conversations listlengths 2 2 | system stringclasses 1
value |
|---|---|
[
{
"role": "user",
"content": "Find the number of possible solutions to the Diophantine equation \\( w + x + y + z = n \\)."
},
{
"role": "assistant",
"content": "To find the number of possible solutions to the Diophantine equation \\( w + x + y + z = n \\) where \\( w, x, y, z \\) are non-negati... | 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": "Determine all local maxima and minima in the interior of the plate defined by \\( x^2 + y^2 \\leq 4 \\) for the temperature function \\( T(x, y) = x^3 + 2y^2 - 4x \\)."
},
{
"role": "assistant",
"content": "To determine all local maxima and minima in the interior of... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given that \\(a + b + c = 1\\), find the minimum value of \\(\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c}\\)."
},
{
"role": "assistant",
"content": "To find the minimum value of \\(\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c}\\) given that \\(a + b + c = 1\\) and assumi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Derek has 10 American coins in his pocket, summing to a total of 53 cents. If he randomly grabs 3 coins from his pocket, what is the probability that they are all different?\n\nGiven that the 10 coins consist of 3 pennies, 4 nickels, and 3 dimes, determine the probability that ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the sum of the reciprocals of the natural-number factors of 6?"
},
{
"role": "assistant",
"content": "To find the sum of the reciprocals of the natural-number factors of 6, we first identify the factors of 6. The factors of 6 are the natural numbers that div... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $c$ is the total cost (in million rupiah) for producing $q$ units of a product, then the average cost per unit is given by\n\\[\\overline{c} = \\frac{5000 + 6q}{q}.\\]\na. If producers continuously increase their production, what is the limit value of this average cost?\nb. ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The radius of the first circle is $1$ inch, that of the second $\\frac{1}{2}$ inch, that of the third $\\frac{1}{4}$ inch, and so on indefinitely. What is the sum of the areas of the circles?"
},
{
"role": "assistant",
"content": "To find the sum of the areas of the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the expression $(\\omega+1)^{3n}$, where $\\omega$ is a cube root of unity."
},
{
"role": "assistant",
"content": "To evaluate the expression \\((\\omega + 1)^{3n}\\) where \\(\\omega\\) is a cube root of unity, we start by using the properties of cube root... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Jenna spent $175.00 on Christmas gifts for 23 nurses at the hospital. She bought candles for some nurses at $5.00 each and scarves for the others at $10.00 each. How many candles and how many scarves did she buy?"
},
{
"role": "assistant",
"content": "To solve the p... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How can you add eight 8's to get the number 1000?"
},
{
"role": "assistant",
"content": "To solve the problem of adding eight 8's to get the number 1000, we need to creatively combine the digits into multi-digit numbers. Here is the clean and detailed solution:\n\n1... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Consider the function \\( f(x) = x^3 + 2x \\) and its inverse function \\( g(x) \\). Use the derivative of \\( f(g(x)) \\) to find the gradient of the tangent to \\( y = g(x) \\) at the point where \\( x = 3 \\)."
},
{
"role": "assistant",
"content": "To find the gr... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If the lengths of the legs of a right triangle are 5 and 12, what is the length of the hypotenuse?"
},
{
"role": "assistant",
"content": "To find the length of the hypotenuse of a right triangle with legs of lengths 5 and 12, we use the Pythagorean theorem, which st... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a certain algebraic exercise book, there are 4 examples on arithmetical progressions, 5 examples on permutation and combination, and 6 examples on binomial theorem. Find the number of ways a teacher can select at least one but not more than 2 examples from each of these sets... | You are a helpful assistant. |
[
{
"role": "user",
"content": "David drives from his home to the airport. He drives 35 miles in the first hour and realizes he will be 1 hour late if he continues at this speed. He then increases his speed by 15 miles per hour for the rest of the trip and arrives 30 minutes early. How many miles is the airpo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Calculate the indefinite integral $\\int \\frac{\\tan^2 x}{\\cos^2 x} \\, dx$."
},
{
"role": "assistant",
"content": "To calculate the indefinite integral \\(\\int \\frac{\\tan^2 x}{\\cos^2 x} \\, dx\\), we start by simplifying the integrand. Recall that \\(\\sec^2 ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Differentiate the function \\( f(x) = \\sin (\\ln x) \\)."
},
{
"role": "assistant",
"content": "To find the derivative of the function \\( f(x) = \\sin (\\ln x) \\), we will use the chain rule. The chain rule states that the derivative of a composite function \\( f... | 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": "Compute $\\sum_{r=1}^{n} r(r-1) \\binom{n}{r}$."
},
{
"role": "assistant",
"content": "To compute the sum \\(\\sum_{r=1}^{n} r(r-1) \\binom{n}{r}\\), we start by expressing \\(r(r-1) \\binom{n}{r}\\) in a more convenient form.\n\nFirst, recall the binomial coefficie... | 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 will use the Rational Root Theorem and synthetic division.\n\n1. **Identify Possible Rational Roots:**\n A... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the equation \\(\\frac{31}{195} = \\frac{1}{A + \\frac{1}{B + \\frac{1}{C + \\frac{1}{D}}}}\\), where \\(A, B, C, D\\) are positive integers, find \\(A^2 + B^2 + C^2 + D^2\\)."
},
{
"role": "assistant",
"content": "To solve the equation \\(\\frac{31}{195} = \\... | 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": "An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 2, find the area of the hexagon."
},
{
"role": "assistant",
"content": "To find the area of the regular hexagon given that the equilateral triangle and the regular he... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny one of those days for a World War III preenactment, but if it's sunny for more ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the equation $10^{2y} = 25$ using only laws of exponents and algebra."
},
{
"role": "assistant",
"content": "To solve the equation \\(10^{2y} = 25\\) using laws of exponents and algebra, follow these steps:\n\n1. Recognize that \\(25\\) can be written as \\(5^... | 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": "In a right triangle, the square of the hypotenuse is equal to twice the product of the legs. What is one of the acute angles of the triangle?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by defining the sides of the right triangle. Let the l... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the degree measure of an angle whose measure is double the measure of its complement?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the degree measure of an angle whose measure is double the measure of its complement, we proceed as f... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given real numbers \\(a, b, c, d, e\\) such that\n\\[a + b + c + d + e = 8\\]\n\\[a^2 + b^2 + c^2 + d^2 + e^2 = 16\\]\nWhat is the largest possible value of \\(e\\)?"
},
{
"role": "assistant",
"content": "Given real numbers \\(a, b, c, d, e\\) such that\n\\[\na + b ... | 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": "If Anna flips 8 coins, what is the probability that she gets more heads than tails?"
},
{
"role": "assistant",
"content": "To find the probability that Anna gets more heads than tails when flipping 8 coins, we need to consider the cases where the number of heads is ... | 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": "Let $ABCDEF$ be a cyclic hexagon such that $AB \\parallel DE$. What result is obtained when applying Pascal's Theorem to the hexagon $ABCDEF$?"
},
{
"role": "assistant",
"content": "To solve the problem of applying Pascal's Theorem to the cyclic hexagon \\(ABCDEF\\)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int \\frac{x^2}{2 + x^6} \\, dx \\)."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\frac{x^2}{2 + x^6} \\, dx\\), we use the substitution method. Let \\( u = x^3 \\). Then, the differential \\( du \\) is given ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Is there a natural number \\( n \\) such that the sum of the digits of \\( n^2 \\) is exactly 2018? If it exists, find the number; if it does not exist, explain the reasons."
},
{
"role": "assistant",
"content": "To determine if there exists a natural number \\( n \... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the line through the point (3,1,-2) that intersects and is perpendicular to the line given by the parametric equations \\( x = -1 + t \\), \\( y = -2 + t \\), \\( z = -1 + t \\)."
},
{
"role": "assistant",
"content": "To find the line through the point \\((3,1,... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a plane, there are \\( n \\) points out of which 4 are collinear. How many distinct triangles can be drawn?"
},
{
"role": "assistant",
"content": "To determine the number of distinct triangles that can be formed from \\( n \\) points in a plane, where 4 of these ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of $(x - y)(x + y)$ if $x = 10$ and $y = 15$?"
},
{
"role": "assistant",
"content": "To find the value of \\((x - y)(x + y)\\) when \\(x = 10\\) and \\(y = 15\\), we can use the difference of squares formula, which states that \\((x - y)(x + y) = x... | 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": "Find the remainder when $8\\cdot10^{18}+1^{18}$ is divided by 9."
},
{
"role": "assistant",
"content": "To find the remainder when \\(8 \\cdot 10^{18} + 1^{18}\\) is divided by 9, we can use modular arithmetic.\n\nFirst, observe that:\n\\[ 10 \\equiv 1 \\pmod{9} \\]... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Four positive integers $A$, $B$, $C$ and $D$ have a sum of 36. If $A+2 = B-2 = C \\times 2 = D \\div 2$, what is the value of the product $A \\times B \\times C \\times D$?"
},
{
"role": "assistant",
"content": "Given four positive integers \\( A \\), \\( B \\), \\(... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many vertices will the simple graph G have if it contains 15 edges, 3 vertices of degree 4, and the other vertices of degree 3?"
},
{
"role": "assistant",
"content": "To determine the number of vertices in the simple graph \\( G \\) with 15 edges, where 3 vertic... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways are there to color the vertices of a square with six colors if rotations are considered the same?"
},
{
"role": "assistant",
"content": "To determine the number of distinct ways to color the vertices of a square using six colors, considering that rotat... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$?"
},
{
"role": "assistant",
"content": "To find \\( g(f(-2)) \\), we need to evaluate the inner function \\( f(-2) \\) first and then use that result as the input for the function \\( g \\).\n\n1. Evalua... | You are a helpful assistant. |
[
{
"role": "user",
"content": "There are 20 houses along the shore of a lake, each painted one of four colors at random. Starting from one of these houses, you walk around the lake and laugh each time you pass a house that is the same color as the previous house you passed. What is the expected number of tim... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Andi, Beni, Coki, Doni, and Edo are playing a game where each player is either a mousedeer (who always tells the truth) or a wolf (who always lies). Andi says that Beni is a mousedeer. Coki says Doni is a wolf, while Edo says Andi is not a wolf. Beni says Coki is not a mousedee... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many other points with integer coordinates lie on the line passing through $P(2,2)$ and $Q(50,998)$ that are strictly between $P$ and $Q$?"
},
{
"role": "assistant",
"content": "To determine the number of integer coordinate points strictly between \\( P(2, 2) \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Shauna takes five tests, each with a maximum of 100 points. Her scores on the first three tests are 76, 94, and 87. In order to average 81 for all five tests, what is the lowest score she can get on one of the other two tests?"
},
{
"role": "assistant",
"content": "... | 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": "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 $r$ such that $\\log_{81} (2r-1) = -1/2$."
},
{
"role": "assistant",
"content": "To solve the equation \\(\\log_{81} (2r-1) = -\\frac{1}{2}\\), we start by converting the logarithmic equation to its exponential form. The equation \\(\\log_{81} (2r-1) = -\\frac{... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute $2011 \\cdot 2012 \\cdot 2013 \\cdot 2014$ modulo 5."
},
{
"role": "assistant",
"content": "To compute \\(2011 \\cdot 2012 \\cdot 2013 \\cdot 2014 \\mod 5\\), we can simplify the problem by reducing each of the numbers modulo 5 first.\n\n1. Calculate each nu... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The positive difference between the two roots of the quadratic equation $3x^2 - 7x - 8 = 0$ can be written as $\\frac{\\sqrt{m}}{n}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$."
},
{
"role": "assistant",
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( u_n = \\sum_{i=1}^n \\frac{1}{i^a} \\) where \\( a > 0 \\). Find the value of \\( a \\) such that \\( u_n \\) converges to a finite limit (not equal to infinity)."
},
{
"role": "assistant",
"content": "To determine the value of \\( a \\) such that the series... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of the 25th term of the arithmetic sequence $2,\n5, 8, \\ldots$?"
},
{
"role": "assistant",
"content": "To find the 25th term of the arithmetic sequence \\(2, 5, 8, \\ldots\\), we start by identifying the first term \\(a_1\\) and the common differe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the limit $\\lim _{n \\rightarrow \\infty} \\sqrt[n]{n^{e}+e^{n}}$."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{n \\to \\infty} \\sqrt[n]{n^e + e^n}\\), we start by analyzing the behavior of the terms inside the radical as \\(n\\)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$."
},
{
"role": "assistant",
"content": "Given the equation \\( x^3 + 4x = 8 \\), we need to determine the value of \\( x^7 + 64x^2 \\).\n\nFirst, rearrange the given equation to expres... | 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": "Using the result from the previous problem, find $\\int{\\frac{x^{3}}{x^{2}+2}}\\,dx$ without performing any further division."
},
{
"role": "assistant",
"content": "To solve the integral \\(\\int \\frac{x^3}{x^2 + 2} \\, dx\\) using the result from the previous pro... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Bill is sent to a donut shop to purchase exactly six donuts. If the shop has four kinds of donuts and Bill is to get at least one of each kind, how many combinations will satisfy Bill's order requirements?"
},
{
"role": "assistant",
"content": "To determine the numb... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of $x$ for which $(2008+x)^2=x^2$?"
},
{
"role": "assistant",
"content": "To solve the equation \\((2008 + x)^2 = x^2\\), we start by expanding the left side of the equation:\n\n\\[\n(2008 + x)^2 = 2008^2 + 2 \\cdot 2008 \\cdot x + x^2\n\\]\n\nSubs... | 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": "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": "How many ways are there to scramble the letters of the word \"contest\"?"
},
{
"role": "assistant",
"content": "To determine the number of distinct ways to scramble the letters of the word \"contest,\" we start by noting that the word consists of 7 letters, with the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the equation \\(3x^2 - 25x + 66 = 0\\) with solutions \\(x = 4\\) and \\(x = 9\\), determine the base of the number system used on the planet Pandora, assuming the symbols for digits in Pandora's system and the decimal system have the same value."
},
{
"role": "as... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A square has an area of $25x^2 + 20x + 4$. Find the restrictions on $x$ such that the side length of the square is positive."
},
{
"role": "assistant",
"content": "To find the restrictions on \\( x \\) such that the side length of the square is positive, given that ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( A = \\{1, 2, 3, 4, 5\\} \\) and \\( B = \\{0, 1, 2\\} \\). Find the number of pairs of functions \\((f, g)\\) where both \\( f \\) and \\( g \\) map the set \\( A \\) into the set \\( B \\) and there are exactly two elements \\( x \\in A \\) such that \\( f(x) = g(x) \\... | 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": "Given \\( f(t) = t + \\ln(t) \\), find \\( \\frac{d(f^{-1})}{dt} \\) at \\( t = 1 + e \\). Provide exact values."
},
{
"role": "assistant",
"content": "To find \\( \\frac{d(f^{-1})}{dt} \\) at \\( t = 1 + e \\) for the function \\( f(t) = t + \\ln(t) \\), we follow ... | 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": "Mr. Brown travels 30 kilometers in 3 hours. Part of the time he travels at 5 km/h and the rest of the time at 11 km/h. How long does Mr. Brown travel at each speed?"
},
{
"role": "assistant",
"content": "To determine how long Mr. Brown traveled at each speed, we can... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many positive three-digit integers are divisible by both 11 and 5?"
},
{
"role": "assistant",
"content": "To find how many positive three-digit integers are divisible by both 11 and 5, we need to determine the numbers divisible by their least common multiple (LC... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many whole numbers lie in the interval between $\\frac{5}{3}$ and $2\\pi$ ?"
},
{
"role": "assistant",
"content": "To determine how many whole numbers lie in the interval between \\(\\frac{5}{3}\\) and \\(2\\pi\\), we follow these steps:\n\n1. Calculate the appr... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $\\displaystyle\\frac{q}{r} = 9$, $\\displaystyle\\frac{s}{r} = 6$, and $\\displaystyle \\frac{s}{t} = \\frac{1}{2}$, then what is $\\displaystyle\\frac{t}{q}$?"
},
{
"role": "assistant",
"content": "To find the value of \\(\\frac{t}{q}\\) given the equations \\(... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} (-1)^n \\cdot \\sin\\left(\\frac{2n+1}{2} \\pi\\right) \\)."
},
{
"role": "assistant",
"content": "To evaluate the limit \\( \\lim_{n \\to \\infty} (-1)^n \\cdot \\sin\\left(\\frac{2n+1}{2} \\pi\\right) \\), we proceed a... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given a polynomial \\( P(x) \\) such that \\( P(2) = 5 \\) and \\( P(3) = 7 \\), find the polynomial \\( u(x) \\) in the expression \\( P(x) = (x-2)(x-3)p(x) + u(x) \\)."
},
{
"role": "assistant",
"content": "To find the polynomial \\( u(x) \\) in the expression \\(... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Eight distinct points are in the plane such that five of them lie on a line \\( \\ell \\), and the other three points lie off the line, in a way such that if some three of the eight points lie on a line, they lie on \\( \\ell \\). How many triangles can be formed using some thr... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A gambler plays a game \\( n \\) times, with the probability of winning each game being \\( p \\). If it is known that the gambler did not win \\( x \\) times, what is the expected number of times the gambler won?"
},
{
"role": "assistant",
"content": "To find the e... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The city has 80 buses. 30% of them are new and cause less pollution. If the city replaces 14 old buses with new ones, what percent of the buses will be new? Round your answer to the nearest tenth."
},
{
"role": "assistant",
"content": "To solve the problem, we need ... | 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": "Find all integers $N$ such that $N^3 = 9k + 2$, where $k$ is an integer."
},
{
"role": "assistant",
"content": "To find all integers \\( N \\) such that \\( N^3 = 9k + 2 \\) where \\( k \\) is an integer, we need to determine if there exists an integer \\( N \\) for... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the equation \\( e^{-2x} - 3e^{-x} + 2 = 0 \\)."
},
{
"role": "assistant",
"content": "To solve the equation \\( e^{-2x} - 3e^{-x} + 2 = 0 \\), we start by making a substitution. Let \\( y = e^{-x} \\). Then, \\( e^{-2x} = (e^{-x})^2 = y^2 \\). Substituting th... | 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": "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": "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": "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": "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": "If the value of $2^x + 2^{-x}$ is $\\sqrt[3]{3}$, find the exact value of $8^x + 8^{-x}$. Give the final result in its simplest form."
},
{
"role": "assistant",
"content": "Given \\(2^x + 2^{-x} = \\sqrt[3]{3}\\), we need to find the exact value of \\(8^x + 8^{-x}\\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the indefinite integral of $r e^{r}$ with respect to $r$?"
},
{
"role": "assistant",
"content": "To find the indefinite integral of \\( r e^{r} \\) with respect to \\( r \\), we use integration by parts. The formula for integration by parts is given by:\n\n\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the values of $\\alpha$ and $\\beta$ that satisfy the equality: \n\\[\n\\lim_{n\\to+\\infty} n^{\\alpha} \\sum_{m=1}^{n}\\frac{1}{\\sqrt{m}} = \\beta\n\\]"
},
{
"role": "assistant",
"content": "To evaluate the values of \\(\\alpha\\) and \\(\\beta\\) that s... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Simplify \\(3(1-5)-[(1-7)(-5)]\\)."
},
{
"role": "assistant",
"content": "To simplify the expression \\(3(1-5) - [(1-7)(-5)]\\), we will follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a right-angled triangle with a perimeter of 60, the length of the perpendicular drawn from one vertex to the hypotenuse is 12. What is the area of the triangle?"
},
{
"role": "assistant",
"content": "To find the area of the right-angled triangle with a perimeter ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "When $f(x) = ax^3 - 6x^2 + bx - 5$ is divided by $x - 1,$ the remainder is $-5.$ When $f(x)$ is divided by $x + 2,$ the remainder is $-53.$ Find the ordered pair $(a,b).$"
},
{
"role": "assistant",
"content": "To solve the problem, we use the Remainder Theorem whi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate $\\int \\frac{1}{\\sqrt{x^4 + 2x^2 + 1}} + \\frac{12\\sqrt{1-x^2}}{1-x^2} \\, dx$."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\frac{1}{\\sqrt{x^4 + 2x^2 + 1}} + \\frac{12\\sqrt{1-x^2}}{1-x^2} \\, dx\\), we will break it into t... | 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 need to follow the order of operations (PEMDAS/BODMAS):\n\n1. **Parentheses/Brackets**: There are no parentheses or... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the limit $\\lim_{n \\to \\infty }(\\frac{1}{n+1}+\\frac{1}{n+2}+\\frac{1}{n+3}+...+\\frac{1}{3n})$."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{n \\to \\infty} \\left( \\frac{1}{n+1} + \\frac{1}{n+2} + \\frac{1}{n+3} + \\cdots + ... | 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": "Evaluate $\\log_{\\sqrt{6}} (216\\sqrt{6})$."
},
{
"role": "assistant",
"content": "To evaluate \\(\\log_{\\sqrt{6}} (216\\sqrt{6})\\), we start by expressing the logarithm in terms of exponents. Let:\n\n\\[\n\\log_{\\sqrt{6}} (216\\sqrt{6}) = x\n\\]\n\nThis implies... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Characterize all graphs with $\\Delta(G) \\leq 2$."
},
{
"role": "assistant",
"content": "To characterize all graphs with \\(\\Delta(G) \\leq 2\\), we need to understand the structure of such graphs. Recall that \\(\\Delta(G)\\) denotes the maximum degree of a graph... | You are a helpful assistant. |
No dataset card yet
- Downloads last month
- 5