conversations listlengths 2 2 | system stringclasses 1
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[
{
"role": "user",
"content": "What is the area, in square units, of a triangle with vertices at $(0,0)$, $(0,5)$, and $(7,12)$? Express your answer as a decimal to the nearest tenth."
},
{
"role": "assistant",
"content": "To find the area of the triangle with vertices at \\((0,0)\\), \\((0,5)\\)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Write down a definite integral that will give the value of the area under the curve \\( y = x^2 \\cos\\left(\\frac{1}{5}x\\right) \\) between \\( x = \\frac{1}{2}\\pi \\) and \\( x = \\frac{5}{2}\\pi \\). (The expression \\( x^2 \\cos\\left(\\frac{1}{5}x\\right) \\) takes no ne... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given that $\\sum_{r=1}^{\\infty} \\frac{1}{r^{2}} = \\frac{\\pi^{2}}{6}$, find the value of $1 - \\frac{1}{2^{2}} + \\frac{1}{3^{2}} - \\frac{1}{4^{2}} + \\cdots$."
},
{
"role": "assistant",
"content": "To find the value of the series \\(1 - \\frac{1}{2^2} + \\frac... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An 18-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Exactly ten slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms?"
},
{
"role": "assistant",
"c... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A competition is attended by 20 participants. In each round, two participants compete. Each participant who loses twice is removed from the competition, and the last participant in the competition is the winner. If it is known that the winner of the competition has never lost, ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet?"
},
{
"role": "assistant",
"content": "To determine the fewest number of 3-inch by 4-inch ti... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the first perfect square? Is it 0 or 1?"
},
{
"role": "assistant",
"content": "To determine the first perfect square, we need to recall the definition of a perfect square. A perfect square is an integer that can be expressed as the square of another integer.... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $a\\star b = ab+2b-a$. If $5\\star x = 37$, find the value of $x$."
},
{
"role": "assistant",
"content": "To solve the problem where the operation \\( a \\star b = ab + 2b - a \\) is given and we need to find \\( x \\) such that \\( 5 \\star x = 37 \\):\n\n1. Su... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can 9 boys and 6 girls be paired up in a boy-girl format?"
},
{
"role": "assistant",
"content": "To determine the number of ways to pair 6 girls with 9 boys in a boy-girl format, we need to follow these steps:\n\n1. **Select 6 boys out of 9**: This can... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Express $142$ as a sum of powers of $2$."
},
{
"role": "assistant",
"content": "To express \\(142\\) as a sum of powers of \\(2\\), we can use the binary representation method. Here are the detailed steps:\n\n1. **Convert \\(142\\) to binary:**\n - Divide \\(142\\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If the sum of the second and fourth terms of an arithmetic sequence is $6$, what is the third term?"
},
{
"role": "assistant",
"content": "To find the third term of an arithmetic sequence where the sum of the second and fourth terms is 6, we can proceed as follows:\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The area of a rectangular rug is $84 \\text{ ft}^2$ and its perimeter is $38 \\text{ ft}$. Find the length and width of the rug."
},
{
"role": "assistant",
"content": "To find the length and width of a rectangular rug with an area of \\(84 \\text{ ft}^2\\) and a per... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The area of triangle $ABC$ is $30 \\text{ cm}^2$. Point $D$ on $\\overline{AC}$ divides the side in the ratio $AD:DC = 2:3$. The length of the perpendicular $\\overline{DE}$ dropped to $\\overline{BC}$ is $9 \\text{ cm}$. Find $BC$."
},
{
"role": "assistant",
"conte... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the linear transformation \\( f: \\mathbb{R} \\rightarrow \\mathbb{R}^2 \\) defined by \\( f(u) = v \\), where \\( u = \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\) and \\( v = \\begin{pmatrix} x + y \\\\ x - y \\end{pmatrix} \\), if the points \\( P', Q', R' \\) are the i... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the first 5 cumulants of the normal one-dimensional distribution, where the cumulants \\( k_n \\) of \\( X \\) are defined by \\( g(t) = \\log(E(\\exp(tX))) \\) and the \\( n \\)-th cumulant is given by the \\( n \\)-th derivative of \\( g \\) at 0."
},
{
"role": "... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $F(a, b, c, d) = a^b + c \\times d$, what is the value of $x$ such that $F(2, x, 4, 11) = 300$?"
},
{
"role": "assistant",
"content": "To find the value of \\( x \\) such that \\( F(2, x, 4, 11) = 300 \\), we start by substituting the given values into the functi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Grady distributed $x$ pieces of candy evenly among nine Halloween bags such that every bag received the greatest possible number of whole pieces of candy, but some candy was left over. What is the greatest possible number of pieces that could have been left over?"
},
{
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In how many ways can the word $MISSISSIPPI$ be arranged such that the vowels are in alphabetical order?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the number of ways to arrange the word \"MISSISSIPPI\" such that the vowels are in alphabet... | You are a helpful assistant. |
[
{
"role": "user",
"content": "At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yel... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The real number \\( 2.525252... \\) is a rational number, so it can be written in the form \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are integers, \\( n \\ne 0 \\). If \\( m \\) and \\( n \\) are relatively prime, what is \\( m + n \\)?"
},
{
"role": "assistan... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a right triangle ABC with hypotenuse AB, side AC = 15. The altitude CH divides AB into segments AH and HB, with HB = 16. Find the area of triangle ABC."
},
{
"role": "assistant",
"content": "To find the area of the right triangle \\( \\triangle ABC \\) with hypot... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the length of rectangle ABCD if the width is 3 inches and its diagonal is 20 inches."
},
{
"role": "assistant",
"content": "To find the length of the rectangle \\(ABCD\\) where the width is 3 inches and the diagonal is 20 inches, we can use the Pythagorean theo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ordered sets of $(a,b,c,d)$ are such that $abcd = -2013^4$?"
},
{
"role": "assistant",
"content": "To determine the number of ordered sets \\((a, b, c, d)\\) such that \\(abcd = -2013^4\\), we start by factorizing 2013 into its prime factors:\n\n\\[ 2013 = ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of $x$ if $|x-1| = |x-2|$? Express your answer as a common fraction."
},
{
"role": "assistant",
"content": "To solve the equation \\( |x - 1| = |x - 2| \\), we need to consider the definition of absolute value and the fact that the equation implies... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the limit of the sequence \\( u_n = n^{\\alpha} \\sin n \\) as \\( n \\to \\infty \\), where \\( \\alpha > 0 \\)."
},
{
"role": "assistant",
"content": "To find the limit of the sequence \\( u_n = n^{\\alpha} \\sin n \\) as \\( n \\to \\infty \\), where \\( \\a... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In triangle $ABC$, $AB = 16$, $AC = 24$, $BC = 19$, and $AD$ is an angle bisector. Find the ratio of the area of triangle $ABD$ to the area of triangle $ACD$. (Express your answer as a fraction in lowest terms.)"
},
{
"role": "assistant",
"content": "To solve the ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the phase shift of the function $f(x) = \\sin(4x + \\pi)$?"
},
{
"role": "assistant",
"content": "To find the phase shift of the function \\( f(x) = \\sin(4x + \\pi) \\), we start by identifying the form of the sine function. The general form for the sine fu... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Give an example of a function that is both odd and bounded."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is $4\\cdot 6+8\\cdot 3-28\\div 2$?"
},
{
"role": "assistant",
"content": "To solve the expression \\(4 \\cdot 6 + 8 \\cdot 3 - 28 \\div 2\\), we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the equation of the directrix of the parabola $x = -\\frac{1}{6} y^2.$"
},
{
"role": "assistant",
"content": "To find the equation of the directrix of the parabola given by \\( x = -\\frac{1}{6} y^2 \\), we start by comparing it to the standard form of a horizo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the smallest positive integer divisible by $10$, $11$, and $12$."
},
{
"role": "assistant",
"content": "To find the smallest positive integer divisible by \\(10\\), \\(11\\), and \\(12\\), we need to determine the least common multiple (LCM) of these numbers. T... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Simplify the expression \\(\\frac{2a}{\\sqrt{a-b} + \\sqrt{a+b}}\\)."
},
{
"role": "assistant",
"content": "To simplify the expression \\(\\frac{2a}{\\sqrt{a-b} + \\sqrt{a+b}}\\), we start by rationalizing the denominator. This involves multiplying both the numerato... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Suppose \\(\\gcd(p, q) = 1\\), \\(\\gcd(p, 10) = 1\\), and \\(\\gcd(q, 10) = 1\\). If the decimal expansion of \\(1/p\\) has period \\(r\\) and the decimal expansion of \\(1/q\\) has period \\(s\\), find the period of the decimal expansion of \\(1/(pq)\\)."
},
{
"role":... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Graph the region defined by \\(0 \\leq \\theta \\leq \\pi\\) and \\(r < -1\\)."
},
{
"role": "assistant",
"content": "To graph the region defined by \\(0 \\leq \\theta \\leq \\pi\\) and \\(r < -1\\), we need to understand the transformation of negative \\(r\\) value... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An octopus told me that his underwater cave is $567_{8}$ years old. How many years is this in base ten?"
},
{
"role": "assistant",
"content": "To convert the octal number \\(567_8\\) to its decimal (base 10) equivalent, we need to express it as a sum of powers of 8.... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the sum:\n\\[\n\\frac{\\sqrt{1+1} - \\sqrt{1}}{\\sqrt{1^2 + 1}} + \\frac{\\sqrt{2+1} - \\sqrt{2}}{\\sqrt{2^2 + 2}} + \\frac{\\sqrt{3+1} - \\sqrt{3}}{\\sqrt{3^2 + 3}} + \\ldots + \\frac{\\sqrt{15+1} - \\sqrt{15}}{\\sqrt{15^2 + 15}}\n\\]"
},
{
"role": "assistant",
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( A \\) be a set of \\( kn \\) elements where \\( k \\) and \\( n \\) are natural numbers. A \"k\" grouping of \\( A \\) is a partition of \\( A \\) into subsets, each containing \\( k \\) elements. Find the number of different \"k\" groupings of \\( A \\)."
},
{
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $\\#$ be the relation defined by $A \\# B = A^2 + B^2$. If $A \\# 5 = 169$, what is the positive value of $A$?"
},
{
"role": "assistant",
"content": "To solve the problem, we start with the given relation \\( A \\# B = A^2 + B^2 \\). We are given that \\( A \\# ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Sunny runs at a steady rate, and Moonbeam runs $m$ times as fast, where $m$ is a number greater than 1. If Moonbeam gives Sunny a head start of $h$ meters, how many meters must Moonbeam run to overtake Sunny?"
},
{
"role": "assistant",
"content": "To solve the probl... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find all continuous functions \\( f \\) on \\([0,1]\\) such that \n\\[\n\\int_{0}^{x} f(t) \\, dt = \\int_{x}^{1} f(t) \\, dt\n\\]\nfor every \\( x \\in (0,1) \\)."
},
{
"role": "assistant",
"content": "To find all continuous functions \\( f \\) on \\([0,1]\\) such ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the derivative $\\frac{d}{dx} \\sqrt{2x-1}$."
},
{
"role": "assistant",
"content": "To find the derivative of \\(\\sqrt{2x - 1}\\) with respect to \\(x\\), we will use the chain rule. The chain rule states that if \\(y = f(g(x))\\), then the derivative \\(y'\\)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\(5 + \\sqrt[3]{7}\\) be a root of a polynomial equation in \\(x\\) of the smallest possible degree with integer coefficients, where the coefficient of the highest degree term is 1. Find the sum of the numerical coefficients of the \\(x^2\\) term and the \\(x\\) term."
}... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute the limit of the sequence $\\{z_n\\}$ where $z_n = i^{n!}$."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. |
[
{
"role": "user",
"content": "At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yel... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The five-digit number $246\\underline{\\hspace{5pt}}8$ is divisible by 9. What is the missing digit?"
},
{
"role": "assistant",
"content": "To determine the missing digit in the five-digit number \\(246\\underline{\\hspace{5pt}}8\\) that makes it divisible by 9, we ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral $\\int{\\sin(x)\\cos(x)dx}$."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\sin(x) \\cos(x) \\, dx\\), we can use substitution. Let \\( u = \\sin(x) \\). Then, the differential \\( du \\) is given by:\n\n\\[\ndu = \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a singles tennis tournament with 10 players, in how many ways can the players be organised into 5 pairs?"
},
{
"role": "assistant",
"content": "To determine the number of ways to organize 10 players into 5 pairs, we can use the concept of combinations and adjust ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How do you factor $x^n - 1$, where $n \\geq 3$, into irreducibles in $\\mathbb{Z}[x]$?"
},
{
"role": "assistant",
"content": "To factor the polynomial \\( x^n - 1 \\) into irreducibles in \\( \\mathbb{Z}[x] \\) for \\( n \\geq 3 \\), we use the concept of cyclotomic... | You are a helpful assistant. |
[
{
"role": "user",
"content": "John recently bought a used car for $\\$5000$ for his pizza delivery job. He gets $\\$10$ for each pizza he delivers, but he has to spend $\\$3$ on gas for each pizza he delivers. What is the minimum whole number of pizzas John must deliver in order to earn back the money he sp... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In how many ways can 4 nuts be placed in a $4 \\times 4$ table such that there is at least one nut in every row and every column?"
},
{
"role": "assistant",
"content": "To solve the problem of placing 4 nuts in a \\(4 \\times 4\\) table such that there is at least o... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can the letters in the word \"MIXED\" be rearranged such that no letter maintains its original position?"
},
{
"role": "assistant",
"content": "To determine the number of ways to rearrange the letters in the word \"MIXED\" such that no letter maintains... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve $\\log_4 x + \\log_2 x^2 = 10$."
},
{
"role": "assistant",
"content": "To solve the equation \\(\\log_4 x + \\log_2 x^2 = 10\\):\n\n1. Convert \\(\\log_4 x\\) to base 2 using the change of base formula:\n \\[\n \\log_4 x = \\frac{\\log_2 x}{\\log_2 4} = \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Calculate the limit \\( \\lim_{h \\rightarrow 0} \\frac{(x+h)^{x+h} - x^x}{h} \\)."
},
{
"role": "assistant",
"content": "To calculate the limit \\(\\lim_{h \\rightarrow 0} \\frac{(x+h)^{x+h} - x^x}{h}\\), we recognize that this expression represents the derivative ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The base $9$ representation of a positive integer is $AB$ and its base $7$ representation is $BA$. What is the integer expressed in base $10$?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to find a positive integer that has the base 9 represe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The line $y = \\frac{3}{2} x - 25$ is parameterized by $(x,y) = (f(t),15t - 7).$ Enter the function $f(t).$"
},
{
"role": "assistant",
"content": "To find the function \\( f(t) \\) that parameterizes the x-coordinate such that the parametric equations \\((x, y) = (... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given \\( x = \\log 36 \\) and \\( y = \\log 125 \\), express \\( \\log 3 \\) in terms of \\( x \\) and \\( y \\). Show your work."
},
{
"role": "assistant",
"content": "Given \\( x = \\log 36 \\) and \\( y = \\log 125 \\), we need to express \\( \\log 3 \\) in term... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the tot... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many points on the Cartesian plane \\( (x, y) \\) satisfy \\( ((x - 20)^{20} - 20)^{20} + ((y - 20)^{20} - 20)^{20} = 0 \\)?"
},
{
"role": "assistant",
"content": "To solve the equation \\(((x - 20)^{20} - 20)^{20} + ((y - 20)^{20} - 20)^{20} = 0\\), we need to ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the equation in $\\mathbb{Z}$: $(x+2012)^{100}=x^{100}$"
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Ten distinct points are identified on the circumference of a circle. How many different convex quadrilaterals can be formed if each vertex must be one of these 10 points?"
},
{
"role": "assistant",
"content": "To determine the number of different convex quadrilatera... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A right cylinder with a height of 5 inches has a radius of 2 inches. What is the area of the curved surface of the cylinder, in square inches? Express your answer in terms of $\\pi$."
},
{
"role": "assistant",
"content": "To find the area of the curved surface of a ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\(ABCD\\) be a parallelogram. Let \\(G\\) and \\(H\\) be the feet of the altitudes from \\(A\\) to \\(CD\\) and \\(BC\\) respectively. If \\(AD = 15\\), \\(AG = 12\\), and \\(AH = 16\\), find the length of \\(AB\\)."
},
{
"role": "assistant",
"content": "To fin... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the parametric equations for the horizontal line and vertical line passing through the point $(-4, 5)$."
},
{
"role": "assistant",
"content": "To find the parametric equations for the horizontal and vertical lines passing through the point \\((-4, 5)\\), we pro... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given that \\(a + b = 17\\) and \\(a, b \\in \\mathbb{Z^+}\\), find the minimum value of \\(a\\) and \\(b\\)."
},
{
"role": "assistant",
"content": "Given that \\(a + b = 17\\) and \\(a, b \\in \\mathbb{Z^+}\\), we need to find the minimum value of \\(a\\) and \\(b\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "On an old-fashioned bicycle the front wheel has a radius of $2.5$ feet and the back wheel has a radius of $4$ inches. If there is no slippage, how many revolutions will the back wheel make while the front wheel makes $100$ revolutions?"
},
{
"role": "assistant",
"c... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The probability that Kim has a math test today is $\\frac{4}{7}$. What is the probability that Kim does not have a math test today? Express your answer as a common fraction."
},
{
"role": "assistant",
"content": "To find the probability that Kim does not have a math... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Factor $x^3+2x^2-5x-6$ over the rationals."
},
{
"role": "assistant",
"content": "To factor the polynomial \\(x^3 + 2x^2 - 5x - 6\\) over the rationals, we start by using the Rational Root Theorem. The Rational Root Theorem states that any rational root, \\(p/q\\), ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What number, when divided by 2, gives the same result as when 2 is subtracted from it?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to find a number \\( x \\) such that when it is divided by 2, the result is the same as when 2 is subtracted f... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Calculate the sum $1 + 3 + 5 + \\cdots + 15 + 17$."
},
{
"role": "assistant",
"content": "To calculate the sum of the sequence \\(1 + 3 + 5 + \\cdots + 15 + 17\\), we can use the formula for the sum of an arithmetic sequence. The sequence is an arithmetic sequence w... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a kingdom, $\\frac{2}{7}$ of the knights are red, and the rest are blue. Additionally, $\\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magic... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the sum ${n\\choose0}^2+{n\\choose1}^2+{n\\choose2}^2+\\dots+{n\\choose{n}}^2$ using a counting argument."
},
{
"role": "assistant",
"content": "To evaluate the sum \\(\\binom{n}{0}^2 + \\binom{n}{1}^2 + \\binom{n}{2}^2 + \\dots + \\binom{n}{n}^2\\) using a... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many different AoPS avatars are possible using 3 colors to color the differently oriented rhombuses, given that there are 3 different orientations possible?"
},
{
"role": "assistant",
"content": "To determine the number of different AoPS avatars possible using 3... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the inverse of the equation \\( y = 0.5x - \\frac{3}{2} \\)."
},
{
"role": "assistant",
"content": "To find the inverse of the function \\( y = 0.5x - \\frac{3}{2} \\), follow these steps:\n\n1. Start with the original function:\n \\[\n y = 0.5x - \\frac{3}... | You are a helpful assistant. |
[
{
"role": "user",
"content": "(ii) Given a set $A = \\{a_1, a_2, a_3, \\ldots, a_n\\}$, find the total number of binary operations on $A$ such that $a_i * a_j \\neq a_i * a_k$ if $j \\neq k$."
},
{
"role": "assistant",
"content": "To find the total number of binary operations on a set \\( A = \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can you choose a pair of cards from a set of 2 cards?"
},
{
"role": "assistant",
"content": "To determine the number of ways to choose a pair of cards from a set of 2 cards, we use the combination formula \\( C(n, k) = \\frac{n!}{k!(n - k)!} \\), where... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Form the integer 24 using the numbers 2, 4, 5, 7. Only addition, subtraction, division, and multiplication can be used. Each number can only be used once."
},
{
"role": "assistant",
"content": "To form the integer 24 using the numbers 2, 4, 5, and 7 with the operati... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the limit \\(\\lim_{x \\to 0^+} \\left\\lfloor \\frac{\\tan x}{x} \\right\\rfloor\\), where \\(\\left\\lfloor \\cdot \\right\\rfloor\\) denotes the greatest integer function."
},
{
"role": "assistant",
"content": "To find the limit \\(\\lim_{x \\to 0^+} \\left\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If \\(a\\) is real, what is the only real number that could be a multiple root of the equation \\(x^3 + ax + 1 = 0\\)?"
},
{
"role": "assistant",
"content": "To determine the only real number that could be a multiple root of the equation \\(x^3 + ax + 1 = 0\\), we u... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways are there for 10 people to be friends, where it is possible that they are all friends with each other or none of them are friends with each other?"
},
{
"role": "assistant",
"content": "To determine the number of ways 10 people can be friends, where ea... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Jordan and his three best friends are on a relay team. His relay team will run a race, where the first runner runs a lap, then the second, then the third, then the fourth.\n\nJordan is the fastest so he will run the fourth lap. How many different orders can the four team memb... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the domain of the function $g(x) = \\frac{3x+1}{x+8}$ ? Enter your answer using interval notation."
},
{
"role": "assistant",
"content": "To determine the domain of the function \\( g(x) = \\frac{3x+1}{x+8} \\), we need to identify the values of \\( x \\) f... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the sum \\( \\sum_{i=0}^{\\infty} \\frac{(-1)^i}{2i+1} \\)."
},
{
"role": "assistant",
"content": "To evaluate the sum \\( \\sum_{i=0}^{\\infty} \\frac{(-1)^i}{2i+1} \\), we start by recognizing that this series is the Taylor series expansion of the arctang... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the remainder when $x^3$ is divided by $x^2 + 5x + 1.$"
},
{
"role": "assistant",
"content": "To find the remainder when \\( x^3 \\) is divided by \\( x^2 + 5x + 1 \\), we can use polynomial long division or express \\( x^3 \\) in terms of the divisor.\n\nFirst... | You are a helpful assistant. |
[
{
"role": "user",
"content": "During a recent period of time, eleven days had some rain. A morning rain was always followed by a clear afternoon. An afternoon rain was always preceded by a clear morning. A total of nine mornings and twelve afternoons were clear. How many days had no rain at all?"
},
{
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the number of quadruples $(x, y, z, w)$ of non-negative integers that satisfy the inequality $w + x + y + z \\le 1992$."
},
{
"role": "assistant",
"content": "To find the number of quadruples \\((w, x, y, z)\\) of non-negative integers that satisfy the inequali... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine whether \\( H = A_4 \\) is a normal subgroup of \\( G = S_4 \\). If it is, write out a Cayley table for the factor group \\( G/H \\)."
},
{
"role": "assistant",
"content": "To determine whether \\( H = A_4 \\) is a normal subgroup of \\( G = S_4 \\) and to... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate $\\log_2\\frac{1}{16}$."
},
{
"role": "assistant",
"content": "To evaluate \\(\\log_2 \\frac{1}{16}\\), we need to determine the exponent \\(x\\) such that \\(2^x = \\frac{1}{16}\\).\n\nFirst, recognize that \\(16\\) can be written as a power of \\(2\\):\n\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$?"
},
{
"role": "assistant",
"content": "To determine the value of \\( c \\) such that \\( x - 5 \\) is a factor of the polynomial \\( P(x) = x^3 + 2x^2 +... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An investment of $\\$$10,000 is made in a government bond that will pay 6$\\%$ interest compounded annually. At the end of five years, what is the total number of dollars in this investment? Express your answer to the nearest whole number."
},
{
"role": "assistant",
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A printer numbers the pages of a book starting with $1$. If the printer uses a total of $3,189$ digits, how many pages does the book have?"
},
{
"role": "assistant",
"content": "To determine the number of pages in the book given that the printer uses a total of 3,18... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given parallelogram ABCD, with AB = 10, AD = 12, and BD = $2\\sqrt{13}$, find AC."
},
{
"role": "assistant",
"content": "To find the length of \\( AC \\) in parallelogram \\( ABCD \\) where \\( AB = 10 \\), \\( AD = 12 \\), and \\( BD = 2\\sqrt{13} \\), we can use t... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the smallest value of $x$ that satisfies the equation $8x^2 - 38x + 35 = 0$? Express your answer as a decimal."
},
{
"role": "assistant",
"content": "To find the smallest value of \\( x \\) that satisfies the equation \\( 8x^2 - 38x + 35 = 0 \\), we can use ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute \\(\\sigma(12)\\), the sum of the divisors of 12."
},
{
"role": "assistant",
"content": "To compute \\(\\sigma(12)\\), the sum of the divisors of 12, we can use the prime factorization method. First, we factorize 12 into its prime factors:\n\n\\[ 12 = 2^2 \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Suppose that the number of people entering a department store in a given day is a random variable with mean 50. Suppose further that the amount of money spent by these customers are random variables with a common mean of 8. Assume also that the amount of money spent by a custom... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock?"
},
{
"role": "assistant",
"content": "To determine the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock, we can follow these steps:\n\n1. *... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve for \\( x \\) in the congruence \\( 5x \\equiv 4 \\pmod{3} \\)."
},
{
"role": "assistant",
"content": "To solve the congruence \\(5x \\equiv 4 \\pmod{3}\\), we first simplify the coefficients modulo 3.\n\n1. **Simplify Coefficients:**\n \\[\n 5 \\equiv 2 \... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the largest prime factor of $$(25!)^3 - (24!)^3$$?"
},
{
"role": "assistant",
"content": "To find the largest prime factor of \\((25!)^3 - (24!)^3\\), we start by factoring the expression.\n\nFirst, note that \\(25! = 25 \\times 24!\\). Therefore, we can rew... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If the sum of the second and fourth terms of an arithmetic sequence is $6$, what is the third term?"
},
{
"role": "assistant",
"content": "To find the third term of an arithmetic sequence where the sum of the second and fourth terms is 6, we can follow these steps:\... | You are a helpful assistant. |
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