", "role": "user" }, { "content": "The number of natural number factors is the product of all prime factors plus one.\nLet's find all th...	[ { "content": "How many natural-number factors does $N$ have if $N = 2^4 \\cdot 3^3 \\cdot 5^2 \\cdot X$? The answer is 180. What is the value of unknown variable X?", "role": "user" }, { "content": "We can use Python's sympy library to solve this problem. Let's put the expression $2^4 \\cdot 3^3 \\c...
5040	[ "7" ]	[ { "content": "What is the product of the whole number values for $c$ that make the quadratic equation $8x^2+15x+c=0$ have two different real roots?", "role": "user" }, { "content": "In order to make the quadratic equation have two different real roots, the discriminant ($b^2-4ac$) must be greater th...	[ { "content": "What is the product of the whole number values for $c$ that make the quadratic equation $8x^2+15x+c=0$ have two different real roots?", "role": "user" }, { "content": "Let's first check when there is a real root:\n```python\nfrom sympy import Symbol, solve\n\nx = Symbol('x')\nc = Symbo...
60	[ "54" ]	[ { "content": "If a number is the product of 6! and 2^5, how many positive divisors does it have?", "role": "user" }, { "content": "We can find the number of positive divisors by finding the total number of prime factors.\n\nThe prime factorization of $6!$ is\n\n<div style=\"border: 1px solid #ccc; p...	[ { "content": "If a number is the product of 6! and 2^5, how many positive divisors does it have?", "role": "user" }, { "content": "First, let's find all the divisors of the number. To do that we have to find all integers that divide this number without remainder.\nThen we can iterate through the div...
131	[ "104" ]	[ { "content": "The city's environmental division is expanding and needs to employ professionals who can handle multiple environmental concerns. Specifically, they need 85 workers to tackle issues related to water, another 73 to address problems with air quality, but they've found that 27 individuals are qualifie...	[ { "content": "The city's environmental division is expanding and needs to employ professionals who can handle multiple environmental concerns. Specifically, they need 85 workers to tackle issues related to water, another 73 to address problems with air quality, but they've found that 27 individuals are qualifie...
6	[ "18" ]	[ { "content": "In a Cartesian plane, the line segment AB has point A located at $(6, 12)$ and point B located at $(0, -6)$. What is the aggregate of the x-coordinate and y-coordinate of the midpoint of AB?", "role": "user" }, { "content": "Let's first check whether point B is to the left or right of ...	[ { "content": "In a Cartesian plane, the line segment AB has point A located at $(6, 12)$ and point B located at $(0, -6)$. What is the aggregate of the x-coordinate and y-coordinate of the midpoint of AB?", "role": "user" }, { "content": "Let's write a python script to find the coordinates of the mi...
18	[ "36" ]	[ { "content": "Given the matrix \\[A = \\begin{pmatrix} 9 & 18 \\\\ -6 & -12 \\end{pmatrix}\\], scalar \\[k = 3\\], and matrix \\[B = \\begin{pmatrix} 2 & 0 \\\\ 0 & 2 \\end{pmatrix}\\], calculate the determinant of matrix \\[D = k \\cdot B + A\\].", "role": "user" }, { "content": "First, we calculat...	[ { "content": "Given the matrix \\[A = \\begin{pmatrix} 9 & 18 \\\\ -6 & -12 \\end{pmatrix}\\], scalar \\[k = 3\\], and matrix \\[B = \\begin{pmatrix} 2 & 0 \\\\ 0 & 2 \\end{pmatrix}\\], calculate the determinant of matrix \\[D = k \\cdot B + A\\].", "role": "user" }, { "content": "Let's first check ...
3000	[ "13300" ]	[ { "content": "Mr. Jesiah is a dairy cow farmer with cows producing 200 gallons of milk every day. The expenses for maintenance of the dairy farm and purchase of feeds is $x per month. Calculate his total income in June if he sells 1 gallon of milk at $3.55. The answer is 18300. What is the value of unknown vari...	[ { "content": "Mr. Jesiah is a dairy cow farmer with cows producing 200 gallons of milk every day. The expenses for maintenance of the dairy farm and purchase of feeds is $x per month. Calculate his total income in June if he sells 1 gallon of milk at $3.55. The answer is 18300. What is the value of unknown vari...
20	[ "14" ]	[ { "content": "Ali had a stock of 800 books in his Room. He sold 60 on Monday, 10 on Tuesday, x on Wednesday, 44 on Thursday and 66 on Friday. 600 books were not sold. What is the value of unknown variable x?", "role": "user" }, { "content": "Let's write this out as an algebraic expression.\n```pytho...	[ { "content": "Ali had a stock of 800 books in his Room. He sold 60 on Monday, 10 on Tuesday, x on Wednesday, 44 on Thursday and 66 on Friday. 600 books were not sold. What is the value of unknown variable x?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\n# ...
\frac{189}{2}\sqrt{3}	[ "38.971" ]	[ { "content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled, with $\\angle AEB=\\angle BEC = \\angle CED = 60^\\circ$, and $AE=24$. [asy]\npair A, B, C, D, E;\nA=(0,20.785);\nB=(0,0);\nC=(9,-5.196);\nD=(13.5,-2.598);\nE=(12,0);\ndraw(A--B--C--D--E--A);\ndraw(B--E);\ndr...	[ { "content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled, with $\\angle AEB=\\angle BEC = \\angle CED = 60^\\circ$, and $AE=24$. [asy]\npair A, B, C, D, E;\nA=(0,20.785);\nB=(0,0);\nC=(9,-5.196);\nD=(13.5,-2.598);\nE=(12,0);\ndraw(A--B--C--D--E--A);\ndraw(B--E);\ndr...
0.5	[ "8.5" ]	[ { "content": "On a Sunday morning, Josephine sold milk in the farm stall. The buyers brought their containers. She filled three containers with two liters each, two containers with 0.75 liters each, and five containers with x liters each. How many liters of milk did Josephine sell in total?\nIf we know the answ...	[ { "content": "On a Sunday morning, Josephine sold milk in the farm stall. The buyers brought their containers. She filled three containers with two liters each, two containers with 0.75 liters each, and five containers with x liters each. How many liters of milk did Josephine sell in total?\nIf we know the answ...
22	[ "76" ]	[ { "content": "Trisha needed to buy some groceries for herself and her dog. She spent $17 on meat, $x on chicken, $43 on all the veggies, $5 on the eggs, and finally $45 on her dog’s food. When she left the store, she had only $35 left. She brought 167 money with her at the beginning. What is the value of unknow...	[ { "content": "Trisha needed to buy some groceries for herself and her dog. She spent $17 on meat, $x on chicken, $43 on all the veggies, $5 on the eggs, and finally $45 on her dog’s food. When she left the store, she had only $35 left. She brought 167 money with her at the beginning. What is the value of unknow...
480	[ "980" ]	[ { "content": "Jessie's community built a new hedge. Each section of the hedge used 30 pieces of concrete blocks that cost $2 per piece. The hedge has eight sections. How much did they spend on the concrete blocks?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```pyth...	[ { "content": "Jessie's community built a new hedge. Each section of the hedge used 30 pieces of concrete blocks that cost $2 per piece. The hedge has eight sections. How much did they spend on the concrete blocks?", "role": "user" }, { "content": "The total cost consists of the cost for the blocks p...
60	[ "40" ]	[ { "content": "If the age difference between Declan's elder son and younger son is 10 years and the elder son is currently 40 years old, how old will the younger son be in 30 years?", "role": "user" }, { "content": "Let's solve this problem using Python's sympy library.\n```python\nimport sympy as sp...	[ { "content": "If the age difference between Declan's elder son and younger son is 10 years and the elder son is currently 40 years old, how old will the younger son be in 30 years?", "role": "user" }, { "content": "Let's solve this problem using Python's sympy library.\n```python\nimport sympy as sp...
17	[ "0" ]	[ { "content": "Within the drama society, there are 60 actors. Among them, 36 are attending math lectures, 27 are taking physics, and there is an overlap of 20 students who are engaged in both subjects. How many members of the drama society are not attending any math or physics lectures?", "role": "user" },...	[ { "content": "Within the drama society, there are 60 actors. Among them, 36 are attending math lectures, 27 are taking physics, and there is an overlap of 20 students who are engaged in both subjects. How many members of the drama society are not attending any math or physics lectures?", "role": "user" },...
3	[ "4" ]	[ { "content": "Determine the number of nonzero terms in the expansion of $$(x+4)(2x^2+3x+9)-3(x^3-2x^2+7x)?$$", "role": "user" }, { "content": "To determine the number of nonzero terms in the expansion, we can expand the expression directly. We'll use sympy to do the computations:\n```python\nfrom sy...	[ { "content": "Determine the number of nonzero terms in the expansion of $$(x+4)(2x^2+3x+9)-3(x^3-2x^2+7x)?$$", "role": "user" }, { "content": "We can compute this directly using Python.\n```python\nimport sympy\n\nx = sympy.Symbol('x')\n\n# Multiply out and expand\nres = ((x+4)(2x*2+3x+9) - 3*(x...
\frac{11}{15}	[ "\\frac{11}{18}" ]	[ { "content": "From the integers between 1 and 6 inclusive, two distinct numbers are chosen at random. Determine the probability that their highest common divisor is 1. Provide your answer as a simplified fraction.", "role": "user" }, { "content": "I will find the answer by enumerating all possible p...	[ { "content": "From the integers between 1 and 6 inclusive, two distinct numbers are chosen at random. Determine the probability that their highest common divisor is 1. Provide your answer as a simplified fraction.", "role": "user" }, { "content": "Let's write a python script to find the fraction of ...
7	[ "\\frac{21}{2}" ]	[ { "content": "Joe’s mother gave him $56 to go to the store. Joe bought x notebooks and 2 books. Each notebook costs $4 and each book costs $7. How much money does Joe have left?\nIf we know the answer to the above question is 14, what is the value of unknown variable x?", "role": "user" }, { "conten...	[ { "content": "Joe’s mother gave him $56 to go to the store. Joe bought x notebooks and 2 books. Each notebook costs $4 and each book costs $7. How much money does Joe have left?\nIf we know the answer to the above question is 14, what is the value of unknown variable x?", "role": "user" }, { "conten...
6	[ "-6+\\sqrt{61}" ]	[ { "content": "In a right-angled triangle, called $GHI$, with angle $H$ being 90 degrees, it is given that the hypotenuse $GI$ is $\\sqrt{61}$ units long, while leg $GH$ is 5 units. A circle, drawn with its center positioned on $GH$, gently touches both the hypotenuse $GI$ and side $HI$. When we name the touchpo...	[ { "content": "In a right-angled triangle, called $GHI$, with angle $H$ being 90 degrees, it is given that the hypotenuse $GI$ is $\\sqrt{61}$ units long, while leg $GH$ is 5 units. A circle, drawn with its center positioned on $GH$, gently touches both the hypotenuse $GI$ and side $HI$. When we name the touchpo...
\frac{8}{7}	[ "8" ]	[ { "content": "The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \\log_b 7^7$. What is $b$?", "role": "user" }, { "content": "I can find the answer by taking logarithms on both sides of the equation\n$7^{x+7} = 8^x.$\n```python\nfrom sympy import Symbol, Eq, log, solve\n\...	[ { "content": "The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \\log_b 7^7$. What is $b$?", "role": "user" }, { "content": "Since $8$ and $7$ are close, we can solve this equation numerically in Python using a root-finding algorithm, which would be faster than algebraic...
208000	[ "1872000" ]	[ { "content": "If the number of people living per cubic yard in a city is 80, how many additional people are there in a city with 9000 cubic yards compared to a city with 6400 cubic yards?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\n# number of people per...	[ { "content": "If the number of people living per cubic yard in a city is 80, how many additional people are there in a city with 9000 cubic yards compared to a city with 6400 cubic yards?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\n# let's represent a cu...
3	[ "2.0625" ]	[ { "content": "Maria has 4 dimes, 4 quarters, and 7 nickels in her piggy bank. Her mom gives her 5 quarters. How much money, in dollars, does Maria have now?", "role": "user" }, { "content": "Let's write this out as an algebraic expression.\n```python\nquarters_in_piggy_bank = 4\nquarters_mom_gave = ...	[ { "content": "Maria has 4 dimes, 4 quarters, and 7 nickels in her piggy bank. Her mom gives her 5 quarters. How much money, in dollars, does Maria have now?", "role": "user" }, { "content": "Let's solve this problem using Python code.\n```python\nquarters = 4\ndimes = 4\nnickels = 7\n# dimes is 0.1 ...
36	[ "108" ]	[ { "content": "Adam is assessing his assets and found that he has 15 identical rare coins. If a cluster of 5 of these coins can be sold for 12 dollars, how much can Adam expect to receive for his entire collection of these coins?", "role": "user" }, { "content": "Let's use math to solve this problem....	[ { "content": "Adam is assessing his assets and found that he has 15 identical rare coins. If a cluster of 5 of these coins can be sold for 12 dollars, how much can Adam expect to receive for his entire collection of these coins?", "role": "user" }, { "content": "Let's use Python code to solve this p...
84	[ "32" ]	[ { "content": "If Dave has $46 and Kyle has $12 less than 3 times what Dave has, how much money does Kyle have after spending one-third of it on snowboarding?", "role": "user" }, { "content": "Let's solve this problem using Python's sympy library.\n```python\nimport sympy as sp\n\n# first, let's deno...	[ { "content": "If Dave has $46 and Kyle has $12 less than 3 times what Dave has, how much money does Kyle have after spending one-third of it on snowboarding?", "role": "user" }, { "content": "Let's solve this problem using Python's sympy library.\n```python\nimport sympy as sp\n\n# Dave has 46 dolla...
14	[ "11" ]	[ { "content": "What is the number of diagonals in a regular seven-sided polygon?", "role": "user" }, { "content": "We'll need to consider the different types of diagonals that can be drawn.\nA \"vertex\" can be connected to the other 5 vertices to make 5 types of diagonals.\nWe can use the definition...	[ { "content": "What is the number of diagonals in a regular seven-sided polygon?", "role": "user" }, { "content": "To find the number of diagonals in a polygon, we need to find the number of edges of the form (v1, v3) where $v1$ and $v3$ are vertices and $(v1, v2)$ and $(v2, v3)$ are not edges.\nThe ...
16	[ "31" ]	[ { "content": "What is the largest four-digit number whose digits add up to X? The answer is 9700. What is the value of unknown variable X?", "role": "user" }, { "content": "We're looking for a four-digit number with digits $a, b, c, d$ such that $1000a + 100b + 10c + d = 9700$.\nLet $a = 9$, $b = 7$...	[ { "content": "What is the largest four-digit number whose digits add up to X? The answer is 9700. What is the value of unknown variable X?", "role": "user" }, { "content": "Let's define our variables and generate all 4-digit numbers using list comprehension.\n```python\nfrom sympy import symbols, so...
6	[ "15" ]	[ { "content": "If Alexa spent a week and 2 days on vacation, which is equivalent to 3/4ths of the time it took Ethan to learn 12 fencing tricks, and Joey spent half as much time as Ethan learning to swim, how many days did it take Joey to learn swimming?", "role": "user" }, { "content": "Let's solve ...	[ { "content": "If Alexa spent a week and 2 days on vacation, which is equivalent to 3/4ths of the time it took Ethan to learn 12 fencing tricks, and Joey spent half as much time as Ethan learning to swim, how many days did it take Joey to learn swimming?", "role": "user" }, { "content": "Let's solve ...
1	[ "-2" ]	[ { "content": "Let $\\omega$ be a complex number such that $\\omega^7 = 1$ and $\\omega \\ne X Compute\n\\[\\omega^{16} + \\omega^{18} + \\omega^{20} + \\dots + \\omega^{54}.\\]\nIf we know the answer to the above question is -1, what is the value of unknown variable X?", "role": "user" }, { "conten...	[ { "content": "Let $\\omega$ be a complex number such that $\\omega^7 = 1$ and $\\omega \\ne X Compute\n\\[\\omega^{16} + \\omega^{18} + \\omega^{20} + \\dots + \\omega^{54}.\\]\nIf we know the answer to the above question is -1, what is the value of unknown variable X?", "role": "user" }, { "conten...
\frac{8}{16575}	[ "\\frac{8}{525150}" ]	[ { "content": "Imagine dealing three cards one after the other from a complete set of 52 cards, without returning them to the deck. What is the likelihood that you will pull a Jack first, followed by a Queen, and then a King?", "role": "user" }, { "content": "The key insight to this problem is that t...	[ { "content": "Imagine dealing three cards one after the other from a complete set of 52 cards, without returning them to the deck. What is the likelihood that you will pull a Jack first, followed by a Queen, and then a King?", "role": "user" }, { "content": "Since we're looking at the probability of...