messages listlengths 2 2 |
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[
{
"role": "user",
"content": "Define the operation $a\\nabla b = 2 + b^a$. What is the value of $(1\\nabla 2) \\nabla 3$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to figure out how to solve this problem step by step. The operation given is $a \\nabla b = 2 + b^a$. The que... |
[
{
"role": "user",
"content": "How many integers $n$ satisfy $(n+3)(n-7) \\le 0$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem again. The question is asking how many integers $ n $ satisfy the inequality $ (n + 3)(n - 7) \\leq 0 $. \n\nFirst, I remember th... |
[
{
"role": "user",
"content": "What is the sum of all values of $x$ for which $\\sqrt{(x+3)^{2}} = 7$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking for the sum of all values of $ x $ that satisfy the equation $ \\sqrt{(x+3)^2... |
[
{
"role": "user",
"content": "Determine the smallest value that $y$ can take on in the equation $y = x^2 + 12x + 5$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the smallest value that $ y $ can take in the equation $ y = x^2 + 12x + 5 $. Let me think about how to app... |
[
{
"role": "user",
"content": "What is the largest prime factor of $5! + 6!$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking for the largest prime factor of $5! + 6!$. \n\nFirst, I need to calculate the values of 5! and 6!. \n\... |
[
{
"role": "user",
"content": "Evaluate $\\log_464$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem again. I need to evaluate the logarithm of 64 with base 4. So, the question is asking for log base 4 of 64. \n\nFirst, I remember that a logarithm log_b(a) represen... |
[
{
"role": "user",
"content": "What is the units digit of the product of all natural numbers from 1 to 99, inclusive?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is asking for the units digit of the product of all natural numbers from 1 to 99. Hmm, so that's 99 fa... |
[
{
"role": "user",
"content": "Given that a line segment starts at (2, 5), is 10 units long, and ends at the point (-6, y) where y is greater than 0, what is the value of y?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to figure out this problem again. So, there's a line segme... |
[
{
"role": "user",
"content": "What is the smallest positive integer n such that the matrix \n\\[\\begin{pmatrix} \\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\ -\\frac{\\sqrt{3}}{2} & \\frac{1}{2} \\end{pmatrix}^n = \\mathbf{I}\\]"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to figu... |
[
{
"role": "user",
"content": "If $x$ has 12 positive factors and 12 and 15 are factors of $x$, what is the value of $x$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem step by step. The problem says that x has 12 positive factors and 12 and 15 are factors o... |
[
{
"role": "user",
"content": "What is the value of $2 \\times 2^5 - 8^{58} \\div 8^{56}$ when computed?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The expression given is $2 \\times 2^5 - 8^{58} \\div 8^{56}$. I need to compute its value.\n\nFir... |
[
{
"role": "user",
"content": "What is the greatest integer $n$ such that $n^2 - 11n +24 \\leq 0$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking for the greatest integer $ n $ such that $ n^2 - 11n + 24 \\leq 0 $. \n\nFirst, I... |
[
{
"role": "user",
"content": "Consider the function $f(x) = x^2 +2\\sqrt{x}$. Evaluate $2f(2) - f(8)$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to solve this problem again step by step to make sure I didn't make any mistakes. The function given is $ f(x) = x^2 + 2\\sqrt{... |
[
{
"role": "user",
"content": "Determine the value of $n$ that satisfies the equation $10^n = 10^{-5} \\times \\sqrt{\\frac{10^{73}}{0.001}}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem step by step. The equation given is $10^n = 10^{-5} \\times \\sqrt{\... |
[
{
"role": "user",
"content": "For what value of $a$ does the equation $3(2x-a) = 2(3x+12)$ have infinitely many solutions $x$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is asking for the value of 'a' such that the equation 3(2x - a) = 2(3x + 12) has infinitely ... |
[
{
"role": "user",
"content": "Compute $\\arcsin X Express your answer in radians. The answer is 0. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to figure out this problem step by step. The question is asking to compute $\\arcsin X$ and... |
[
{
"role": "user",
"content": "If $y = -x^2 + X and $x$ is a real number, The maximum value possible for $y$ is 5. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to figure out the problem again. So, we have the equation y equals negative x... |
[
{
"role": "user",
"content": "What is the integer value of $n$, where $0 \\le n \\le 7$, such that $n \\equiv -3737 \\pmod{8}$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is to find the integer $ n $ between 0 and 7 (inclusive) such that $ n \\equiv -3737 \\pmod... |
[
{
"role": "user",
"content": "Three positive integers $a,$ $b,$ and $x$ form an O'Hara triple $(a,b,x)$ if $\\sqrt{a}+\\sqrt{b}=x.$ For example, $(1,4,3)$ is an O'Hara triple because $\\sqrt{1}+\\sqrt{4}=3.$\n\nIf $(36,25,x)$ is an O'Hara triple, determine the value of $x.$"
},
{
"role": "assistant"... |
[
{
"role": "user",
"content": "Compute $\\dbinom{14}{11}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to compute the binomial coefficient 14 choose 11. Hmm, binomial coefficients... right, they represent the number of ways to choose k elements from a set of n elements... |
[
{
"role": "user",
"content": "Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem again. So, we have two functions: f(x) = x³ + 3 and g(x) = 2x² + 2x + 1. The question is asking for g(f(-2)). \n\nFirst, I... |
[
{
"role": "user",
"content": "The sides of this parallelogram measure 7,9, $8y-1$ and $2x+3$ units, consecutively. What is the value of $x+y$?\n\n[asy]draw((0,0)--(21,0)--(30,25)--(9,25)--cycle);\nlabel(\"$8y-1$\",(10,0),S);\nlabel(\"9\",(25.5,12.5),E);\nlabel(\"7\",(19.5,25),N);\nlabel(\"$2x+3$\",(4.5,12.5... |
[
{
"role": "user",
"content": "What is the square of the greatest integer less than or equal to the square root of 17?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking for the square of the greatest integer that is less than or e... |
[
{
"role": "user",
"content": "Mark borrows $10$ dollars from Emily with a simple interest rate of $15\\%$ everyday. What is the least integer number of days after which Mark will have to pay her back at least twice as much as he borrowed?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so ... |
[
{
"role": "user",
"content": "Let $p$ and $q$ satisfy $pq=9$ and $p+q=6$. What is the value of $p^2 + q^2$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem says that p and q satisfy pq = 9 and p + q = 6. We need to find p² + q².\n\nFirst, I remember there's a formula... |
[
{
"role": "user",
"content": "How many perfect squares are there between 20 and 150?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to figure out how many perfect squares are between 20 and 150. Let me start by recalling what a perfect square is. A perfect square is an integer t... |
[
{
"role": "user",
"content": "The number of white balls and red balls in a jar is in the ratio of $3:2$. If there are X white balls, how many red balls are there?\nIf we know the answer to the above question is 6, what is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<t... |
[
{
"role": "user",
"content": "If $c$ is a constant that makes $x^2 + 100x + c$ equal to the square of a binomial, what is the value of $c$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is asking for the value of $ c $ in the quadratic expression $ x^2 + 100x + c $... |
[
{
"role": "user",
"content": "Let $F_n$ be the $n$th Fibonacci number, where as usual $F_1 = F_2 = 1$ and $F_{n + X = F_n + F_{n - 1}.$ Then\n\\[\\prod_{k = 2}^{100} \\left( \\frac{F_k}{F_{k - 1}} - \\frac{F_k}{F_{k + 1}} \\right) = \\frac{F_a}{F_b}\\]for some positive integers $a$ and $b.$ Enter the orde... |
[
{
"role": "user",
"content": "Determine the smallest positive integer that satisfies the congruence $4x \\equiv 13 \\pmod{27}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the smallest positive integer x that satisfies the congruence 4x ≡ 13 mod 27. Let me start by re... |
[
{
"role": "user",
"content": "What is the sum of the the roots of the equation $4x^3 + 5x^2 - 8x = 0$? Express your answer as a decimal to the nearest hundredth."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the sum of the roots of the equation $4x^3 + 5x^2 - 8x = 0$. L... |
[
{
"role": "user",
"content": "Given the functions $f(x) = 3x^2 - 2x + 4$ and $g(x) = x^2 - kx - 6$, if $f(10) - g(10) = 10$, what is the value of $k$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem again step by step. The problem says that for the functions... |
[
{
"role": "user",
"content": "Determine the distance between the vertices of the hyperbola defined by the equation $\\frac{x^2}{99} - \\frac{y^2}{36} = 1$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the distance between the vertices of the hyperbola given by the equa... |
[
{
"role": "user",
"content": "Engineers use the formula $L=\\frac{25T^4}{H^2}$ to find the crushing load for square pillars. If $T = X and $H = 8$, The value of $L$ is 100. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is ab... |
[
{
"role": "user",
"content": "If the area of a region in the plane is 4, and we apply the matrix \\[\\begin{pmatrix} 2 & -1 \\\\ 7 & 2 \\end{pmatrix}\\]to the region, resulting in the region $S'$, what is the area of $S'$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so the problem is a... |
[
{
"role": "user",
"content": "For the opening home game of the baseball season, the Madd Batters minor league baseball team offered the following incentives to its fans:\n\nEvery 75th fan who entered the stadium got a coupon for a free hot dog.\n\nEvery 30th fan who entered the stadium got a coupon for a fr... |
[
{
"role": "user",
"content": "A professor is assigning grades to a class of X students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign grades to all his students?\nIf we know the answer to the above question is 59049, what is the value of unknown variab... |
[
{
"role": "user",
"content": "Joe is studying a bacteria population. There are X bacteria present at 3:00 p.m. and the population doubles every 3 minutes. Assuming none of the bacteria die, how many bacteria are present at 3:15 p.m. The same day is 640. What is the value of unknown variable X?"
},
{
... |
[
{
"role": "user",
"content": "If $f$ is a linear function with $f(6)-f(2)=12$, what is the value of $f(12)-f(2)$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem says that f is a linear function, and that f(6) minus f(2) equals 12. We need to find f(12) minus f(2). \... |
[
{
"role": "user",
"content": "Kim earned scores of 87, 83 and 88 on her first three mathematics examinations. If Kim receives a score of X on the fourth exam, Her average will increase by 1. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see.... |
[
{
"role": "user",
"content": "Amy works for 36 hours per week for 10 weeks during the summer, making $\\$3000$. If she works for 30 weeks during the school year at the same rate of pay and needs to make another X$, how many hours per week must she work?\nIf we know the answer to the above question is 12, wh... |
[
{
"role": "user",
"content": "If $S$, $H$, and $E$ are all distinct non-zero digits less than $5$ and the following is true, find the sum of the three values $S$, $H$, and $E$, expressing your answer in base X$. $$\\begin{array}{c@{}c@{}c@{}c} &S&H&E_5\\\\ &+&H&E_5\\\\ \\cline{2-4} &S&E&S_5\\\\ \\end{array}... |
[
{
"role": "user",
"content": "Find the units digit of the sum, $$ 1! + 2! + 3! + \\cdots + 2006!. $$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the units digit of the sum from 1! up to 2006!. Let me start by recalling what the units digit means. The units digit of a ... |
[
{
"role": "user",
"content": "How many sides does a regular polygon have if its interior angles measure 162 degrees?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking how many sides a regular polygon has if each of its interior a... |
[
{
"role": "user",
"content": "Determine the number of integers that satisfy the inequality $(x + 3)^2 \\leq 1$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking for the number of integers that satisfy the inequality $(x + 3)^2 \... |
[
{
"role": "user",
"content": "What is the value of $(-2) \\$$ $3$ if $a \\$$ $b$ is defined as $a(b + 1) + ab$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to figure out the value of (-2) $ 3, where the operator $ is defined as a(b + 1) + ab. Let me start by understanding the... |
[
{
"role": "user",
"content": "The measure of each exterior angle of a regular polygon is X degrees. What is the sum of the measures of the interior angles, In degrees, the answer is 1800. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. Th... |
[
{
"role": "user",
"content": "The complement of an angle is $5^{\\circ}$ more than four times the angle. What is the number of degrees in the measure of the angle?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem says the complement of an angle is 5 degrees more than ... |
[
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with angle $A$ congruent to angle $B$. The measure of angle $C$ is 30 degrees more than the measure of angle $A$. What is the number of degrees in the measure of angle $C$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackl... |
[
{
"role": "user",
"content": "Find the smallest positive integer $b$ for which $x^2+bx+2008$ factors into a product of two binomials, each having integer coefficients."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem step by step. The question is asking for t... |
[
{
"role": "user",
"content": "Let \\[f(n) =\n\\begin{cases}\nn^2-1 & \\text{ if }n < 4,\n\\\\ 3n-2 & \\text{ if }n \\geq 4.\n\\end{cases}\n\\]Find $f(f(f(2)))$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The function f(n) is defined piecewise, s... |
[
{
"role": "user",
"content": "The fifth term of an arithmetic sequence is $11$. If the difference between two consecutive terms is $1$, what is the product of the first two terms?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem step by step. So, the problem ... |
[
{
"role": "user",
"content": "If the bottom row of a stack of logs has 12 logs and each successive row has one less log than the previous row, with three logs at the top, how many logs are in the entire stack?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is about ... |
[
{
"role": "user",
"content": "For what value of $n$ is $3^3-5=4^2+n$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to figure out the problem again. The question is asking for the value of n in the equation 3³ - 5 = 4² + n. Alright, first, I need to solve for n here.\n\nStarti... |
[
{
"role": "user",
"content": "A bowling ball is a solid ball with a spherical surface and diameter 30 cm. To custom fit a bowling ball for each bowler, three holes are drilled in the ball. Bowler Kris has holes drilled that are 8 cm deep and have diameters of 2 cm, X cm, and 3 cm. Assuming the three holes a... |
[
{
"role": "user",
"content": "If $(x + y)^2 = 25$ and $xy = X$, The value of $x^2 + y^2$ is 13. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem step by step. So, we have two equations here:\n\n1. $(x + y)^2 = 25$\n2. ... |
[
{
"role": "user",
"content": "Let $\\bold{v} = \\begin{pmatrix} 5 \\\\ -3 \\end{pmatrix}$ and $\\bold{w} = \\begin{pmatrix} X \\\\ -2 \\end{pmatrix}$. Find the area of the parallelogram with vertices $\\bold{0}$, $\\bold{v}$, $\\bold{w}$, and $\\bold{v} + \\bold{w}$.\nIf we know the answer to the above que... |
[
{
"role": "user",
"content": "The area of the floor in a square room is 225 square feet. The homeowners plan to cover the floor with rows of 6-inch by 6-inch tiles. How many tiles will be in each row?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem says the area of t... |
[
{
"role": "user",
"content": "If $y=\\frac{12x^4+4x^3+9x^2+5x+3}{3x^4+2x^3+8x^2+3x+1}$, at what value of $y$ will there be a horizontal asymptote?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to figure out the value of y where there's a horizontal asymptote for the given funct... |
[
{
"role": "user",
"content": "What is the number of square meters in the area of a circle with diameter 4 meters? Express your answer in terms of $\\pi$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the area of a circle with a diameter of 4 meters. Let me start by reca... |
[
{
"role": "user",
"content": "What are all the solutions to the equation $\\sqrt{5+2z} = 11$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve the equation √(5 + 2z) = 11. Hmm, first step is to eliminate the square root. To do that, I remember that squaring both sides of ... |
[
{
"role": "user",
"content": "A wooden model of a square pyramid has a base edge of 12 cm and an altitude of 8 cm. A cut is made parallel to the base of the pyramid that separates it into two pieces: a smaller pyramid and a frustum. Each base edge of the smaller pyramid is 6 cm and its altitude is 4 cm. How... |
[
{
"role": "user",
"content": "Calculate the value of the expression: $\\frac{1}{2} \\times 4 \\times \\frac{1}{8} \\times 16 \\times \\frac{1}{32} \\times 64 \\times \\frac{1}{128} \\times 256 \\times \\frac{1}{512} \\times 1024$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. ... |
[
{
"role": "user",
"content": "The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem again. So, we have two positive integers. Their sum of... |
[
{
"role": "user",
"content": "Suppose $\\cos Q = 0.4$ in the diagram below. What is $QR$?\n\n\n\n[asy]\npair P,Q,R;\nP = (0,0);\nQ = (6,0);\nR = (0,6*tan(acos(0.4)));\ndraw(P--Q--R--P);\ndraw(rightanglemark(Q,P,R,18));\nlabel(\"$P$\",P,SW);\nlabel(\"$Q$\",Q,SE);\nlabel(\"$R$\",R,N);\nlabel(\"$12$\",Q/2,S);... |
[
{
"role": "user",
"content": "Given vectors $\\mathbf{a} = \\begin{pmatrix} 5 \\\\ 1 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 2 \\\\ 4 \\end{pmatrix}$, what is the area of the triangle formed by the points $\\mathbf{0}$, $\\mathbf{a}$, and $\\mathbf{b}$?"
},
{
"role": "assistant",
"co... |
[
{
"role": "user",
"content": "What is $3^{-1} + 3^{-2} \\pmod{25}$? Express your answer as an integer from $0$ to X$, inclusive. The answer is 6. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem step by step. The quest... |
[
{
"role": "user",
"content": "If $\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = X then find\n\\[\\begin{vmatrix} a - c & b - d \\\\ c & d \\end{vmatrix}. The answer is 5. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem sa... |
[
{
"role": "user",
"content": "The sides of triangle ABC measure 6 units, 8 units, and 10 units. The width of a rectangle with an equal area to the triangle is 4 units. What is the perimeter of this rectangle, in units?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem ... |
[
{
"role": "user",
"content": "To make a profit in an 8-hour workday, a widget company has to pay $\\$500$ in maintenance fees and $\\$15$ per hour to each worker. Each worker produces 5 widgets per hour, which are sold for $\\$3.10$ each. What is the minimum number of workers the company needs to hire?"
}... |
[
{
"role": "user",
"content": "If $\\det \\mathbf{A} = 2$ and $\\det \\mathbf{B} = 12,$ then find $\\det (\\mathbf{A} \\mathbf{B}).$"
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to figure this out again. The problem says that the determinant of matrix A is 2 and the determinan... |
[
{
"role": "user",
"content": "What is the value of the whole number N, when N/3 is between 7.5 and 8?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to solve this problem again. The question is asking for the whole number N such that when you divide it by 3, the result is betwe... |
[
{
"role": "user",
"content": "What is the sum of the integers from $-30$ to $50$, inclusive?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is to find the sum of all integers from -30 to 50, inclusive. Alright, first step, I need to figure out how many terms there a... |
[
{
"role": "user",
"content": "Suppose that $3$ is a factor of $a$, $a$ is a divisor of $12$, and $a$ is positive. What is the number of possible values of $a$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem again. The question is asking for the number of positive... |
[
{
"role": "user",
"content": "Compute $35^{1723} - 16^{1723}$ modulo 6."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to compute $35^{1723} - 16^{1723}$ modulo 6. Let me start by breaking down each term individually.\n\nFirst, I remember that when dealing with modular arithmeti... |
[
{
"role": "user",
"content": "Determine the sum of all positive factors of 24."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to figure out the sum of all positive factors of 24. Hmm, I remember that factors are numbers that divide another number without leaving a remainder. So ... |
[
{
"role": "user",
"content": "Given that the area of rectangle ABCD, with vertices A(0, 0), B(0, 4), C(x, 4), and D(x, 0), is 28 square units and x is greater than 0, what is the value of x?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. So, we have... |
[
{
"role": "user",
"content": "A parabola with equation $y = x^2 + bx + c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is about finding the value of $ c $ in the parabola equation $ y = x^2 + bx + c $, g... |
[
{
"role": "user",
"content": "What is the product of 2.4 and 0.2?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to find the product of 2.4 and 0.2. Hmm, decimals can sometimes be tricky, but I remember that there's a way to multiply them without converting to fractions.... |
[
{
"role": "user",
"content": "Express the fraction 5/8 as a decimal."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to convert the fraction 5/8 into a decimal. Let me start by recalling how to do this. Fractions can be converted to decimals by dividing the numerator by the denom... |
[
{
"role": "user",
"content": "How many positive integers less than $250$ are multiples of $5$, but not multiples of $10$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to figure out how many positive integers less than 250 are multiples of 5 but not multiples of 10. Hmm, so fir... |
[
{
"role": "user",
"content": "Sami remembers that the digits in her new three-digit area code contain a 9, 8, and 7, but she can't recall the order. How many possibilities are there for her to try?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so Sami is trying to figure out the number o... |
[
{
"role": "user",
"content": "If a square has a diagonal measuring $10\\sqrt{2}$ centimeters, what is the area of the square in square centimeters?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to figure out the area of the square when given the diagonal. So, the problem says t... |
[
{
"role": "user",
"content": "Two chords, $AB$ and $CD,$ meet inside a circle at $P.$ If $AP = X and $CP = 8,$ then what is $\\frac{BP}{DP}$?\nIf we know the answer to the above question is \\frac{8}{3}, what is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOk... |
[
{
"role": "user",
"content": "The center of the circle with equation $x^2+y^2=8x-6y-20$ is the point $(x,y)$. What is $x+y$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so the problem is asking for the value of x + y where the center of the circle given by the equation x² + y² = 8x - 6... |
[
{
"role": "user",
"content": "Determine the smallest prime number that divides evenly into $5^{23} + 7^{17}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the smallest prime number that divides evenly into $5^{23} + 7^{17}$. Let me start by recalling some number theory... |
[
{
"role": "user",
"content": "Suppose $173\\cdot 927\\equiv n\\pmod{50}$, where $0\\le n< 50$.\n\nWhat is the value of $n$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to find the value of $ n $ such that $ 173 \\cdot 927 \\equiv n \\pmod{50} $, where $ 0 \\leq n < 50... |
[
{
"role": "user",
"content": "What is the area enclosed by the region defined by the equation $x^2+y^2+10x+24y=0$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to figure this out again. The problem is to find the area enclosed by the circle defined by the equation $ x^2 + y^2... |
[
{
"role": "user",
"content": "Find $3 \\cdot 5 \\cdot 7 + X \\div 3.$\nIf we know the answer to the above question is 110, what is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem is to find the value of X in the equation 3 multiplied ... |
[
{
"role": "user",
"content": "What is the value of $x$ if $-2x - 7 = 7x + 2$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to solve the equation -2x - 7 = 7x + 2. Let me start by looking at the equation again to make sure I understand it correctly. It says that when you take -... |
[
{
"role": "user",
"content": "What is the value of $\\lfloor0.999\\rfloor+\\lceil2.001\\rceil$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking for the value of the floor of 0.999 plus the ceiling of 2.001. \n\nFirst, I need to... |
[
{
"role": "user",
"content": "What is the least positive whole number divisible by four different prime numbers?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to figure out the least positive whole number divisible by four different prime numbers. Hmm, primes are numbers greate... |
[
{
"role": "user",
"content": "Suppose that $a$ and $b$ are integers such that $$3b = 8 - 2a.$$How many of the first six positive integers must be divisors of $2b + 12$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is about determini... |
[
{
"role": "user",
"content": "What is the value of x in the equation $4^{\\log_7x}=16$?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem step by step. The equation given is $4^{\\log_7 x} = 16$. I need to find the value of $x$. \n\nFirst, I remember that when... |
[
{
"role": "user",
"content": "Compute $\\dbinom{8}{2}$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. I need to compute the combination $\\dbinom{8}{2}$. Hmm, combinations... right, that's when you choose 2 items out of 8 without considering the order. The formula for combinat... |
[
{
"role": "user",
"content": "If the average of six numbers is X, The sum of the six numbers is 24.6. What is the value of unknown variable X?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's see. The problem says that the average of six numbers is X, and the sum of these six numbers ... |
[
{
"role": "user",
"content": "Find the value of $x$ such that $\\sqrt{1 - 3x} = 7$."
},
{
"role": "assistant",
"content": "<think>\nOkay, let's try to solve this problem again. The equation is √(1 - 3x) = 7. Alright, so first, I remember that when you have a square root equation, you can elimina... |
[
{
"role": "user",
"content": "For what value of $n$ is the four-digit number $712n$, with units digit $n$, divisible by 18?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let's tackle this problem step by step. The question is asking for the value of $ n $ in the four-digit number 712n, w... |
[
{
"role": "user",
"content": "For how many positive integers $x$ is $x^2 + 6x + 9$ between 20 and 40?"
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try to figure this out again. The problem is asking for how many positive integers $ x $ make the expression $ x^2 + 6x + 9 $ between... |
[
{
"role": "user",
"content": "A regular polygon has sides of length 5 units and an exterior angle of 120 degrees. What is the perimeter of the polygon, in units?"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the perimeter of a regular polygon. The problem says the polyg... |
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