conversations listlengths 2 2 | system stringclasses 1
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[
{
"role": "user",
"content": "Find the matrix $\\mathbf{P}$ such that for any vector $\\mathbf{v},$ $\\mathbf{P} \\mathbf{v}$ is the projection of $\\mathbf{v}$ onto the vector $\\begin{pmatrix} 2 \\\\ -2 \\\\ -1 \\end{pmatrix}.$"
},
{
"role": "assistant",
"content": "To find the matrix \\(\\mat... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the smallest digit that is never found in the units place of an even number?"
},
{
"role": "assistant",
"content": "To determine the smallest digit that is never found in the units place of an even number, we first identify the possible digits for the units ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the greatest common divisor of 12 and 20."
},
{
"role": "assistant",
"content": "To find the greatest common divisor (GCD) of 12 and 20, we can use the prime factorization method. \n\nFirst, we determine the prime factorization of each number:\n- The prime fact... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The perimeter of a rectangle is 56 meters. The ratio of its length to its width is 4:3. What is the length in meters of a diagonal of the rectangle?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by using the given information about the perime... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the following differential equation by separation of variables (or otherwise):\n\\[ \\frac{dy}{dx} - 1 = e^{x-y} \\]"
},
{
"role": "assistant",
"content": "To solve the differential equation \\(\\frac{dy}{dx} - 1 = e^{x-y}\\), we start by making a substitution... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers such that $a + b = 8$. Let $c$ be the minimum possible value of $x^2 + ax + b$ over all real numbers $x$. Find the maximum possible value of $c$ over all such $a$ and $b$."
},
{
"role": "assistant",
"content": "To find the maximum pos... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Two more than three times $B$ is equal to 20. What is the value of $B$?"
},
{
"role": "assistant",
"content": "To solve the problem where \"two more than three times \\( B \\) is equal to 20,\" we start by translating the words into a mathematical equation. The phra... | You are a helpful assistant. |
[
{
"role": "user",
"content": "At a women's doubles tennis tournament, there were three teams of two women. After the tournament, each woman shook hands once with each of the other players except her partner. What is the number of handshakes that occurred?"
},
{
"role": "assistant",
"content": "... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Points \\( X \\) and \\( Y \\) lie on a circle of radius 2 and center at \\( O \\), and the measure of \\( \\angle XOY \\) is \\( 30^\\circ \\). What is the area of the smaller region bounded by the circle and the chord joining \\( X \\) to \\( Y \\)? Express your answer in ter... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ squares, respectively. If this pattern continues, how many squares does figure $100$ have?"
},
{
"role": "assistant",
"content": "To determine the number of squares in figure 100, we first need to id... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a cube-shaped box with an edge length of 5, there are two balls. The radius of one ball is 2. This ball touches the base and two side faces of the cube. The other ball touches the first ball, the base, and the other two side faces of the cube. Find the radius of the second b... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Investigate the convergence or divergence of the series $\\sum_{k=1}^\\infty \\frac{\\sin \\frac{k\\pi}{2}}{k}$."
},
{
"role": "assistant",
"content": "To determine the convergence or divergence of the series \\(\\sum_{k=1}^\\infty \\frac{\\sin \\frac{k\\pi}{2}}{k}\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int \\frac{dx}{x(1+(\\ln x)^2)} \\) without using trigonometric substitution."
},
{
"role": "assistant",
"content": "To evaluate the integral \\( \\int \\frac{dx}{x(1+(\\ln x)^2)} \\), we use the substitution \\( u = \\ln x \\). This subs... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\\cdot 4\\cdot3\\cdot2\\cdot 1= 120$.)"
},
{
"role": "assistant",
"content": "To determine the numbe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute:\n\\[\n\\frac{1 \\cdot 1! + 2 \\cdot 2! + 3 \\cdot 3! + \\ldots + 7 \\cdot 7! + 1}{6!}\n\\]"
},
{
"role": "assistant",
"content": "To compute the expression\n\\[\n\\frac{1 \\cdot 1! + 2 \\cdot 2! + 3 \\cdot 3! + \\ldots + 7 \\cdot 7! + 1}{6!},\n\\]\nwe start... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the quadratic equation \\(3x^2 - 2x - 8 = 0\\) step by step."
},
{
"role": "assistant",
"content": "To solve the quadratic equation \\(3x^2 - 2x - 8 = 0\\), we will use the quadratic formula:\n\n\\[ x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\]\n\nHere, the co... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $p$ and $q$ be the two distinct solutions to the equation $$(x-3)(x+3) = 21x - 63.$$If $p > q$, what is the value of $p - q$?"
},
{
"role": "assistant",
"content": "To solve the equation \\((x-3)(x+3) = 21x - 63\\), we start by simplifying the left side using th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $f(x)=-3x^2+x-4$, $g(x)=-5x^2+3x-8$, and $h(x)=5x^2+5x+1$. Express $f(x)+g(x)+h(x)$ as a single polynomial, with the terms in order by decreasing degree."
},
{
"role": "assistant",
"content": "To solve the problem of expressing \\( f(x) + g(x) + h(x) \\) as a si... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle. The distance $x$ covered in time $t$ satisfies the formula $x = at^2 + bt + c$, where $a$, $b$, and $c$ are numerical values determined b... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Among all the solutions, real and complex, of the system of equations\n\n\\[ x^2 + y^2 + x + y = 8 \\]\n\\[ xy + x + y = 5 \\]\n\nwhat is the least possible real part of \\( x \\)?"
},
{
"role": "assistant",
"content": "To solve the system of equations:\n\n\\[ x^2 +... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the sum of the geometric series \\(3 + 6 + 12 + \\ldots + 3072\\)."
},
{
"role": "assistant",
"content": "To find the sum of the geometric series \\(3 + 6 + 12 + \\ldots + 3072\\), we start by identifying the first term \\(a\\) and the common ratio \\(r\\).\n\n... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What number must we add to $3-5i$ to get $2+7i$? Here, $i^2 = -1.$"
},
{
"role": "assistant",
"content": "To find the number \\( z \\) that must be added to \\( 3 - 5i \\) to get \\( 2 + 7i \\), we can set up the equation:\n\n\\[\n(3 - 5i) + z = 2 + 7i\n\\]\n\nTo is... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine the type of discontinuous point at \\( x = 0 \\) for the function \\( f(x) = \\frac{1}{x} \\sin\\left(\\frac{1}{x}\\right) \\)."
},
{
"role": "assistant",
"content": "To determine the type of discontinuous point at \\( x = 0 \\) for the function \\( f(x) =... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find all possible matrices $A$ which satisfy the equation\n\\[ \\begin{bmatrix}3 & 4 \\\\ 6 & 8\\end{bmatrix} A = \\begin{bmatrix}8 \\\\ 16\\end{bmatrix}. \\]"
},
{
"role": "assistant",
"content": "To find all possible matrices \\( A \\) that satisfy the equation\n\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( X \\) and \\( Y \\) be topological spaces, and let \\( \\phi: X \\rightarrow Y \\) be a homeomorphism. Define an equivalence relation \\( \\sim \\) on \\( Y \\) and a relation \\( \\bowtie \\) on \\( X \\) such that \\( x_1 \\bowtie x_2 \\) if and only if \\( \\phi(x_1)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $\\sqrt{5 + x} + \\sqrt{20 - x} = 7$, what is the value of $(5 + x)(20 - x)$?"
},
{
"role": "assistant",
"content": "To solve the problem, we start with the given equation:\n\n\\[\n\\sqrt{5 + x} + \\sqrt{20 - x} = 7\n\\]\n\nLet \\( A = \\sqrt{5 + x} \\) and \\( B... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Expand the product\n$$(1+\\frac{2}{3})(1+\\frac{2}{4})(1+\\frac{2}{5})\\cdots(1+\\frac{2}{98})$$"
},
{
"role": "assistant",
"content": "To solve the given product \\((1+\\frac{2}{3})(1+\\frac{2}{4})(1+\\frac{2}{5})\\cdots(1+\\frac{2}{98})\\), we start by rewriting e... | 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": "Determine if the function \\( g: \\mathbb{N} \\to \\mathbb{N} \\) defined by \\( g(x) = \\left\\lfloor \\frac{3x+1}{3} \\right\\rfloor \\) is bijective and find its inverse if it exists."
},
{
"role": "assistant",
"content": "To determine if the function \\( g: \\ma... | 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": "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": "If the person sitting next to you has \\( c \\) correct answers and you have \\( m \\) more correct answers than they do, and the total number of correct answers between the two of you is \\( t \\), express \\( c \\) in terms of \\( m \\) and \\( t \\)."
},
{
"role": "a... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A giant, circular and perfectly flat pizza must be shared by 22 people. What is the least number of cuts to be made on the pizza so that each person can have a piece of pizza? (The pieces are not necessarily the same shape or with the same area.)"
},
{
"role": "assistan... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The lengths of the sides of a triangle are 6 cm, 7 cm and 9 cm. In a similar triangle whose perimeter is 110 cm, what is the length of the longest side, in centimeters?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by determining the perimet... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Express the value of the following expression as a fraction. $$1+\\cfrac{2}{3+\\cfrac{4}{5}}$$"
},
{
"role": "assistant",
"content": "To solve the expression \\(1 + \\cfrac{2}{3 + \\cfrac{4}{5}}\\), we will break it down step by step.\n\n1. **Evaluate the inner frac... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the expression:\n\\[\n\\left(\\left(\\left(3^{1-\\frac{1}{2}}\\right)^{1-\\frac{1}{3}}\\right)^{1-\\frac{1}{4}}\\right)^{\\cdots \\text{n terms}}\n\\]"
},
{
"role": "assistant",
"content": "To evaluate the expression\n\n\\[\n\\left(\\left(\\left(3^{1-\\frac... | 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": "Ms. Marks is selecting students for a committee that must consist of three seniors and five juniors. Six senior volunteers are available. What is the least number of junior volunteers needed if Ms. Marks wants at least 100 different possible ways to pick the committee?"
},
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$."
},
{
"role": "assistant",
"content": "To find the largest value of \\( c \\) such that \\(-2\\) is in the range of the function \\( f(x) = x^2 + 3x + c \\), we need to determine the m... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the set of all matrices with respect to the standard basis for \\(\\mathbb{R}^2\\) for all linear operators that take all vectors of the form \\([0, y]\\) to vectors of the form \\([0, y']\\)."
},
{
"role": "assistant",
"content": "To find the set of all matric... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The perimeter of a rectangular garden is 60 feet. If the length of the field is twice the width, what is the area of the field, in square feet?"
},
{
"role": "assistant",
"content": "To find the area of the rectangular garden, we start by using the given information... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games and every team in the other division $M$ games, with $N > 2M$ and $M > 4$. Each team plays a total of 76 games. How many games does a team play within its own divis... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve for $\\log(i)$ given the expression $Bz \\cdot \\log(i) - Bh \\cdot \\log(i)$ and the result $\\log(i) = \\frac{1}{Bz - Bh}$."
},
{
"role": "assistant",
"content": "To solve for \\(\\log(i)\\) given the expression \\(Bz \\cdot \\log(i) - Bh \\cdot \\log(i)\\) ... | 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": "Identify the pattern and find the next term in the series: 11, 13, 18, 28, 45, 71, ..."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of \\( k \\) such that the equation \\( x^2 - kx + 4 = 0 \\) has a repeated real solution? Provide the steps to find the value of \\( k \\)."
},
{
"role": "assistant",
"content": "To find the value of \\( k \\) such that the equation \\( x^2 - kx +... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The denominator of a fraction is 7 less than 3 times the numerator. If the fraction is equivalent to $2/5$, what is the numerator of the fraction?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to find the numerator of a fraction where the den... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given a point on a line and the slope of the line, find a point on the line that is a specific distance from the given point. What formula would be used to determine this new point?"
},
{
"role": "assistant",
"content": "To find a point on a line at a specific dista... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In parallelogram $ABCD$, $AB = 38$ cm, $BC = 3y^3$ cm, $CD = 2x +4$ cm, and $AD = 24$ cm. What is the product of $x$ and $y$?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by using the properties of a parallelogram. In a parallelogram, opposi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The line \\( y = mx + 2k \\) is tangent to the curve \\( y = \\frac{1}{x} \\) at the point \\( (p, 5p) \\). Find \\( |k| \\)."
},
{
"role": "assistant",
"content": "To solve the problem of finding \\( |k| \\) given that the line \\( y = mx + 2k \\) is tangent to the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $\\log_{25}(x-4)=\\frac{1}{2}$, find $\\frac{1}{\\log_{x}3}$."
},
{
"role": "assistant",
"content": "To solve the problem, we start with the given equation:\n\n\\[\n\\log_{25}(x-4) = \\frac{1}{2}\n\\]\n\nWe convert this logarithmic equation to its exponential for... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Factor the expression $x(x+2)+(x+2)$."
},
{
"role": "assistant",
"content": "To factor the expression \\( x(x+2) + (x+2) \\), we can follow these steps:\n\n1. **Identify the common factor**: Notice that both terms in the expression \\( x(x+2) + (x+2) \\) have a comm... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve by completing the square: \\( x^2 + 6x + 2 = 0 \\)"
},
{
"role": "assistant",
"content": "To solve the quadratic equation \\( x^2 + 6x + 2 = 0 \\) by completing the square, follow these steps:\n\n1. Start with the given equation:\n \\[\n x^2 + 6x + 2 = 0\n... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If \\( f(x) \\) belongs to the interval \\([-2, 3]\\), determine the interval to which \\( f(x-2) \\) belongs."
},
{
"role": "assistant",
"content": "To determine the interval to which \\( f(x-2) \\) belongs, given that \\( f(x) \\) belongs to the interval \\([-2, 3... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the remainder when $2 \\times 12 \\times 22 \\times 32 \\times \\ldots \\times 72 \\times 82 \\times 92$ is divided by $5$."
},
{
"role": "assistant",
"content": "To find the remainder when the product \\(2 \\times 12 \\times 22 \\times 32 \\times \\ldots \\tim... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Caroline can make eleven lassis out of two mangoes. How many lassis can she make out of twelve mangoes?"
},
{
"role": "assistant",
"content": "To determine how many lassis Caroline can make with 12 mangoes, given that she can make 11 lassis with 2 mangoes, we can se... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given two real numbers $1<p<q$ so that $\\frac{1}{p} + \\frac{1}{q} = 1$ and $pq = \\frac{9}{2}$, what is $q$?"
},
{
"role": "assistant",
"content": "Given two real numbers \\(1 < p < q\\) such that \\(\\frac{1}{p} + \\frac{1}{q} = 1\\) and \\(pq = \\frac{9}{2}\\), ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find $q(x)$ if the graph of $\\frac{x^3-2x^2-5x+3}{q(x)}$ has vertical asymptotes at $2$ and $-2$, no horizontal asymptote, and $q(3) = 15$."
},
{
"role": "assistant",
"content": "To find the polynomial \\( q(x) \\) such that the function \\( \\frac{x^3 - 2x^2 - 5x ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Simplify $(2x - 5)(x + 7) - (x + 5)(2x - 1)$."
},
{
"role": "assistant",
"content": "To simplify the expression \\((2x - 5)(x + 7) - (x + 5)(2x - 1)\\), we will expand both products and then combine like terms.\n\nFirst, we expand \\((2x - 5)(x + 7)\\):\n\\[\n(2x - ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Write the expression $\\frac{4+3c}{7}+2$ as a single fraction."
},
{
"role": "assistant",
"content": "To combine the expression \\(\\frac{4+3c}{7} + 2\\) into a single fraction, follow these steps:\n\n1. **Identify the common denominator**: The first term \\(\\frac{... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find $\\tan \\frac{9 \\pi}{4}.$"
},
{
"role": "assistant",
"content": "To find \\(\\tan \\frac{9\\pi}{4}\\), we start by recognizing that the tangent function has a period of \\(\\pi\\). However, for simplification, we can reduce the angle \\(\\frac{9\\pi}{4}\\) to ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the axis of symmetry of the parabola \\(4x^2 - 8x = y\\)."
},
{
"role": "assistant",
"content": "To find the axis of symmetry of the parabola given by the equation \\(4x^2 - 8x = y\\), we can use two methods: the formula for the axis of symmetry and converting ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( f_n \\) denote the number of ways to toss a fair coin \\( n \\) times such that no two consecutive heads occur. Find the recursion for \\( f_n \\)."
},
{
"role": "assistant",
"content": "To find the recursion for \\( f_n \\), which denotes the number of ways... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many pairs of positive rational numbers \\((a, b)\\) are there that satisfy the equation \\(\\sqrt{a} + \\sqrt{b} = \\sqrt{2 + \\sqrt{3}}\\)?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the number of pairs of positive rational numbers ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If Jeff picks one letter randomly from the alphabet, what is the probability that the letter is in the word `PROBABILITY'?"
},
{
"role": "assistant",
"content": "To find the probability that Jeff picks a letter randomly from the alphabet that is in the word \"PROBAB... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( K \\) be a commutative field. Find all prime ideals of the polynomial ring \\( K[X] \\)."
},
{
"role": "assistant",
"content": "To find all prime ideals of the polynomial ring \\( K[X] \\) where \\( K \\) is a commutative field, we proceed as follows:\n\n1. ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Joe must determine the greatest whole number of feet from which a ceramic ball can be dropped without breaking. He has two identical ceramic balls and can drop them from any whole numbered height. If he must determine this height with no more than 12 drops, what is the greatest... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A sphere has a surface area of \\(676\\pi\\). What is its volume? Express your answer in terms of \\(\\pi\\)."
},
{
"role": "assistant",
"content": "To find the volume of a sphere given its surface area, we start with the formula for the surface area of a sphere:\n\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If \\((1 + x + x^2)^n = a_0 + a_1 x + \\cdots + a_{2n} x^{2n}\\), find \\(a_0 + a_2 + a_4 + \\cdots + a_{2n}\\)."
},
{
"role": "assistant",
"content": "To find the sum of the even-indexed coefficients in the expansion of \\((1 + x + x^2)^n\\), we start by evaluating... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Is it possible for an infinite group \\( G \\) to have a subgroup \\( H \\neq G \\) which is of finite index in \\( G \\)? Provide an example."
},
{
"role": "assistant",
"content": "Certainly! Here is the clean, final correct solution:\n\nTo determine if it is possi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many common points can at least two lines have in a set of 3 plan lines?"
},
{
"role": "assistant",
"content": "To determine the maximum number of common points where at least two lines intersect in a set of three plane lines, we need to consider the possible co... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Two years ago there were 20 trailer homes on Elm Street with an average age of 18 years. At that time, a group of brand new trailer homes was then added to Elm Street. Today, the average age of all the trailer homes on Elm Street is 14 years. How many new trailer homes were add... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the product of the least common multiple and the greatest common factor of $20$ and $90$?"
},
{
"role": "assistant",
"content": "To find the product of the least common multiple (LCM) and the greatest common factor (GCF) of 20 and 90, we can use the relation... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In how many ways can a dog trainer buy 8 dogs, all of which are either cocker spaniels, Irish setters, or Russian wolfhounds, if she wants at least one from each category?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the number of ways a do... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?"
},
{
"role": "assistant",
"content": "To determine the total number of people in the line, we need to use the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The 5-digit number $52\\,28\\square$ is a multiple of 6. Which digit is represented by $\\square$?"
},
{
"role": "assistant",
"content": "To determine the digit represented by \\(\\square\\) in the 5-digit number \\(5228\\square\\) such that the number is a multipl... | You are a helpful assistant. |
[
{
"role": "user",
"content": "You and nine friends initially spend $4000$ dollars on tickets to attend a concert. Six friends cancel last minute due to the flu. You and the remaining friends split the original cost of $4000$ dollars equally. What percent of the total cost does each remaining individual have... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the equation \\( e^{1-5x} = 3.5 \\)."
},
{
"role": "assistant",
"content": "To solve the equation \\( e^{1-5x} = 3.5 \\), we follow these steps:\n\n1. **Take the natural logarithm of both sides:**\n \\[\n \\ln(e^{1-5x}) = \\ln(3.5)\n \\]\n\n2. **Simplify... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Factor the following expression: $37a^2 +111a$."
},
{
"role": "assistant",
"content": "To factor the expression \\(37a^2 + 111a\\), we start by identifying the greatest common factor (GCF) of the terms.\n\n1. **Identify the GCF of the coefficients:**\n - The coeff... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral $\\int_{0}^{t} sgn(x) \\ dx$, where $sgn(x) = -1$ for $x < 0$, $sgn(0) = 0$, and $sgn(x) = 1$ for $x > 0$."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int_{0}^{t} \\text{sgn}(x) \\, dx\\), we need to consider the behavi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "At a birthday party, everyone who gets an invitation card can come alone or bring a friend. Everyone invited was present at the event. The ratio of the number of invitation cards to the number of guests attending is $3:5$. What is the ratio of the number of invitation cards to ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the number of combinations of a given length \\( k \\) that use elements from a set of \\( n \\) elements, allowing repetition."
},
{
"role": "assistant",
"content": "To find the number of combinations of length \\( k \\) from a set of \\( n \\) elements, allow... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the expression \\[ \\frac{a+2}{a+1} \\cdot \\frac{b-1}{b-2} \\cdot \\frac{c + 8}{c+6} , \\] given that $c = b-10$, $b = a+2$, $a = 4$, and none of the denominators are zero."
},
{
"role": "assistant",
"content": "To evaluate the expression \\(\\frac{a+2}{a+... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Molly flips a fair coin five times, and she is very surprised to flip a head each time. What is the probability she will flip a tail on her next flip of the coin? Express your answer as a common fraction."
},
{
"role": "assistant",
"content": "To solve the problem o... | 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": "If the reciprocal of \\( x+1 \\) is \\( x-1 \\), determine \\( x \\)."
},
{
"role": "assistant",
"content": "To solve the problem where the reciprocal of \\( x + 1 \\) is \\( x - 1 \\), we start by setting up the equation:\n\n\\[\n\\frac{1}{x + 1} = x - 1\n\\]\n\nNe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The sum of two numbers is $45$. Their difference is $3$. What is the lesser of the two numbers?"
},
{
"role": "assistant",
"content": "To solve the problem where the sum of two numbers is 45 and their difference is 3, we can use a system of equations. Let the two nu... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Expand the product ${3(x+4)(x+5)}$."
},
{
"role": "assistant",
"content": "To expand the product \\(3(x+4)(x+5)\\), we will follow these steps:\n\n1. First, expand the binomials \\((x+4)\\) and \\((x+5)\\) using the distributive property (FOIL method).\n2. Combine l... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Integrate \\(\\frac{\\sin x}{x}\\) with respect to \\(x\\)."
},
{
"role": "assistant",
"content": "The integral of \\(\\frac{\\sin x}{x}\\) with respect to \\(x\\) cannot be expressed in terms of elementary functions. Instead, it is represented by a special function... | You are a helpful assistant. |
[
{
"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": "To find the measure of ang... | 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": "Khali has to shovel snow off the sidewalk in front of his house. The sidewalk is 20 feet long and 2 feet wide. If the snow is $\\frac{1}{2}$ foot deep, how many cubic feet of snow does Khali have to shovel off the sidewalk?"
},
{
"role": "assistant",
"content": "To ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "For some positive integer $k$, when 60 is divided by $k^2$, the remainder is 6. What is the remainder when 100 is divided by $k$?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by noting that when 60 is divided by \\( k^2 \\), the remainder is... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve for natural numbers \\( n \\) in the equation \\( n^3 - n = n! \\)."
},
{
"role": "assistant",
"content": "To solve the equation \\( n^3 - n = n! \\) for natural numbers \\( n \\), we will proceed step-by-step.\n\nFirst, note that natural numbers are positive ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( F(x) = x^{\\cos x + \\sin x} \\). Find \\( F'(1) \\)."
},
{
"role": "assistant",
"content": "To find \\( F'(1) \\) where \\( F(x) = x^{\\cos x + \\sin x} \\), we use logarithmic differentiation.\n\n1. Let \\( y = x^{\\cos x + \\sin x} \\). Taking the natural... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find $\\ln{-1}$."
},
{
"role": "assistant",
"content": "To find \\(\\ln(-1)\\), we need to use the properties of complex logarithms. The natural logarithm function \\(\\ln(x)\\) is not defined for negative real numbers in the real number system, but it can be extend... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A merchant buys goods at $25\\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\\%$ on the marked price and still clear a profit of $25\\%$ on the selling price. What percent of the list price must he mark the goods?"
},
{
"role":... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral \\(\\int e^{1/z} \\, dz\\) along the boundary of the rectangle centered at \\(1 + 2i\\) with width 5 and height 6. The boundary is divided into four contours."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int e^{1/z} \\, ... | 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": "How many perfect squares are two-digit and divisible by $3?$"
},
{
"role": "assistant",
"content": "To determine how many two-digit perfect squares are divisible by 3, we first identify the range of integers whose squares are two-digit numbers. The smallest integer ... | You are a helpful assistant. |
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