conversations listlengths 2 2 | system stringclasses 1
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[
{
"role": "user",
"content": "How many ways can 3 indistinguishable balls be arranged in 4 distinguishable urns?"
},
{
"role": "assistant",
"content": "To determine the number of ways to arrange 3 indistinguishable balls into 4 distinguishable urns, we can use the \"stars and bars\" theorem. Thi... | 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": "How many ordered triples of positive integers $(x, y, z)$ satisfy the equation $x^{2018} + y^{2018} = z^{2018}$?"
},
{
"role": "assistant",
"content": "To determine the number of ordered triples of positive integers \\((x, y, z)\\) that satisfy the equation \\(x^{20... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a case with five different men, if nine victims randomly guess which voice belongs to the criminal, what is the probability that all nine victims would select the same person?"
},
{
"role": "assistant",
"content": "To find the probability that all nine victims ra... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In how many ways can a postman deliver seven letters to seven neighbors such that exactly two of the neighbors receive the wrong letters?"
},
{
"role": "assistant",
"content": "To determine the number of ways a postman can deliver seven letters to seven neighbors su... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute the two rightmost digits in the sum: $4! + 8! + 12! + 16! + 20!$."
},
{
"role": "assistant",
"content": "To find the two rightmost digits of the sum \\(4! + 8! + 12! + 16! + 20!\\), we need to compute the sum modulo 100.\n\nFirst, we calculate each factorial... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Eight eligible bachelors and seven beautiful models randomly purchase single seats in a 15-seat row of a theatre. On average, how many pairs of adjacent seats are occupied by a bachelor and a model?"
},
{
"role": "assistant",
"content": "To solve the problem of find... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Simplify the following: $(2y-1)\\cdot(4y^{10}+2y^9+4y^8+2y^7).$ Express your answer as a polynomial with the degrees of the terms in decreasing order."
},
{
"role": "assistant",
"content": "To simplify the expression \\((2y-1) \\cdot (4y^{10} + 2y^9 + 4y^8 + 2y^7)\... | 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": "By purchasing rupees 50 shares for rupees 80 each, a man gets a 4% profit on his investment. What is his dividend if he buys 60 shares?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the dividend for 60 shares given that the shares... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate $\\int{\\sin{(\\log{x})}}{dx}$."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\sin(\\log x) \\, dx\\), we proceed as follows:\n\n1. **Substitution**: Let \\( t = \\log x \\). Then \\( dt = \\frac{1}{x} \\, dx \\) or equivalently ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the matrix of the quadratic form \\( f \\) in the basis \\( W \\) and the matrix \\( L_{W}^{V} \\) of the linear transformation \\( L \\) from basis \\( V \\) to basis \\( W \\), how can the matrix of the quadratic form \\( f \\circ L \\) be presented?"
},
{
"role... | 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": "In triangle PQR, PQ has length 6, QR has length 15, and angle PQR is 60 degrees. S is a point on PR such that QS bisects angle PQR. Find the length of PS."
},
{
"role": "assistant",
"content": "To find the length of \\( PS \\) in triangle \\( PQR \\) where \\( PQ = ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A particular novel contains $40,\\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write?"
},
{
"role": "assistant",
"content": "To determine the average number of words per hour the author wrote, we need to di... | 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": "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": "Find the maximal dimension of a subspace of \\( M_n(K) \\) consisting only of nilpotent matrices, where \\( K = \\mathbb{R} \\) or \\( \\mathbb{C} \\)."
},
{
"role": "assistant",
"content": "To find the maximal dimension of a subspace of \\( M_n(K) \\) consisting on... | You are a helpful assistant. |
[
{
"role": "user",
"content": "There are 8 balls in a box, 4 of which are blue, and 4 of which are red. They are otherwise identical. Daniel removes four of the balls without replacement. What is the probability that Daniel removes more blue balls than red balls?"
},
{
"role": "assistant",
"conte... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many vertical asymptotes does the function \\( f(x) = \\frac{x^2 - 6x - 16}{x^3 - 5x^2 - 2x + 24} \\) have?"
},
{
"role": "assistant",
"content": "To determine the number of vertical asymptotes for the function \\( f(x) = \\frac{x^2 - 6x - 16}{x^3 - 5x^2 - 2x + ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the value of $\\frac{2014^3-2013^3-1}{2013 \\times 2014}$."
},
{
"role": "assistant",
"content": "To find the value of \\(\\frac{2014^3 - 2013^3 - 1}{2013 \\times 2014}\\), we can use the algebraic identity for the sum of cubes. Specifically, if \\(a + b + c = ... | 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": "Given the function \\( f(a, b, c) = \\frac{a}{b+c+1} + \\frac{b}{c+a+1} + \\frac{c}{a+b+1} + (1-a)(1-b)(1-c) \\), compute the first partial derivative \\(\\frac{\\partial f}{\\partial a}\\)."
},
{
"role": "assistant",
"content": "To find the first partial derivative... | 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": "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": "Let \\( n \\in \\mathbb{N}, n \\ge 2 \\). Determine which of the following statements are true:\na) Any finite group \\( G \\) of order \\( n \\) is isomorphic to a subgroup of \\( GL_n(\\mathbb{R}) \\)."
},
{
"role": "assistant",
"content": "To determine whether an... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If a leaky faucet collects 1/2 cup of water from 8am to 4pm, how many gallons of water would this amount to in 1 day, given that there are 16 cups in a gallon?"
},
{
"role": "assistant",
"content": "To determine how many gallons of water a leaky faucet collects in o... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given that \\( f(x) \\) is continuous and satisfies \\( f'(1) = f(-\\frac{\\pi}{2}) \\), determine the conditions on \\( a \\) and \\( b \\) for the function \\( \\varphi(x) = a \\sin x + b \\cos x \\)."
},
{
"role": "assistant",
"content": "To determine the conditi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The nonzero roots of the equation $x^2 + 6x + k = 0$ are in the ratio $2:1$. What is the value of $k$?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the value of \\( k \\) in the quadratic equation \\( x^2 + 6x + k = 0 \\) given that the non... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A circle is inscribed in a triangle with sides of lengths 8, 13, and 17. Let the segments of the side of length 8 made by a point of tangency be r and s, with r < s. Find the ratio of r:s."
},
{
"role": "assistant",
"content": "To find the ratio of the segments \\( ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the radius of convergence of the power series $\\sum_{n=0}^\\infty n^2 x^n$."
},
{
"role": "assistant",
"content": "To find the radius of convergence of the power series \\(\\sum_{n=0}^\\infty n^2 x^n\\), we can use the ratio test. The ratio test involves compu... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Ryosuke is picking up his friend from work. The odometer reads 74,568 when he picks his friend up, and it reads 74,592 when he drops his friend off at his house. Ryosuke's car gets 28 miles per gallon and the price of one gallon of gas is $\\$4.05$. What was the cost of the gas... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Circles with centers of $(2,2)$ and $(17,10)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles?"
},
{
"role": "assistant",
"content": "To find the distance between the closest points of the two circles with centers... | 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": "Let \\( x, y, z, w \\) be positive real numbers, and \\( n \\) be a positive integer. If the sum of the coefficients in the expansion of \\( (x+y+z)^n \\) is 6561, find the sum of the coefficients in the expansion of \\( (x+y+z+w)^{\\frac{n}{2}} \\)."
},
{
"role": "assi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int_0^\\frac{1}{2} \\frac{\\sqrt{x}}{\\sqrt{1 - x}} \\, dx \\) using the substitution \\( \\sqrt{x} = \\sin{y} \\)."
},
{
"role": "assistant",
"content": "To evaluate the integral \\( \\int_0^\\frac{1}{2} \\frac{\\sqrt{x}}{\\sqrt{1 - x}} ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Simplify $\\frac{36}{54}$."
},
{
"role": "assistant",
"content": "To simplify the fraction \\(\\frac{36}{54}\\), we need to find the greatest common divisor (GCD) of the numerator (36) and the denominator (54). \n\nFirst, we determine the prime factorizations of bot... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Convert $1357_{10}$ to base 5."
},
{
"role": "assistant",
"content": "To convert the decimal number \\(1357_{10}\\) to base 5, we follow these steps:\n\n1. **Divide the number by 5 and record the remainder.**\n2. **Use the quotient from the previous division as the ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A positive integer \\( n \\) not exceeding 100 is chosen such that if \\( n \\leq 50 \\), the probability of choosing \\( n \\) is \\( p \\), and if \\( n > 50 \\), the probability of choosing \\( n \\) is \\( 3p \\). What is the probability that a perfect square is chosen?"
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the fourth power of $\\sqrt{1+\\sqrt{1+\\sqrt{1}}}$?"
},
{
"role": "assistant",
"content": "To find the fourth power of \\(\\sqrt{1+\\sqrt{1+\\sqrt{1}}}\\), we start by simplifying the expression step by step.\n\n1. Compute the innermost expression:\n \\[\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the greatest possible positive integer value of $x$ if $\\displaystyle\\frac{x^4}{x^2} < 10$?"
},
{
"role": "assistant",
"content": "To solve the inequality \\(\\frac{x^4}{x^2} < 10\\) for the greatest possible positive integer value of \\(x\\), we start by ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many prime numbers are between 30 and 40?"
},
{
"role": "assistant",
"content": "To determine how many prime numbers are between 30 and 40, we first list the numbers in this range: 31, 32, 33, 34, 35, 36, 37, 38, 39.\n\nA prime number is a number greater than 1 ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a triangle \\( \\triangle ABC \\) with circumcenter \\( O \\), given that \\( \\angle BOC = 120^\\circ \\) and \\( \\angle AOB = 140^\\circ \\), find the measure of \\( \\angle ABC \\)."
},
{
"role": "assistant",
"content": "To find the measure of \\(\\angle ABC\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The median of the set $\\{n, n + 5, n + 6, n + 9, n + 15\\}$ is 9. What is the mean?"
},
{
"role": "assistant",
"content": "To find the mean of the set \\(\\{n, n + 5, n + 6, n + 9, n + 15\\}\\) given that the median is 9, we proceed as follows:\n\n1. **Identify the... | 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": "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": "It can be shown that for any positive integer $n,$\n\\[\\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix}^n = \\begin{pmatrix} F_{n + 1} & F_n \\\\ F_n & F_{n - 1} \\end{pmatrix},\\]where $F_n$ denotes the $n$th Fibonacci number.\n\nCompute $F_{784} F_{786} - F_{785}^2.$"
},
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the domain and range of the function \\( y = 10^{\\sqrt{1-x^2}} \\), where \\( x, y \\in \\mathbb{R} \\)."
},
{
"role": "assistant",
"content": "To find the domain and range of the function \\( y = 10^{\\sqrt{1-x^2}} \\), we will follow these steps:\n\n### Step... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In how many ways can Myrtle package 5 lollipops from 9 different flavors if she can repeat flavors?"
},
{
"role": "assistant",
"content": "To determine the number of ways Myrtle can package 5 lollipops from 9 different flavors, where repetition of flavors is allowed... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Aimee flips a coin 8 times. Each time she flips heads, she writes down a 1, and each time she flips tails, she writes down a 0. What is the expected value of the binary number formed by the 8 flips?"
},
{
"role": "assistant",
"content": "To find the expected value o... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many prime numbers are between 30 and 40?"
},
{
"role": "assistant",
"content": "To determine how many prime numbers are between 30 and 40, we first list the numbers in this range: 31, 32, 33, 34, 35, 36, 37, 38, 39.\n\nA prime number is a number greater than 1 ... | 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": "What is the sum of all values of $y$ for which the expression $\\frac{y+6}{y^2-5y+4}$ is undefined?"
},
{
"role": "assistant",
"content": "To determine the sum of all values of \\( y \\) for which the expression \\(\\frac{y+6}{y^2 - 5y + 4}\\) is undefined, we need ... | 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": "Evaluate the product \\(\\sin\\left(\\frac{\\pi}{199}\\right) \\cdot \\sin\\left(\\frac{2\\pi}{199}\\right) \\cdot \\sin\\left(\\frac{3\\pi}{199}\\right) \\cdots \\sin\\left(\\frac{198\\pi}{199}\\right)\\)."
},
{
"role": "assistant",
"content": "To evaluate the prod... | You are a helpful assistant. |
[
{
"role": "user",
"content": "While shopping at the local mall, Ann walked slowly (at a constant speed) down a descending escalator, taking 10 steps to reach the last step. She then ran up the same (descending) escalator with a speed five times greater than the speed with which she had walked down, taking 2... | 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": "Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7."
},
{
"role": "assistant",
"content": "To find the units digit of \\( n \\) given that \\( mn = 21^6 \\) and \\( m \\) has a units digit of 7, we can proceed as follows:\n\n1. **Deter... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A circle passes through vertices A and D and touches side BC of a square with side length 2. What is the radius of the circle?"
},
{
"role": "assistant",
"content": "Given a square with side length 2, vertices \\( A(0,0) \\), \\( B(2,0) \\), \\( C(2,2) \\), and \\( ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $\\log_b 2 = a$ and $\\log_b 3 = c$, then compute $\\log_b 3456b$."
},
{
"role": "assistant",
"content": "To compute \\(\\log_b 3456b\\) given \\(\\log_b 2 = a\\) and \\(\\log_b 3 = c\\), we proceed as follows:\n\n1. Use the logarithm property \\(\\log_b (xy) = \... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the value of $t$ that satisfies $\\frac{1}{t+2} + \\frac{2t}{t+2} - \\frac{3}{t+2} = 3$."
},
{
"role": "assistant",
"content": "To solve the equation \\(\\frac{1}{t+2} + \\frac{2t}{t+2} - \\frac{3}{t+2} = 3\\), we start by combining the fractions on the left si... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the probability of having 4 boys and 4 girls in a family of 8 children?"
},
{
"role": "assistant",
"content": "To find the probability of having exactly 4 boys and 4 girls in a family of 8 children, we can use the binomial probability formula. Here are the d... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A recent study found that 60$\\%$ of men and 80$\\%$ of women surveyed support increased funding for particular medical research. The study surveyed 100 men and 900 women. What was the overall percent of the people surveyed who supported increased funding?"
},
{
"role":... | 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": "Find all limit points of the sequence \\( x_n = \\cos(n) \\)."
},
{
"role": "assistant",
"content": "To find all limit points of the sequence \\( x_n = \\cos(n) \\), we need to analyze the behavior of the sequence \\( \\cos(n) \\) where \\( n \\) is an integer.\n\n#... | 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. |
[
{
"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": "If $1+2x+3x^2 + \\dotsb=9$, find $x$."
},
{
"role": "assistant",
"content": "To solve the problem \\(1 + 2x + 3x^2 + \\dotsb = 9\\), we start by recognizing that the series can be written as \\(\\sum_{n=0}^{\\infty} (n+1)x^n\\). This series converges for \\(|x| < 1\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Amanda has an 8-digit phone number such that the first four digits are 1 seven, 1 eight, and 2 nines. The fifth digit is either 2, 3, or 5. The sixth digit is any digit, the seventh digit is 9 minus the sixth digit, and the eighth digit is any odd digit. If the total possible n... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the limit $\\lim_{h\\to0}\\left[\\frac{1}{h}\\int^{x^2+h}_{x^2}\\log[3+\\arcsin(t)]\\,dt\\right]$ in terms of $x$."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{h\\to0}\\left[\\frac{1}{h}\\int^{x^2+h}_{x^2}\\log[3+\\arcsin(t)]\\,dt\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Contractor Steve agreed to complete a job in 30 days. After 6 days he found that the 8 people assigned to the work had already done $\\frac{1}{3}$ of the job. If everyone works at the same rate, what is the least number of people he must keep on the job to ensure that the job w... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Three independent meteorologists, each of whom is wrong 1/4 of the time, all predict that it will be sunny tomorrow. What is the probability that it will be sunny tomorrow?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the probability that i... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve for $x$: $4x^{1/3}-2 \\cdot \\frac{x}{x^{2/3}}=7+\\sqrt[3]{x}$."
},
{
"role": "assistant",
"content": "To solve the equation \\(4x^{1/3} - 2 \\cdot \\frac{x}{x^{2/3}} = 7 + \\sqrt[3]{x}\\), we start by simplifying the terms involving \\(x\\).\n\nFirst, simplif... | 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": "Express $\\frac{165_7}{11_2}+\\frac{121_6}{21_3}$ in base 10."
},
{
"role": "assistant",
"content": "To solve the problem \\(\\frac{165_7}{11_2} + \\frac{121_6}{21_3}\\) and express the result in base 10, we will follow these steps:\n\n1. Convert \\(165_7\\) to base... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine the median and the mode of Jerelyn’s scores from her first six tests, which were 92, 78, 86, 92, 95, and 91."
},
{
"role": "assistant",
"content": "To determine the median and the mode of Jerelyn’s scores from her first six tests, which were 92, 78, 86, 92... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given that $\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}}=\\frac{\\pi^{2}}{6}$, find $p(6)$, where $p(x)$ is a unique polynomial with rational coefficients satisfying $p(\\pi)=\\sum_{n=1}^{\\infty}\\frac{d(n)}{n^2}$, and $d(n)$ denotes the number of divisors of $n$."
},
{
"role... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find a formula or rule for the sequence: $1, 2, 6, 30, 210, 2310, \\ldots$"
},
{
"role": "assistant",
"content": "To find a formula or rule for the sequence \\(1, 2, 6, 30, 210, 2310, \\ldots\\), we observe the pattern in the sequence and derive the rule as follows:... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Abigail, Beatrice, and Carson combine their eggs to sell them at the market. If Abigail has 37 eggs, Beatrice has 49 eggs, and Carson has 14 eggs, and if eggs can only be sold in cartons of 12, how many eggs will be left over if all cartons are sold?"
},
{
"role": "assi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the sum \\( S_{n} = \\sum_{i=1}^{n} \\frac{a_{i}}{(a_{1}+1)(a_{2}+1) \\cdots (a_{i}+1)} \\), where \\( a_{n} \\) is a sequence."
},
{
"role": "assistant",
"content": "To evaluate the sum \\( S_{n} = \\sum_{i=1}^{n} \\frac{a_{i}}{(a_{1}+1)(a_{2}+1) \\cdots (... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the remainder when $x^{203} - 1$ is divided by $x^4 - 1$?"
},
{
"role": "assistant",
"content": "To find the remainder when \\( x^{203} - 1 \\) is divided by \\( x^4 - 1 \\), we start by noting that \\( x^4 \\equiv 1 \\mod (x^4 - 1) \\). This equivalence all... | 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": "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. |
[
{
"role": "user",
"content": "How do you derive the function \\( f(x^n) \\) where \\( n \\) is a positive integer?"
},
{
"role": "assistant",
"content": "To find the derivative of the function \\( f(x^n) \\) where \\( n \\) is a positive integer, we use the chain rule. The chain rule states that... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can the letters in $AMERICA$ be arranged such that the $R$ and $I$ are not right next to each other?"
},
{
"role": "assistant",
"content": "To find the number of ways to arrange the letters in \"AMERICA\" such that the letters R and I are not next to e... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Calculate the intersection \\(\\bigcap_{e>0} (x - e, x + e)\\)."
},
{
"role": "assistant",
"content": "To find the intersection \\(\\bigcap_{e>0} (x - e, x + e)\\), we need to determine the set of all points that belong to every interval \\((x - e, x + e)\\) for all... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is $0.1 \\div 0.004$?"
},
{
"role": "assistant",
"content": "To solve the problem \\(0.1 \\div 0.004\\), we can use the method of eliminating the decimals by multiplying both the dividend and the divisor by the same power of 10. Here are the steps:\n\n1. Identi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find an explicit formula for the sequence $0, 3, 26, 255, 3124, 46655, \\ldots$."
},
{
"role": "assistant",
"content": "To find an explicit formula for the sequence \\(0, 3, 26, 255, 3124, 46655, \\ldots\\), we will follow a systematic approach to identify the patte... | 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": "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": "What numeral is in the 100th decimal place in the decimal representation of $\\frac{6}{7}$?"
},
{
"role": "assistant",
"content": "To determine the numeral in the 100th decimal place of the decimal representation of \\(\\frac{6}{7}\\), we start by finding the repeat... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can Snow White redistribute the presents so that each of the \\( n \\) dwarves gets a different present than what they had before?"
},
{
"role": "assistant",
"content": "To determine the number of ways Snow White can redistribute the presents such that... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the equation \\(3x^4 - 12 = 0\\)."
},
{
"role": "assistant",
"content": "To solve the equation \\(3x^4 - 12 = 0\\), follow these steps:\n\n1. Start by isolating the term involving \\(x\\):\n \\[\n 3x^4 - 12 = 0\n \\]\n Add 12 to both sides:\n \\[\n ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the function \\( F(x, y) = x^3 + 3x^2y - 2xy^5 \\), find the first partial derivative with respect to \\( x \\) when \\( y \\) is held constant."
},
{
"role": "assistant",
"content": "To find the first partial derivative of the function \\( F(x, y) = x^3 + 3x^... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Is the cube function \\( f(x) = x^3 \\) increasing or strictly increasing?"
},
{
"role": "assistant",
"content": "To determine whether the cube function \\( f(x) = x^3 \\) is increasing or strictly increasing, we need to analyze the behavior of the function based on... | 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": "Richard has 3 bananas, 4 apples, and 2 peaches in a basket. If he randomly chooses two pieces of fruit without replacement, what is the probability that he chooses two different kinds of fruit?"
},
{
"role": "assistant",
"content": "To find the probability that Rich... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the limit $\\underset{x\\to \\infty }{\\mathop{\\lim }}\\,\\left( \\sin \\left( x+1 \\right)-\\sin x \\right)$."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{x \\to \\infty} \\left( \\sin(x+1) - \\sin(x) \\right)\\), we will use a t... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\n\\sqrt{x} &\\text{ if }x>4, \\\\\nx^2 &\\text{ if }x \\le 4.\n\\end{array}\n\\right.\\]Find $f(f(f(2)))$."
},
{
"role": "assistant",
"content": "To find \\( f(f(f(2))) \\) for the given piecewise function \\( f(x) \\):\n\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute the integral \\( I = \\int \\frac{1}{2\\tan(x) + 1} \\, dx \\)."
},
{
"role": "assistant",
"content": "To compute the integral \\( I = \\int \\frac{1}{2\\tan(x) + 1} \\, dx \\), we start by rewriting the integrand in terms of sine and cosine:\n\n\\[\n2\\tan(... | You are a helpful assistant. |
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