conversations listlengths 2 2 | system stringclasses 1
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[
{
"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": "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": "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": "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 of the function \\( f(x) = \\frac{x^2 - 6x - 16}{x^3 - 5x^2 - 2x + 2... | 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": "The four-digit numeral $3AA1$ is divisible by 9. What digit does $A$ represent?"
},
{
"role": "assistant",
"content": "To determine the digit \\( A \\) in the four-digit number \\( 3AA1 \\) such that the number is divisible by 9, we use the rule that a number is div... | 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": "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": "Find the polynomial whose roots are three more than those of the polynomial \\(x^3 - 6x^2 + 3x - 10\\)."
},
{
"role": "assistant",
"content": "To find the polynomial whose roots are three more than those of the polynomial \\(x^3 - 6x^2 + 3x - 10\\), we need to subst... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the parametric equations $x=2t-\\sin t$ and $y=2-\\cos t$ for $0 < t < 2\\pi$, find the value of $(x,y)$ when $\\frac{dy}{dx}=0$."
},
{
"role": "assistant",
"content": "Given the parametric equations \\(x = 2t - \\sin t\\) and \\(y = 2 - \\cos t\\) for \\(0 < ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "(ii) Given a set $A = \\{a_1, a_2, a_3, \\ldots, a_n\\}$, find the total number of binary operations on $A$ such that $a_i * a_j \\neq a_i * a_k$ if $j \\neq k$."
},
{
"role": "assistant",
"content": "To find the total number of binary operations on a set \\( A = \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "How many ways can 8 queens be arranged on an 8x8 chessboard so that no two queens can attack each other?"
},
{
"role": "assistant",
"content": "To solve the problem of determining how many ways 8 queens can be arranged on an 8x8 chessboard so that no two queens can ... | 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": "What is the perimeter of the right triangle whose exradius of the hypotenuse is 30?"
},
{
"role": "assistant",
"content": "To find the perimeter of the right triangle whose exradius of the hypotenuse is 30, we start by recalling the formula for the exradius correspo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many spheres are there that are tangent to all lines containing the edges of a triangular pyramid?"
},
{
"role": "assistant",
"content": "To determine the number of spheres that are tangent to all lines containing the edges of a triangular pyramid (tetrahedron),... | 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": "Three-fourths of the parrots on Bird Island are green, and the remainder are blue. If there are 92 parrots total on Bird Island, how many of those parrots are blue?"
},
{
"role": "assistant",
"content": "To determine the number of blue parrots on Bird Island, we sta... | 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": "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": "Evaluate the limit $\\lim_{x\\rightarrow 6} f(x)$ for the piecewise function:\n\\[ f(x) = \\begin{cases} \nx^2 - 4, & 0 < x < 6 \\\\ \n4, & x = 6 \\\\ \n8x + 2, & x > 6 \n\\end{cases} \\]"
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{x \\to ... | 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": "Find the number of ordered triples of sets $(A, B, C)$ such that $A \\cup B \\cup C = \\{ 1, 2, \\cdots, 2003 \\}$ and $A \\cap B \\cap C = \\varnothing$."
},
{
"role": "assistant",
"content": "To find the number of ordered triples of sets \\((A, B, C)\\) such that ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In the \"Let's Make a Deal\" game, what is the probability of winning the good prize if you switch doors after being shown one of the junky prizes?"
},
{
"role": "assistant",
"content": "To determine the probability of winning the good prize by switching doors in th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Goldilocks has a $1/2$ chance of choosing the correct path at each fork. If she chooses the wrong path more than two times, she gets lost. What is the probability she gets lost if she encounters five forks?"
},
{
"role": "assistant",
"content": "To determine the pro... | 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": "If $23=x^4+\\frac{1}{x^4}$, then what is the value of $x^2+\\frac{1}{x^2}$?"
},
{
"role": "assistant",
"content": "To find the value of \\( x^2 + \\frac{1}{x^2} \\) given that \\( x^4 + \\frac{1}{x^4} = 23 \\), we start by letting \\( y = x^2 + \\frac{1}{x^2} \\). W... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a singles tennis tournament with 10 players, in how many ways can the players be organised into 5 pairs?"
},
{
"role": "assistant",
"content": "To determine the number of ways to organize 10 players into 5 pairs, we can use the concept of combinations and adjust ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "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": "Find the critical points of the system of differential equations given by:\n\\[ x' = y(x^2 + 1) \\]\n\\[ y' = 2xy^2 \\]"
},
{
"role": "assistant",
"content": "To find the critical points of the system of differential equations given by:\n\\[ x' = y(x^2 + 1) \\]\n\\[... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find a simple expression for the sum \\(\\sum_{k=1}^{\\infty} \\frac{1}{k^2 + 2}\\) in terms of elementary functions."
},
{
"role": "assistant",
"content": "To find a simple expression for the sum \\(\\sum_{k=1}^{\\infty} \\frac{1}{k^2 + 2}\\), we can use the known ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Into a box are put ten smaller boxes. Each of these smaller boxes is either left empty or filled with ten still smaller empty boxes. Of all the boxes, exactly six have other boxes inserted into them. How many boxes remain empty?"
},
{
"role": "assistant",
"content":... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the equation \\(x^2 + y^2 + z^2 = 1\\), find the maximum value of \\(x + 2y + 3z\\)."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If \\( f(x) \\) has a period of \\( \\frac{1}{2} \\) and \\( g(x) \\) has a period of \\( \\frac{2}{5} \\), what is the period of \\( f(x) + g(x) \\)?"
},
{
"role": "assistant",
"content": "To find the period of the function \\( f(x) + g(x) \\), where \\( f(x) \\) h... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Jonah is inside a whale at sea. At the dawn of each day, there is a 25% chance that he will write in his diary, and at noon, there is a 1/3 chance that the whale will swallow him. If he remains stuck indefinitely, what is the probability that Jonah will be swallowed without wri... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the differential equation \\( \\frac{dy}{dx} = -2x(y+3) \\) with the initial condition \\( y(0) = 1 \\)."
},
{
"role": "assistant",
"content": "To solve the differential equation \\(\\frac{dy}{dx} = -2x(y + 3)\\) with the initial condition \\(y(0) = 1\\), we f... | 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": "Compute $(4+8)^2+(4^2+8^2)$."
},
{
"role": "assistant",
"content": "To compute \\((4+8)^2 + (4^2 + 8^2)\\), we will break it down into manageable parts and solve each part step-by-step.\n\nFirst, we calculate \\((4+8)^2\\):\n\\[\n4 + 8 = 12\n\\]\n\\[\n12^2 = 12 \\ti... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Morgan walks \\( x \\) yards north, then walks \\( (2x-1) \\) yards west. She is now \\( (2x+1) \\) yards from her starting point. What is the distance, in yards, she travels?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by noting that Morga... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( A \\) be a set of \\( kn \\) elements where \\( k \\) and \\( n \\) are natural numbers. A \"k\" grouping of \\( A \\) is a partition of \\( A \\) into subsets, each containing \\( k \\) elements. Find the number of different \"k\" groupings of \\( A \\)."
},
{
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Stan drove 300 miles in 5 hours, 20 minutes. Next, he drove 360 miles in 6 hours, 40 minutes. What was Stan's average speed in miles per hour for the total trip?"
},
{
"role": "assistant",
"content": "To find Stan's average speed for the total trip, we need to calcu... | You are a helpful assistant. |
[
{
"role": "user",
"content": "For what value of \\( x \\) does \\( \\frac{3x}{2x} = 0 \\)?"
},
{
"role": "assistant",
"content": "To solve the equation \\(\\frac{3x}{2x} = 0\\), we start by analyzing the expression.\n\n1. **Identify the condition for the expression to be defined:**\n The expre... | 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": "How many integers satisfy the inequality $(x+3)^{2}\\leq1$?"
},
{
"role": "assistant",
"content": "To solve the inequality \\((x + 3)^2 \\leq 1\\), we start by rewriting it as a compound inequality:\n\n\\[\n-1 \\leq x + 3 \\leq 1\n\\]\n\nNext, we isolate \\(x\\) by ... | 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": "Find the maximum value of \\(|z|\\) if \\(\\text{Re}(z) = 4\\) and \\(\\text{Im}(z^3) = 0\\)."
},
{
"role": "assistant",
"content": "To find the maximum value of \\( |z| \\) given that \\(\\text{Re}(z) = 4\\) and \\(\\text{Im}(z^3) = 0\\), we start by expressing \\(... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the derivative of \\( y = \\frac{3t - 7}{t^2 + 5t - 4} \\) using the quotient rule."
},
{
"role": "assistant",
"content": "To find the derivative of \\( y = \\frac{3t - 7}{t^2 + 5t - 4} \\) using the quotient rule, follow these steps:\n\n1. **Identify the funct... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How do you evaluate the integral $\\int \\frac{1}{1-x} dx$ and show that it equals $-\\ln(1-x)$?"
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\frac{1}{1 - x} \\, dx\\), we use the substitution method. Let \\(u = 1 - x\\). Then, the diffe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yel... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( A \\) be a set with \\( n(A) = 2 \\). Denote \\( P^1(A) = P(A) \\) and \\( P^k(A) = P(P^{k-1}(A)) \\) for every natural number \\( k \\geq 1 \\). Determine the smallest \\( k \\) such that \\( n(P^k(A)) \\geq 65000 \\)."
},
{
"role": "assistant",
"content": ... | 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": "What is the domain of the function $g(x) = \\frac{3x+1}{x+8}$ ? Enter your answer using interval notation."
},
{
"role": "assistant",
"content": "To determine the domain of the function \\( g(x) = \\frac{3x+1}{x+8} \\), we need to identify the values of \\( x \\) f... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "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 number of triplets $(x, y, z)$ of positive integers that satisfy the equation $x + y + z = 2013$."
},
{
"role": "assistant",
"content": "To find the number of triplets \\((x, y, z)\\) of positive integers that satisfy the equation \\(x + y + z = 2013\\), we... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Suppose $a$ and $b$ are unknown but neither of them are equal to 2, and there are exactly 65 unattainable scores. Determine $a$ and $b$."
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the values of \\(a\\) and \\(b\\) such that the... | 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": "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": "If 25,197,624 hot dogs are packaged in sets of 4, how many will be left over?"
},
{
"role": "assistant",
"content": "To determine how many hot dogs will be left over if 25,197,624 hot dogs are packaged in sets of 4, we need to find the remainder when 25,197,624 is d... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the first perfect square? Is it 0 or 1?"
},
{
"role": "assistant",
"content": "To determine the first perfect square, we need to recall the definition of a perfect square. A perfect square is an integer that can be expressed as the square of another integer.... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine whether \\( H = A_4 \\) is a normal subgroup of \\( G = S_4 \\). If it is, write out a Cayley table for the factor group \\( G/H \\)."
},
{
"role": "assistant",
"content": "To determine whether \\( H = A_4 \\) is a normal subgroup of \\( G = S_4 \\) and to... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A 3-dimensional cube contains \"cubes\" of smaller dimensions, i.e., faces (2-cubes), edges (1-cubes), and vertices (0-cubes). How many 3-cubes are in a 5-cube?"
},
{
"role": "assistant",
"content": "To determine the number of 3-dimensional cubes (3-cubes) in a 5-di... | You are a helpful assistant. |
[
{
"role": "user",
"content": "There are 4 representatives from each of 4 companies at a convention. At the start of the convention, every person shakes hands once with every person except the other representatives from their company. How many handshakes are there?"
},
{
"role": "assistant",
"con... | 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": "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": "Given that \\(a + b = 17\\) and \\(a, b \\in \\mathbb{Z^+}\\), find the minimum value of \\(a\\) and \\(b\\)."
},
{
"role": "assistant",
"content": "Given that \\(a + b = 17\\) and \\(a, b \\in \\mathbb{Z^+}\\), we need to find the minimum value of \\(a\\) and \\(b\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "A right cylinder with a height of 5 inches has a radius of 2 inches. What is the area of the curved surface of the cylinder, in square inches? Express your answer in terms of $\\pi$."
},
{
"role": "assistant",
"content": "To find the area of the curved surface of a ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "How many paths are there from $(0,0)$ to $(4,5)$ on a grid that do not pass through the point $(2,2)$?"
},
{
"role": "assistant",
"content": "To find the number of paths from \\((0,0)\\) to \\((4,5)\\) that do not pass through the point \\((2,2)\\), we can break the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a club with 25 members, how many ways can a president, secretary, and treasurer be chosen if each member can hold at most one office?"
},
{
"role": "assistant",
"content": "To solve the problem of choosing a president, secretary, and treasurer from a club of 25 m... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In the expansion of $(x^2 + y - z^3)^8$, find the term that involves $x^{10} z^6$ using the binomial theorem approach."
},
{
"role": "assistant",
"content": "To find the term in the expansion of \\((x^2 + y - z^3)^8\\) that involves \\(x^{10} z^6\\), we use the mult... | You are a helpful assistant. |
[
{
"role": "user",
"content": "My school's math club has 6 boys and 8 girls. I need to select a team to send to the state math competition. We want 6 people on the team. In how many ways can I select the team to have 3 boys and 3 girls?"
},
{
"role": "assistant",
"content": "To determine the n... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute $\\cos 135^\\circ$."
},
{
"role": "assistant",
"content": "To compute \\(\\cos 135^\\circ\\), we can use the properties of the unit circle and the reference angle.\n\n1. **Identify the quadrant**: The angle \\(135^\\circ\\) is in the second quadrant because ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the derivative of \\(3 \\cos x\\)."
},
{
"role": "assistant",
"content": "To find the derivative of \\(3 \\cos x\\), we use the constant multiple rule and the known derivative of the cosine function. The constant multiple rule states that the derivative of a co... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Using the balance-scale shown, determine how many ♦ are necessary to replace the ? in the last picture. The given balance equations are:\n1. ▼♦ = ○\n2. ▼ = ♦◘\n3. ○○ = ◘◘◘\n4. ▼ = ?\n\nExplain your reasoning."
},
{
"role": "assistant",
"content": "To determine how m... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A pile of gold and silver coins contains more silver coins than gold, but the total weight of each type is the same. When enough silver coins are removed to make the number of silver and gold coins equal, the removed silver coins weigh one-third of the weight of the remaining g... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Integrate the function \\( 4x(x^2 + 1)e^{x^2 + 1} \\)."
},
{
"role": "assistant",
"content": "To integrate the function \\( 4x(x^2 + 1)e^{x^2 + 1} \\), we will use the method of substitution followed by integration by parts.\n\n1. **Substitution:**\n Let \\( u = x... | You are a helpful assistant. |
[
{
"role": "user",
"content": "On Monday, John receives $100$ dollars from his father. Over the next week, everyday there is a $50\\%$ chance he will receive a gift of $10$ dollars from a magical Shamu. What is the probability that at the end of the week, John will have exactly $130$ dollars?"
},
{
"... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Suppose there are $5$ red points and $4$ blue points on a circle. Let $\\frac{m}{n}$ be the probability that a convex polygon whose vertices are among the $9$ points has at least one blue vertex, where $m$ and $n$ are relatively prime. Find $m + n$."
},
{
"role": "assis... | You are a helpful assistant. |
[
{
"role": "user",
"content": "One day Max says to Liz, \"Out of the 25 people taking either English or French, you and I are the only two taking both.'' Liz, being mathematically inclined, responds by pointing out that there are exactly twice as many people in the English class as there are in the French cl... | 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": "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": "If $\\sin x = 3 \\cos x,$ then what is $\\sin x \\cos x$?"
},
{
"role": "assistant",
"content": "Given the equation \\(\\sin x = 3 \\cos x\\), we need to find the value of \\(\\sin x \\cos x\\).\n\nFirst, we use the Pythagorean identity \\(\\sin^2 x + \\cos^2 x = 1\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given 5 towns, what is the minimum amount of road needed to connect each town to every other town?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the minimum number of roads required to connect 5 towns such that each town is connected to ever... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Write down a definite integral that will give the value of the area under the curve \\( y = x^2 \\cos\\left(\\frac{1}{5}x\\right) \\) between \\( x = \\frac{1}{2}\\pi \\) and \\( x = \\frac{5}{2}\\pi \\). (The expression \\( x^2 \\cos\\left(\\frac{1}{5}x\\right) \\) takes no ne... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "Of 5 employees, 3 are to be assigned an office and 2 are to be assigned a cubicle. If 3 of the employees are men and 2 are women, what is the probability that the offices will be assigned to 2 of the men and 1 of the women?"
},
{
"role": "assistant",
"content": "To ... | 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": "The population of Central Java is $25\\%$ of the population of the island of Java and $15\\%$ of the population of Indonesia. What percentage of Indonesia's population lives outside Java?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the demand function for boneless chicken breast \\( q = -0.5p + 6 \\), where \\( p \\) is the price per pound and \\( q \\) is the average number of pounds purchased per week per customer, determine the price at which the demand is unit elastic."
},
{
"role": "ass... | 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": "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": "If $2^8=16^x$, find $x$."
},
{
"role": "assistant",
"content": "To solve the equation \\(2^8 = 16^x\\) for \\(x\\), we start by expressing 16 as a power of 2. We know that \\(16 = 2^4\\), so we can rewrite the equation as:\n\n\\[\n2^8 = (2^4)^x\n\\]\n\nNext, we use ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A circle has a radius of \\( \\log_{10}(a^2) \\) and a circumference of \\( \\log_{10}(b^4) \\). What is \\( \\log_ab \\)?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by using the given information about the circle's radius and circumferenc... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the sum of the infinite terms of the series \\( 1 + \\frac{1}{1+2} + \\frac{1}{1+2+3} + \\cdots \\)."
},
{
"role": "assistant",
"content": "To find the sum of the infinite series \\(1 + \\frac{1}{1+2} + \\frac{1}{1+2+3} + \\cdots\\), we start by identifying the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Calculate the limit \\( \\lim_{h \\rightarrow 0} \\frac{(x+h)^{x+h} - x^x}{h} \\)."
},
{
"role": "assistant",
"content": "To calculate the limit \\(\\lim_{h \\rightarrow 0} \\frac{(x+h)^{x+h} - x^x}{h}\\), we recognize that this expression represents the derivative ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute $10^{-1}\\pmod{1001}$. Express your answer as a residue from $0$ to $1000$, inclusive."
},
{
"role": "assistant",
"content": "To compute \\(10^{-1} \\pmod{1001}\\), we need to find an integer \\(x\\) such that \\(10x \\equiv 1 \\pmod{1001}\\). This means we ... | You are a helpful assistant. |
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