conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "A person crosses a 600 m long street in 2 minutes. What is his speed in km per hour?"
},
{
"from": "gpt",
"value": "To determine the speed of the person in kilometers per hour, we need to follow these steps:\n\n1. **Convert the distance from meters to kilometers:**\n... |
[
{
"from": "human",
"value": "What is the slope of the line containing the midpoint of the segment with endpoints at (2,3) and (4,5) and the midpoint of the segment with endpoints at (7,3) and (8,7)? Express your answer in simplest form."
},
{
"from": "gpt",
"value": "To find the slope of the lin... |
[
{
"from": "human",
"value": "Ian spent half the money he made on doing online surveys. If he worked 8 hours doing surveys and on average he's able to earn $18 per hour doing surveys, how much money does he have left?"
},
{
"from": "gpt",
"value": "To determine how much money Ian has left, we nee... |
[
{
"from": "human",
"value": "Given the function $f(x) = \\begin{cases} x+4, & x\\leqslant 0 \\\\ 2^{x}, & x > 0 \\end{cases}$, find the solution set for the inequality $f(x)\\leqslant 2$."
},
{
"from": "gpt",
"value": "To solve the inequality \\( f(x) \\leqslant 2 \\) for the piecewise function ... |
[
{
"from": "human",
"value": "What is the area, in square units, of a triangle with sides measuring $5$, $5$, and $6$ units? Express your answer in simplest radical form."
},
{
"from": "gpt",
"value": "To find the area of a triangle with sides measuring \\(5\\), \\(5\\), and \\(6\\) units, we can... |
[
{
"from": "human",
"value": "As shown in the figure, it is known that the area of triangle $S_{\\triangle B E C}$ is 30 square centimeters. Point $E$ is the quarter point on side $A B$ near point $B$. What is the area of parallelogram $D E C F$ in square centimeters?"
},
{
"from": "gpt",
"value"... |
[
{
"from": "human",
"value": "2 persons can finish a job in 8 days . first person alone can finish the work in 24 days . how many days does the second person take to finish the job ?"
},
{
"from": "gpt",
"value": "To determine how many days the second person takes to finish the job, we can follow... |
[
{
"from": "human",
"value": "The function $g$ is defined on positive integers as follows:\n\\[g(n) = \\left\\{\n\\begin{array}{cl}\nn + 8 & \\text{if $n < 7$}, \\\\\ng(n - 3) & \\text{if $n \\ge 7$}.\n\\end{array}\n\\right.\\]\nFind the maximum value of the function."
},
{
"from": "gpt",
"value"... |
[
{
"from": "human",
"value": "A rectangular floor that is $12$ feet wide and $20$ feet long is tiled with $240$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?\n$\\textbf{(A) } 27 \\qquad\\... |
[
{
"from": "human",
"value": "For a natural number $n>1$ , consider the $n-1$ points on the unit circle $e^{\\frac{2\\pi ik}{n}}\\ (k=1,2,...,n-1) $ . Show that the product of the distances of these points from $1$ is $n$ ."
},
{
"from": "gpt",
"value": "To find the product of the distan... |
[
{
"from": "human",
"value": "In the seven-digit integer $4B6,\\!825,\\!1$, what is the largest digit $B$ so that the seven-digit integer will be divisible by 3?"
},
{
"from": "gpt",
"value": "To determine the largest digit \\( B \\) such that the seven-digit integer \\( 4B6,825,1 \\) is divisibl... |
[
{
"from": "human",
"value": "A can complete a project in 20 days and B can complete the same project in some days. If A and B start working on the project together and A quits 5 days before the project is completed, the project will be completed in 15 days. In how many days can B complete the project alone?... |
[
{
"from": "human",
"value": "The two-digit number $\\overline{ab}$, written in the decimal system, when squared results in a three-digit number: $\\overline{ab^{2}} = \\overline{cde}$.\n\nReversing the digits of the original two-digit number and then squaring it also results in a three-digit number, and thi... |
[
{
"from": "human",
"value": "Given an ellipse $C_1$ with its center at the origin, whose foci are the vertices of the hyperbola $C_{2}: \\frac{x^2}{2} - y^2 = 1$. The line $x + \\sqrt{2}y = 0$ intersects the ellipse $C_1$ at points $A$ and $B$, with the coordinates of point $A$ being $(-\\sqrt{2}, 1)$. Let ... |
[
{
"from": "human",
"value": "A certain number of bricks, each measuring 25 cm x 11 cm x 6 cm, are needed to build a wall of 2 m x 3 m x some thickness. If 72.72727272727273 bricks are needed, what is the thickness of the wall in centimeters?"
},
{
"from": "gpt",
"value": "To determine the thickn... |
[
{
"from": "human",
"value": "On the sides $AB, BC,$ and $AC$ of triangle $ABC$, points $C_1, A_1,$ and $B_1$ are chosen arbitrarily, respectively. Let $K_1, K_2,$ and $K_3$ be the midpoints of $AA_1, BB_1,$ and $CC_1$. Prove that these points cannot lie on a single line."
},
{
"from": "gpt",
"va... |
[
{
"from": "human",
"value": "\nObserve the following equations:\n$$\n\\begin{array}{l}\n\\mathrm{C}_{5}^{1}+\\mathrm{C}_{5}^{5}=2^{3}-2, \\\\\n\\mathrm{C}_{9}^{1}+\\mathrm{C}_{9}^{5}+\\mathrm{C}_{9}^{9}=2^{7}+2^{3}, \\\\\n\\mathrm{C}_{13}^{1}+\\mathrm{C}_{13}^{5}+\\mathrm{C}_{13}^{9}+\\mathrm{C}_{13}^{13}=2... |
[
{
"from": "human",
"value": "A rectangular box with a square base of side length $s$ and a height of $2s$ is centrally placed on a square sheet of wrapping paper such that each vertex of the box's base lies on the midlines of the wrapping paper. The wrapping paper is folded along the edges of the box such t... |
[
{
"from": "human",
"value": "Paul had 108 books and 55 pens. After selling some books and pens in a garage sale he had 66 books and 59 pens left. How many books did he sell in the garage sale?"
},
{
"from": "gpt",
"value": "To determine how many books Paul sold in the garage sale, we need to fin... |
[
{
"from": "human",
"value": "Evaluate the product \\( (2.5)(50.5 + 0.15) \\).\nA) 120.5\nB) 126.6\nC) 130.0\nD) 135.5\nE) 126.625"
},
{
"from": "gpt",
"value": "To evaluate the product \\( (2.5)(50.5 + 0.15) \\), we will follow these steps:\n\n1. First, simplify the expression inside the parenth... |
[
{
"from": "human",
"value": "A regular polygon has interior angles of 150 degrees. Find out how many sides it has if its exterior angles are in the ratio of 2:1 with another polygon's exterior angles."
},
{
"from": "gpt",
"value": "To determine the number of sides of the regular polygon with int... |
[
{
"from": "human",
"value": "Given: An isosceles triangle has two sides of lengths 2 and 4, respectively. What is the perimeter of the isosceles triangle ( )?\nA: 6\nB: 8\nC: 10\nD: 8 or 10"
},
{
"from": "gpt",
"value": "To determine the perimeter of the isosceles triangle, we need to consider... |
[
{
"from": "human",
"value": "How many ways are there to remove an $11\\times11$ square from a $2011\\times2011$ square so that the remaining part can be tiled with dominoes ( $1\\times 2$ rectangles)?"
},
{
"from": "gpt",
"value": "To determine how many ways there are to remove an \\(11 \\t... |
[
{
"from": "human",
"value": "To pay for a book that costs 20 yuan using 10 yuan, 5 yuan, and 1 yuan notes, the number of different payment methods is ( ) \nA: 3 \nB: 5 \nC: 9 \nD: 12"
},
{
"from": "gpt",
"value": "To determine the number of different payment methods to pay for a book that c... |
[
{
"from": "human",
"value": "A club has 12 members, 4 boys and 8 girls. Two of the members are chosen at random. What is the probability that they are both girls?"
},
{
"from": "gpt",
"value": "To determine the probability that two randomly chosen members from a club of 12 members (4 boys and 8 ... |
[
{
"from": "human",
"value": "Improving the vehicle passage capacity of a mountain tunnel can effectively alleviate traffic conditions. Generally, the vehicle flow speed $v$ (unit: kilometers per hour) in the tunnel is a function of the vehicle flow density $x$ (unit: vehicles per kilometer, representing the... |
[
{
"from": "human",
"value": "A computer manufacturer produces a certain electronic component at a cost of $80 per component. Shipping costs for delivering the components are $3 per unit. Further, the manufacturer has costs of $16,500 a month related to the electronic component regardless of how many it prod... |
[
{
"from": "human",
"value": "Let $\\mathbf{v} = \\begin{pmatrix} 2 \\\\ 1 \\\\ -1 \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 3 \\end{pmatrix}.$ The columns of a matrix are $\\mathbf{u},$ $\\mathbf{v},$ and $\\mathbf{w},$ where $\\mathbf{u}$ is a unit vector. Find the largest possibl... |
[
{
"from": "human",
"value": "Given that \\( -5 \\leq x \\leq -3 \\) and \\( 3 \\leq y \\leq 6 \\), what is the largest possible value of \\( \\frac{x+y}{x} \\)?"
},
{
"from": "gpt",
"value": "To find the largest possible value of the expression \\(\\frac{x+y}{x}\\) given the constraints \\(-5 \\... |
[
{
"from": "human",
"value": "The Diving Club offers 2 beginning diving classes on weekdays and 4 beginning classes on each day of the weekend. Each class has room for a certain number of people. In 3 weeks, 270 people can take classes. How many people can each class accommodate?"
},
{
"from": "gpt",... |
[
{
"from": "human",
"value": "The standard equation of circle C, whose center is on the x-axis and which is tangent to the line $l: y = 2x + 1$ at point P(0, 1), is \\_\\_\\_\\_\\_\\_."
},
{
"from": "gpt",
"value": "To find the standard equation of the circle \\( C \\) whose center is on the x-ax... |
[
{
"from": "human",
"value": "As shown in Figure 2.4.9, chords $AB$ and $CD$ of circle $O$ intersect at $K$. Tangents to the circle are drawn from the ends of each chord, intersecting at points $P$ and $Q$. Prove that $OK \\perp PQ$."
},
{
"from": "gpt",
"value": "To prove that \\(OK \\perp PQ\\)... |
[
{
"from": "human",
"value": "The sum of the length of the real axis and the focal distance of the hyperbola $12x^{2}-4y^{2}=3$ is ( )\nA: $\\dfrac {3}{2}$\nB: $\\dfrac {5}{2}$\nC: $3$\nD: $2\\sqrt {3}$"
},
{
"from": "gpt",
"value": "To solve the problem, we need to determine the length of the r... |
[
{
"from": "human",
"value": "Point \\( P \\) on a plane is equidistant from the four vertices of a convex quadrilateral. Where is point \\( P \\) located?\nA. Inside the quadrilateral\nB. On the boundary of the quadrilateral\nC. Outside the quadrilateral\nD. Any of the above three scenarios"
},
{
"f... |
[
{
"from": "human",
"value": "Paul is playing a game with $N$ marbles and a bag, starting with one pile of marbles. In the first round, Paul removes one marble from the pile, puts it in his bag, and splits the remaining marbles into two non-empty piles. Generally, in each subsequent round, Paul chooses a pil... |
[
{
"from": "human",
"value": "Two of the altitudes of an acute triangle divide the sides into segments of lengths $7,4,3$, and $y$ units, respectively. Determine the value of $y$."
},
{
"from": "gpt",
"value": "Given an acute triangle with altitudes dividing the sides into segments of lengths \\(... |
[
{
"from": "human",
"value": "Given the line $l: x - y + 3 = 0$ and the circle $C: (x - a)^2 + (y - 2)^2 = 4$, the length of the chord intercepted by the circle on line $l$ is $2\\sqrt{2}$. Determine the value of $a$."
},
{
"from": "gpt",
"value": "To determine the value of \\(a\\) for which the ... |
[
{
"from": "human",
"value": "James takes a spinning class 3 times a week. He works out for 1.5 hours each class and burns 7 calories per minute. How many calories does he burn per week?"
},
{
"from": "gpt",
"value": "To determine how many calories James burns per week from his spinning classes, ... |
[
{
"from": "human",
"value": "In the humanities sequence, let $S_n$ be the sum of the first $n$ terms of a sequence $\\{a_n\\}$. If the sequence $\\{a_n\\}$ satisfies $a_1=1$ and $S_{n+1} = \\frac {(n+1)a_n}{n} + S_{n}$, then find the sum of the first $n$ terms of the sequence $\\{a_n\\}$, $S_n=$ ______."
... |
[
{
"from": "human",
"value": "Given the sequence \\(\\{a_n\\}\\) with the first term 2, and it satisfies\n\\[ 6 S_n = 3 a_{n+1} + 4^n - 1. \\]\n\nFind the maximum value of \\(S_n\\)."
},
{
"from": "gpt",
"value": "Given the sequence \\(\\{a_n\\}\\) with the first term \\(a_1 = 2\\) and the recurr... |
[
{
"from": "human",
"value": "Evaluate the expression $2a^2 - 3b^2 + ab$ where $a = 4$ and $b = -2$."
},
{
"from": "gpt",
"value": "To evaluate the expression \\(2a^2 - 3b^2 + ab\\) where \\(a = 4\\) and \\(b = -2\\), we will substitute the given values of \\(a\\) and \\(b\\) into the expression ... |
[
{
"from": "human",
"value": "Compute \\( (1 - i)^4 \\)."
},
{
"from": "gpt",
"value": "To compute \\((1 - i)^4\\), we can use the binomial theorem or simplify step by step. Here, we will simplify step by step.\n\nFirst, let's find \\((1 - i)^2\\):\n\\[\n(1 - i)^2 = (1 - i)(1 - i) = 1 \\cdot 1 + ... |
[
{
"from": "human",
"value": "Given a point $P$ on the sine curve $y=\\sin x$, the tangent line at point $P$ is line $l$. Then, the range of the inclination angle of line $l$ is $(\\quad)$ \nA: $\\left[0, \\dfrac {\\pi}{4}\\right]\\cup\\left[ \\dfrac {3\\pi}{4},\\pi\\right)$ \nB: $\\left[0,\\pi\\right)$ \... |
[
{
"from": "human",
"value": "Roll a die twice, observe the points shown each time, and let the point number of the first roll be $m$ and that of the second roll be $n$. Considering vectors $\\overrightarrow{p} = (m, n)$ and $\\overrightarrow{q} = (3, 6)$, find the probability that vectors $\\overrightarrow{... |
[
{
"from": "human",
"value": "Draw a pentagon and connect all of its vertices with each other as shown in the figure. How many triangles are there in the resulting figure?"
},
{
"from": "gpt",
"value": "To determine the number of triangles formed by connecting all the vertices of a pentagon, we c... |
[
{
"from": "human",
"value": "Given that one of the asymptotes of the hyperbola $\\dfrac{x^2}{a^2} - \\dfrac{y^2}{2} = 1$ passes through the point $(\\sqrt{2}, 1)$, then the coordinates of one of the foci of this hyperbola are $(\\quad)$.\nA: $(\\sqrt{2}, 0)$\nB: $(2, 0)$\nC: $(\\sqrt{6}, 0)$\nD: $(\\sqrt{10... |
[
{
"from": "human",
"value": "The $\\$6.80$ in Jason's wallet consists of dimes and nickels. There are ten more nickels than dimes. How many nickels does Jason have?"
},
{
"from": "gpt",
"value": "To determine how many nickels Jason has, we start by defining variables for the number of dimes and ... |
[
{
"from": "human",
"value": "Given $a$, $b$, $c$, $d \\in \\mathbb{R}$ and satisfying $\\frac{a+3\\ln a}{b} = \\frac{d-3}{2c} = 1$, the minimum value of $(a-c)^{2}+(b-d)^{2}$ is \\_\\_\\_\\_\\_\\_."
},
{
"from": "gpt",
"value": "Given the equations \\(\\frac{a+3\\ln a}{b} = 1\\) and \\(\\frac{d-... |
[
{
"from": "human",
"value": "Three tour guides are leading eight tourists. The guides decide to split up. Each tourist must choose one of the guides, with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?"
},
{
"from": "... |
[
{
"from": "human",
"value": "During a chess tournament, Audrey and Thomas played 5 games against each other. In each game, some pieces were lost. The number of pieces that Audrey lost in each game were 6, 8, 4, 7, and 10, while the number of pieces that Thomas lost in each game were 5, 6, 3, 7, and 11. Find... |
[
{
"from": "human",
"value": "Evaluate \\[\\frac{4}{\\log_5{2500^3}} + \\frac{2}{\\log_2{2500^3}},\\] giving your answer as a fraction in lowest terms."
},
{
"from": "gpt",
"value": "To evaluate the expression \\(\\frac{4}{\\log_5{2500^3}} + \\frac{2}{\\log_2{2500^3}}\\), we start by using the ch... |
[
{
"from": "human",
"value": "In a certain town, there is a ratio of NY Yankees fans to NY Mets fans. The ratio of NY Mets fans to Boston Red Sox fans is 4 : 5. There are 330 baseball fans in the town, each of whom is a fan of exactly one of those three teams. There are 88 NY Mets fans in this town. What is ... |
[
{
"from": "human",
"value": "Let $r$ be the speed in miles per hour at which a wheel, $13$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\\frac{1}{4}$ of a second, the speed $r$ is increased by $3$ miles per hour. Then $r$ is:\n$\\text{(A) } 11 \\quad \\te... |
[
{
"from": "human",
"value": "Given the complex numbers \\( z_1 \\) and \\( z_2 \\) that satisfy \\(\\left|z_1\\right| = \\left|z_1 - 2 z_2\\right|\\) and \\( z_1 \\overline{z_2} = \\sqrt{3}(1 - \\mathrm{i}) \\), where \\(\\mathrm{i}\\) is the imaginary unit, find \\(\\frac{z_1}{z_2}\\)."
},
{
"from"... |
[
{
"from": "human",
"value": "Marie now does four equally time-consuming tasks consecutively without breaks. She starts the first task at 1:00 PM and finishes the third task at 3:50 PM. When does she complete the fourth task?\n$\\textbf{(A)}\\; \\text{4:30 PM} \\qquad\\textbf{(B)}\\; \\text{4:40 PM} \\qquad\... |
[
{
"from": "human",
"value": "If the ratio of two numbers is unknown and the LCM of the numbers is 180, then the numbers are 45 and 60. What is the ratio of the two numbers?"
},
{
"from": "gpt",
"value": "To find the ratio of the two numbers, we start by noting that the numbers given are 45 and 6... |
[
{
"from": "human",
"value": "Adam and Simon start on bicycle trips from the same point at the same time. Adam travels north at 10mph and Simon travels west at 8mph. After how many hours are they 50 miles apart?"
},
{
"from": "gpt",
"value": "To determine after how many hours Adam and Simon are 5... |
[
{
"from": "human",
"value": "a 300 m long train crosses a platform in 27 sec while it crosses a signal pole in 18 sec . what is the length of the platform ?"
},
{
"from": "gpt",
"value": "To determine the length of the platform, we need to follow these steps:\n\n1. **Calculate the speed of the t... |
[
{
"from": "human",
"value": "a sum of rs . 2769 is lent into two parts so that the interest on the first part for 8 years at 3 % per annum may be equal to the interest on the second part for 3 years at 5 % per annum . find the second sum ?"
},
{
"from": "gpt",
"value": "Let's denote the first pa... |
[
{
"from": "human",
"value": "Solve\n\\[\\frac{x-1}{x+2} \\ge 0.\\]"
},
{
"from": "gpt",
"value": "To solve the inequality \\(\\frac{x-1}{x+2} \\ge 0\\), we need to determine the values of \\(x\\) for which the expression \\(\\frac{x-1}{x+2}\\) is non-negative (i.e., greater than or equal to zero... |
[
{
"from": "human",
"value": "How many pairs \\((a,b)\\) of non-zero real numbers satisfy the equation\n\\[\n\\frac{1}{a} + \\frac{1}{b} = \\frac{1}{2a + 3b}\n\\]\nA) none \nB) 1 pair \nC) 2 pairs \nD) One pair for each \\( b \\neq 0 \\) \nE) Two pairs for each \\( b \\neq 0 \\)"
},
{
"from": "gp... |
[
{
"from": "human",
"value": "Calculate the product of the trinomial $(2x^2 + 3x + 1)$ and the binomial $(x - 4)$.\nA) $2x^3 - 5x^2 - 11x - 4$\nB) $2x^3 - 5x^2 + 11x + 4$\nC) $2x^3 - 8x^2 + 10x - 4$\nD) $x^3 - 5x^2 - 11x - 4$"
},
{
"from": "gpt",
"value": "To calculate the product of the trinomia... |
[
{
"from": "human",
"value": "\n1) Prove that for all real numbers \\( a, b, c > 0 \\), we have\n\n\\[ a(a-b)(a-c) + b(b-c)(b-a) + c(c-a)(c-b) \\geqslant 0 \\]\n\n2) Prove that for any real number \\( t \\) and all real numbers \\( a, b, c > 0 \\), we have\n\n\\[ a^{t}(a-b)(a-c) + b^{t}(b-c)(b-a) + c^{t}(c-a... |
[
{
"from": "human",
"value": "Arrange the five subjects: Chinese, Mathematics, English, Physics, and Chemistry into five class periods within a day. If Mathematics must be scheduled before Chemistry, then the number of different arrangements is ( )\nA: 48\nB: 24\nC: 60\nD: 120"
},
{
"from": "gpt"... |
[
{
"from": "human",
"value": "A circle is tangent to the lines $3x - 4y = 40$ and $3x - 4y = -20.$ The center of the circle lies on the line $x - 2y = 0.$ Find the center of the circle."
},
{
"from": "gpt",
"value": "To find the center of the circle, we start by noting that the circle is tangent ... |
[
{
"from": "human",
"value": "Given the circle $M$: $x^2 - 4x + y^2 = 0$, find the radius $r$ of the circle $M$."
},
{
"from": "gpt",
"value": "To find the radius of the circle given by the equation \\(x^2 - 4x + y^2 = 0\\), we need to rewrite the equation in the standard form of a circle's equat... |
[
{
"from": "human",
"value": "Let \\\\(a\\\\) and \\\\(b\\\\) be in \\\\((0, +\\infty)\\\\) and \\\\(a \\neq b\\\\). Prove that \\\\(a^{3}+b^{3} > a^{2}b+ab^{2}\\\\)."
},
{
"from": "gpt",
"value": "To prove that \\(a^3 + b^3 > a^2b + ab^2\\) for \\(a, b \\in (0, +\\infty)\\) and \\(a \\neq b\\), ... |
[
{
"from": "human",
"value": "The inverse function of $y=3^{x+1}$ ($x\\in \\mathbb{R}$) is."
},
{
"from": "gpt",
"value": "To find the inverse function of \\( y = 3^{x+1} \\), we need to express \\( x \\) in terms of \\( y \\). Here are the steps:\n\n1. Start with the given function:\n \\[\n ... |
[
{
"from": "human",
"value": "Convert \\(435_{10}\\) to base 3."
},
{
"from": "gpt",
"value": "To convert the decimal number \\(435_{10}\\) to base 3, we need to repeatedly divide the number by 3 and keep track of the remainders. Here are the steps:\n\n1. **Divide 435 by 3:**\n \\[\n 435 \\di... |
[
{
"from": "human",
"value": "Let $A = (0,6)$ and $B = (0,10)$. Points $A'$ and $B'$ are on the line $y = x$, and $\\overline{AA'}$ and $\\overline{BB'}$ intersect at $C=(3,6)$. What is the length of $\\overline{A'B'}$?"
},
{
"from": "gpt",
"value": "To find the length of \\(\\overline{A'B'}\\), ... |
[
{
"from": "human",
"value": "If 1/5th of a number decreased by 5 is 5, what is the number?"
},
{
"from": "gpt",
"value": "Let's denote the unknown number by \\( x \\).\n\nAccording to the problem, 1/5th of the number decreased by 5 is equal to 5. We can write this relationship as an equation:\n\... |
[
{
"from": "human",
"value": "I have $n$ books, and $k$ of them are my favorites. I place the books randomly on a shelf (so that any order of the $n$ books is equally likely). For which values of $k$ will the probability that the $k$ favorite books end up next to each other be maximized and minimized?"
},
... |
[
{
"from": "human",
"value": "What is the 150th letter in the pattern ABCDABCDABCD...?"
},
{
"from": "gpt",
"value": "To determine the 150th letter in the repeating pattern \"ABCDABCDABCD...\", we first need to identify the length of the repeating sequence. The sequence \"ABCD\" consists of 4 let... |
[
{
"from": "human",
"value": "Given a regression equation $y = 3 - 5x$, when the variable $x$ increases by one unit,\n\nA. $y$ decreases by an average of 5 units. \nB. $y$ increases by an average of 3 units. \nC. $y$ decreases by an average of 3 units. \nD. $y$ increases by an average of 5 units."
},
... |
[
{
"from": "human",
"value": "Paul had 51 books and 106 pens. After selling some books and pens in a garage sale, he had 82 books and some pens left. He sold 92 pens in the garage sale. How many pens did Paul have left after the garage sale?"
},
{
"from": "gpt",
"value": "To determine how many pe... |
[
{
"from": "human",
"value": "What is the total volume in cubic meters of four boxes if each box is a cube with edge length 5 meters?"
},
{
"from": "gpt",
"value": "To determine the total volume of four boxes, each of which is a cube with an edge length of 5 meters, we need to follow these steps:... |
[
{
"from": "human",
"value": "Investigate the absolute convergence of the improper multiple integrals:\n\na) \\(\\iint_{\\Omega} \\frac{\\varphi(x, y)}{\\left(x^{2}+y^{2}\\right)^{p}} \\, dx \\, dy\\), where \\(\\Omega: x^{2}+y^{2} \\leqslant 1\\) and \\(\\Omega: x^{2}+y^{2} \\geqslant 1\\); \\(\\varphi(x, y... |
[
{
"from": "human",
"value": "when the no . 7 y 86038 is exactly divisible by 11 , then the smallest whole no . in place of y ?"
},
{
"from": "gpt",
"value": "To determine the smallest whole number \\( y \\) such that the number \\( 7y86038 \\) is exactly divisible by 11, we need to use the rule ... |
[
{
"from": "human",
"value": "Coach Grunt is preparing the 6-person starting lineup for his basketball team, the Grunters. There are 15 players on the team. Two of them, Ace and Zeppo, are league All-Stars and will definitely be in the starting lineup. How many different starting lineups are possible?"
},
... |
[
{
"from": "human",
"value": "If two numbers are randomly chosen without replacement from $\\{2, 3, 6, 9\\}$, what is the probability that their product will be a multiple of 12? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "To determine the probability that the product... |
[
{
"from": "human",
"value": "Given that $a$ and $b$ are real numbers, and $a \\neq b$, $a < 0$, then $a$ \\_\\_\\_\\_\\_\\_ $2b- \\frac {b^{2}}{a}$. (Fill in \"$>$\", \"$<$\", or \"$=$\")"
},
{
"from": "gpt",
"value": "To determine the relationship between \\(a\\) and \\(2b - \\frac{b^2}{a}\\), ... |
[
{
"from": "human",
"value": "Suppose \\( x \\) varies inversely with \\( y^3 \\). If \\( x = 8 \\) when \\( y = 1 \\), find the value of \\( x \\) when \\( y = 2 \\)."
},
{
"from": "gpt",
"value": "Given that \\( x \\) varies inversely with \\( y^3 \\), we can express this relationship mathemati... |
[
{
"from": "human",
"value": "You have exactly $4.40 in quarters (25¢ coins), dimes (10¢ coins), and nickels (5¢ coins). You have the same number of each type of coin. How many dimes do you have?\n(A) 20\n(B) 11\n(C) 10\n(D) 12\n(E) 4"
},
{
"from": "gpt",
"value": "To determine how many dimes you... |
[
{
"from": "human",
"value": "Let \\(ABCD\\) be an isosceles trapezoid such that \\(AD = BC\\), \\(AB = 3\\), and \\(CD = 8\\). Let \\(E\\) be a point in the plane such that \\(BC = EC\\) and \\(AE \\perp EC\\). Compute \\(AE\\)."
},
{
"from": "gpt",
"value": "To solve for \\(AE\\) in the given i... |
[
{
"from": "human",
"value": "A list of $2023$ positive integers has a unique mode, which occurs exactly $15$ times. What is the least number of distinct values that can occur in the list?\nA. $144$\nB. $145$\nC. $146$\nD. $147$"
},
{
"from": "gpt",
"value": "To determine the least number of dist... |
[
{
"from": "human",
"value": "A man is 45 years older than his son. In 10 years, his age will be four times the age of his son. Additionally, the son's age will double in 15 years. What is the present age of the son?"
},
{
"from": "gpt",
"value": "Let's denote the present age of the son as \\( x ... |
[
{
"from": "human",
"value": "The domain of the function $f(x)= \\dfrac {\\sqrt {x+2}}{2x-1}$ is __________."
},
{
"from": "gpt",
"value": "To determine the domain of the function \\( f(x) = \\dfrac{\\sqrt{x+2}}{2x-1} \\), we need to identify the values of \\( x \\) for which the function is defi... |
[
{
"from": "human",
"value": "Please write down a rational number that is smaller than $-\\frac{1}{2}$: ______. (Write down one number)"
},
{
"from": "gpt",
"value": "To find a rational number that is smaller than \\(-\\frac{1}{2}\\), we need to identify a number that lies to the left of \\(-\\fr... |
[
{
"from": "human",
"value": "In a shop, every piece of candy is priced with a whole number of cents. Jerry can buy exactly $20$ pieces of red candy, $18$ pieces of green candy, $24$ pieces of blue candy, or $n$ pieces of orange candy. Each piece of orange candy costs $25$ cents. Determine the smallest possi... |
[
{
"from": "human",
"value": "How many distinguishable rearrangements of the letters in $CONTEST$ have both the vowels first? (For instance, $OETCNST$ is one such arrangement but $OTETSNC$ is not.)\n$\\mathrm{(A)\\ } 60 \\qquad \\mathrm{(B) \\ }120 \\qquad \\mathrm{(C) \\ } 240 \\qquad \\mathrm{(D) \\ } 72... |
[
{
"from": "human",
"value": "At the opening of a trading day at a certain stock exchange, the price per share of stock K was $15. If the price per share of stock K was $16 at the closing of the day, what was the percent increase in the price per share of stock K for that day?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "There are 280 doctors and nurses at a hospital. If the ratio of doctors to nurses is 5 to 9, how many nurses are at the hospital?"
},
{
"from": "gpt",
"value": "To determine the number of nurses at the hospital, we start by defining the variables based on the given r... |
[
{
"from": "human",
"value": "Given that \\( n \\) is a natural number less than 50, find all \\( n \\) such that \\( 3n + 5 \\) and \\( 5n + 4 \\) have a common divisor greater than 1."
},
{
"from": "gpt",
"value": "To find all natural numbers \\( n \\) less than 50 such that \\( 3n + 5 \\) and ... |
[
{
"from": "human",
"value": "The negation of the proposition \"For all $x \\in \\mathbb{R}, x^2 - 2x + 3 \\leq 0$\" is ( )\nA: For all $x \\in \\mathbb{R}, x^2 - 2x + 3 \\geq 0$\nB: There exists $x \\in \\mathbb{R}, x^2 - 2x + 3 > 0$\nC: For all $x \\in \\mathbb{R}, x^2 - 2x + 3 \\leq 0$\nD: There exists $x... |
[
{
"from": "human",
"value": "Prove that\n\na) \\(2^{41}+1\\) is divisible by 83;\n\nb) \\(2^{70}+3^{70}\\) is divisible by 13;\n\nc) \\(2^{60}-1\\) is divisible by 20801."
},
{
"from": "gpt",
"value": "To solve these problems, we will use properties of modular arithmetic and Fermat's Little Theo... |
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