conversations listlengths 2 2 |
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[
{
"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 $ 2 per unit . further , the manufacturer has costs of $ 16,200 a month related to the electronic component regardless of how many i... |
[
{
"from": "human",
"value": "In a school's soccer championship, each team must play one match against every other team. The winning team of each match gets 2 points, each team gets 1 point in the case of a tie, and the losing team gets 0 points. It is known that one team has the most points (each of the oth... |
[
{
"from": "human",
"value": "Given distinct points $A$, $B$, and $C$ on line $e$ in this order, find the locus of points $P$ such that the incircle of triangle $ACP$ touches line $e$ at point $B$."
},
{
"from": "gpt",
"value": "To find the locus of points \\( P \\) such that the incircle of tria... |
[
{
"from": "human",
"value": "Compare the sizes of the following numbers: $85_{(9)}$, $210_{(6)}$, $1000_{(4)}$, $111111_{(2)}$."
},
{
"from": "gpt",
"value": "To compare the sizes of the numbers \\(85_{(9)}\\), \\(210_{(6)}\\), \\(1000_{(4)}\\), and \\(111111_{(2)}\\), we need to convert each nu... |
[
{
"from": "human",
"value": "Verify that the function \\( z = x \\ln \\frac{y}{x} \\) satisfies the equation \\( x \\frac{\\partial z}{\\partial x} + y \\frac{\\partial z}{\\partial y} = z \\)."
},
{
"from": "gpt",
"value": "To verify that the function \\( z = x \\ln \\frac{y}{x} \\) satisfies t... |
[
{
"from": "human",
"value": "Given that $a$ is a real number, if $\\frac{a-i}{1+i}$ is a pure imaginary number, where $i$ is the imaginary unit, then $a=$\n\nA: $1$\n\nB: $-1$\n\nC: $\\sqrt{2}$\n\nD: $-\\sqrt{2}$"
},
{
"from": "gpt",
"value": "To determine the value of \\(a\\) such that \\(\\fra... |
[
{
"from": "human",
"value": "A rectangle was made with a 40 centimeters (cm) wire. If the length of one side is 8 centimeters (cm), what is the length of the other side in centimeters (cm)?"
},
{
"from": "gpt",
"value": "To determine the length of the other side of the rectangle, we start by rec... |
[
{
"from": "human",
"value": "Points $A, B, C,$ and $D$ are on a circle of diameter $1$. Point $X$ is on diameter $\\overline{AD}$. It is given that $BX = CX$ and $3\\angle{BAC} = \\angle{BXC} = 60^\\circ$. Determine the length $AX$ in terms of trigonometric functions with the options:\n$\\text{(A) } \\cos(1... |
[
{
"from": "human",
"value": "Ed booked a hotel while he was on vacation. Staying at the hotel cost was some amount per hour every night and $2 per hour every morning. Ed had $80 and he stayed in the hotel for 6 hours last night and 4 hours this morning. He was left with $63 after paying for his stay at the ... |
[
{
"from": "human",
"value": "The $y$-intercepts, $R$ and $S$, of two perpendicular lines intersecting at the point $B(8,6)$ have a product of -24. What is the area of triangle $BRS$?\n$\\textbf{(A)}\\ 20\\qquad\\textbf{(B)}\\ 40\\qquad\\textbf{(C)}\\ 50\\qquad\\textbf{(D)}\\ 60\\qquad\\textbf{(E)}\\ 72$"
... |
[
{
"from": "human",
"value": "Eight distinct pieces of candy are to be distributed among four bags: red, blue, white, and green. The red bag and the blue bag together must receive at least two pieces of candy, and the green bag must receive at least one piece of candy. The white bag may remain empty. How man... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be positive real numbers such that $a + b = 2.$ Find the set of all possible values of $\\frac{1}{a} + \\frac{1}{b}.$"
},
{
"from": "gpt",
"value": "Given that \\(a\\) and \\(b\\) are positive real numbers such that \\(a + b = 2\\), we need to find th... |
[
{
"from": "human",
"value": "Two circles are defined by the equations $x^{2}+y^{2}-6x+16y-48=0$ and $x^{2}+y^{2}+4x-8y-44=0$. Determine the number of common tangents for these circles.\n\nA: $4$ tangents\nB: $3$ tangents\nC: $2$ tangents\nD: $1$ tangent"
},
{
"from": "gpt",
"value": "To determin... |
[
{
"from": "human",
"value": "The monotonic increasing interval of the function $$y=\\log_{0.5}(x^{2}-4x+3)$$ is \\_\\_\\_\\_\\_\\_."
},
{
"from": "gpt",
"value": "To determine the monotonic increasing interval of the function \\( y = \\log_{0.5}(x^2 - 4x + 3) \\), we need to follow these steps:\... |
[
{
"from": "human",
"value": "Let \\( X = \\{1, 2, 3, \\ldots, 17\\} \\). Find the number of subsets \\( Y \\) of \\( X \\) with odd cardinalities."
},
{
"from": "gpt",
"value": "To determine the number of subsets \\( Y \\) of the set \\( X = \\{1, 2, 3, \\ldots, 17\\} \\) with odd cardinalities,... |
[
{
"from": "human",
"value": "If $z^2 - 6z + 20 = 0$, how many possible values are there for $|z|$?"
},
{
"from": "gpt",
"value": "To determine the number of possible values for \\( |z| \\) given the equation \\( z^2 - 6z + 20 = 0 \\), we start by solving the quadratic equation for \\( z \\).\n\n... |
[
{
"from": "human",
"value": "Determine the sum of the solutions for $x$ in the system of equations $y = 7$ and $x^2 + y^2 = 225$."
},
{
"from": "gpt",
"value": "To determine the sum of the solutions for \\( x \\) in the system of equations \\( y = 7 \\) and \\( x^2 + y^2 = 225 \\), we start by s... |
[
{
"from": "human",
"value": "Charlotte is a dog walker and is creating a schedule for all of her clients this week. Each dog is walked separately. She has time to walk dogs for 8 hours a day from Monday to Friday, and for 4 hours on both Saturday and Sunday. On Monday, she needs to walk 4 poodles, 2 Chihuah... |
[
{
"from": "human",
"value": "Given the functions $f(x)= \\frac {1}{2}ax^{2}-(2a+1)x+2\\ln x(a∈R).g(x)=x^{2}-2x$, if for any $x\\_1∈(0,2]$, there exists $x\\_2∈(0,2]$, such that $f(x\\_1) < g(x\\_2)$, then the range of values for $a$ is _____ ."
},
{
"from": "gpt",
"value": "To solve the problem,... |
[
{
"from": "human",
"value": "Given that the solution set for the inequality $x^2 + ax + b \\leq 0$ is $A = [m, n]$, and the solution set for the inequality $$\\frac{(x + 2)(x + 1)}{x - 1} > 0$$ is $B$, if $A \\cup B = (-2, +\\infty)$ and $A \\cap B = (1, 3]$, find the value of $m + n$."
},
{
"from":... |
[
{
"from": "human",
"value": "Solve the equation \\( \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} = 1 \\) in integers."
},
{
"from": "gpt",
"value": "To solve the equation \\(\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} = 1\\) in integers, we need to find all possible integer solutions \\((a, b, c)\\... |
[
{
"from": "human",
"value": "In a chess tournament, each pair of players plays exactly one game. The winner of each game receives 2 points, the loser receives 0 points, and in case of a draw, both players receive 1 point each. Four scorers have recorded the total score of the tournament, but due to negligen... |
[
{
"from": "human",
"value": "Julie, Letitia, and Anton decided to have dinner together at a nearby restaurant. Julie orders $10 worth of food, Letitia orders $20 worth of food, and Anton orders $30 worth of food. They each pay $4 in tips. What is the percentage of the tip they want to split for the whole me... |
[
{
"from": "human",
"value": "At the end of year x, automobile installment credit accounted for 36% of all outstanding consumer installment credit. At that time, automobile finance companies extended $35 billion of credit, or 1/3 of the automobile installment credit. How many billion dollars of consumer inst... |
[
{
"from": "human",
"value": "Compute $$\\sum_{n=1}^{\\infty} \\frac{4n-3}{3^n}.$$"
},
{
"from": "gpt",
"value": "To compute the sum \\(\\sum_{n=1}^{\\infty} \\frac{4n-3}{3^n}\\), we can break it down into two separate sums:\n\n\\[\n\\sum_{n=1}^{\\infty} \\frac{4n-3}{3^n} = \\sum_{n=1}^{\\infty} ... |
[
{
"from": "human",
"value": "Let $\\displaystyle{z}$ be a complex number with $\\displaystyle{\\left| {z + 1} \\right| > 2}$ . Prove that $\\displaystyle{\\left| {{z^3} + 1} \\right| > 1}$ .\n\n*Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.*"
},
{
"from": "gpt",
"value": "To pr... |
[
{
"from": "human",
"value": "In a program flowchart, the symbol \"parallelogram\" is used for ( )\nA: Output\nB: Assignment\nC: Decision\nD: End of algorithm"
},
{
"from": "gpt",
"value": "To determine the correct use of the \"parallelogram\" symbol in a program flowchart, let's analyze the com... |
[
{
"from": "human",
"value": "Let $p$: $x > 1$, and $q$: $x > 0$. Then, $p$ is $q$'s ( ).\n\nA: Necessary but not sufficient condition \nB: Sufficient but not necessary condition \nC: Necessary and sufficient condition \nD: Neither sufficient nor necessary condition"
},
{
"from": "gpt",
"value... |
[
{
"from": "human",
"value": "A table lamp costs 86 yuan, an electric fan costs 185 yuan, and a bicycle costs 445 yuan.\n(1) How much money is needed to buy all three items?\n(2) Mom has 300 yuan. Is it enough to buy a table lamp and an electric fan?"
},
{
"from": "gpt",
"value": "To solve the pr... |
[
{
"from": "human",
"value": "A point is randomly selected from the part of the number line from 0 to 7. What is the probability that the point is closer to 6 than to 0? Express your answer as a fraction."
},
{
"from": "gpt",
"value": "To determine the probability that a randomly selected point f... |
[
{
"from": "human",
"value": "Milton spilled some ink on his homework paper again. This time, he can't read the coefficient of $x$, but he remembers that the quadratic equation has two distinct negative, integer solutions. What is the sum of all possible distinct integers that could be under the ink stain if... |
[
{
"from": "human",
"value": "Given the function $f(x) = |x-a| + |2x-1|$ where $a \\in \\mathbb{R}$.\n\n(1) When $a=1$, determine the solution set of $f(x) \\leqslant 2$;\n\n(2) If the solution set of $f(x) \\leqslant |2x+1|$ includes the interval $\\left[\\frac{1}{2}, 1\\right]$, find the range of values fo... |
[
{
"from": "human",
"value": "Given the function $f(x)=\\frac{1}{3}x^{3}+ax^{2}+bx$ $(a,b\\in \\mathbb{R})$ attains a maximum value of $9$ at $x=-3$.\n\n$(I)$ Find the values of $a$ and $b$;\n\n$(II)$ Find the maximum and minimum values of the function $f(x)$ in the interval $[-3,3]$."
},
{
"from": "... |
[
{
"from": "human",
"value": "Find the maximum value of the function \\( y = 2 \\sin \\frac{x}{2} \\left(1 - \\sin \\frac{x}{2}\\right) \\left(1 + \\sin \\frac{x}{2}\\right)^{2} \\) for \\( x \\in (0, \\pi) \\)."
},
{
"from": "gpt",
"value": "To find the maximum value of the function \\( y = 2 \\... |
[
{
"from": "human",
"value": "Walter fell from the eighth platform of some scaffolding and fell past David after falling some meters. He fell for an additional three more times that depth before hitting the ground, and the platforms are evenly spaced out in height. David was on the sixth platform. How many m... |
[
{
"from": "human",
"value": "Solve the equation $\\frac{3}{x-3}=\\frac{4}{x-4}$.\nA) \\(x = 1\\)\nB) \\(x = 3\\)\nC) \\(x = 0\\)\nD) \\(x = 4\\)"
},
{
"from": "gpt",
"value": "To solve the equation \\(\\frac{3}{x-3} = \\frac{4}{x-4}\\), we start by eliminating the denominators. We can do this by... |
[
{
"from": "human",
"value": "If the function $y=f(x)$ is differentiable, then \"$f′(x)=0$ has real roots\" is the $(\\ \\ \\ \\ \\ \\ )$ for \"$f(x)$ has extreme values\".\nA: Necessary but not sufficient condition\nB: Sufficient but not necessary condition\nC: Necessary and sufficient condition\nD: Necessa... |
[
{
"from": "human",
"value": "Determine the values of $a$, $b$, and $c$ such that the equations $x^3 + Cx - 24 = 0$ and $x^3 + Dx^2 + 96 = 0$ share two roots. Furthermore, find $a + b + c$, where $a \\sqrt[b]{c}$ is the product of these common roots in simplified form."
},
{
"from": "gpt",
"value... |
[
{
"from": "human",
"value": "Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two d... |
[
{
"from": "human",
"value": "Alice buys three burgers, two sodas, and one fry for \\$4.20, and Bill buys two burgers, one soda, and two fries for \\$3.40. Determine the cost of one fry if a burger costs 80 cents and a soda costs 60 cents."
},
{
"from": "gpt",
"value": "To determine the cost of o... |
[
{
"from": "human",
"value": "The letters of the word MATHEMATICS are rearranged to form distinct strings of the same 11 letters. What proportion of these strings do not contain the string MATH?"
},
{
"from": "gpt",
"value": "To determine the proportion of distinct strings formed from the letters... |
[
{
"from": "human",
"value": "The expression $5y^2 + 3y - 44$ can be written as $(5y + a)(y + b)$, where $a$ and $b$ are integers. What is $a - b$?"
},
{
"from": "gpt",
"value": "To factor the quadratic expression \\(5y^2 + 3y - 44\\) into the form \\((5y + a)(y + b)\\), we need to find integers ... |
[
{
"from": "human",
"value": "A certain beverage is packed in boxes, with each box containing $6$ cans. Among them, there are $2$ cans that are not up to standard. A quality inspector randomly selects $2$ cans from the box and finds that both are acceptable products. The probability of this occurrence is (__... |
[
{
"from": "human",
"value": "$(1)$ Calculate: $(-2)^2-\\sqrt[3]{27}+\\sqrt{16}+(-1)^{2023}$; \n$(2)$ Solve the equation: $\\frac{x}{x-1}=\\frac{5}{x^2-1}+1$."
},
{
"from": "gpt",
"value": "To solve the given problems, we will proceed step by step.\n\n**Problem 1: Calculate \\((-2)^2 - \\sqrt[3]... |
[
{
"from": "human",
"value": "The function $f: R \\longrightarrow R$ is continuous. Prove that if for every real number $x$ , there exists a positive integer $n$ , such that $\\underbrace{(f \\circ f \\circ ... \\circ f)}_{n}(x) = 1$ ,\nthen $f(1) = 1$ ."
},
{
"from": "gpt",
"value": "To pro... |
[
{
"from": "human",
"value": "The points $(0,0)\\,$, $(a,11)\\,$, and $(b,37)\\,$ are the vertices of an equilateral triangle. Find the value of $ab\\,$.\n"
},
{
"from": "gpt",
"value": "To determine the value of \\(ab\\) for the vertices of an equilateral triangle at \\((0,0)\\), \\((a,11)\\), a... |
[
{
"from": "human",
"value": "Find the equation of the circle with its center on the x-axis, a radius of 5, and passing through point A (2, -3)."
},
{
"from": "gpt",
"value": "To find the equation of the circle with its center on the x-axis, a radius of 5, and passing through point \\( A (2, -3) ... |
[
{
"from": "human",
"value": "The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 and 40. Find the length of the arc of the semicircle that is enclosed between the points where it touches the legs."
},
{
"... |
[
{
"from": "human",
"value": "Solomon collected three times as many cans as Juwan. Levi collected half of what Juwan collected. Solomon collected some cans. The boys collected 99 cans in all. How many cans did Solomon collect?"
},
{
"from": "gpt",
"value": "Let's define the number of cans Juwan c... |
[
{
"from": "human",
"value": "For how many odd $n$ in $\\{1, 3, 5, ..., 99\\}$ is the tens digit of $n^2$ odd?\n$\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ 10 \\qquad\\textbf{(C)}\\ 20 \\qquad\\textbf{(D)}\\ 30 \\qquad\\textbf{(E)}\\ 40$"
},
{
"from": "gpt",
"value": "To determine how many odd \\( ... |
[
{
"from": "human",
"value": "$A$ and $B$ are two towns $b$ km apart. Peter cycles at a speed of $7$ km/h from $A$ to $B$ and at the same time John cycles from $B$ to $A$ at a speed of $5$ km/h. If they meet after $p$ hours, find $p$."
},
{
"from": "gpt",
"value": "To determine the time \\( p \\)... |
[
{
"from": "human",
"value": "How many ways are there to choose 4 cards from a standard deck of 52 cards, if all four cards must be of different suits?"
},
{
"from": "gpt",
"value": "To determine the number of ways to choose 4 cards from a standard deck of 52 cards such that all four cards are of... |
[
{
"from": "human",
"value": "The sequence $\\{a_{n}\\}_{n \\ge 1}$ is defined by \\[a_{n}= 1+2^{2}+3^{3}+\\cdots+n^{n}.\\] Prove that there are infinitely many $n$ such that $a_{n}$ is composite."
},
{
"from": "gpt",
"value": "To prove that there are infinitely many \\( n \\) such that \\(... |
[
{
"from": "human",
"value": "When all her clothes are washed and in her closet, Brie has 12 blouses, 6 skirts, and 8 slacks. Today in her hamper, she has 75% of her blouses, 50% of her skirts, and 25% of her slacks. How many pieces of clothing will she need to put in the washer?"
},
{
"from": "gpt... |
[
{
"from": "human",
"value": "Find the matrix $\\mathbf{N}$ that swaps the rows of a matrix. In other words,\n\\[\\mathbf{N} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} c & d \\\\ a & b \\end{pmatrix}.\\] If no such matrix $\\mathbf{N}$ exists, then enter the zero matrix."
},
{
... |
[
{
"from": "human",
"value": "Consider a finite set $T$ of positive integers. Removing the greatest integer from $T$ results in an average of $42$. If the smallest integer is also removed, the average of the remaining integers becomes $48$. Returning the greatest integer to the set increases the average to $... |
[
{
"from": "human",
"value": "A four-digit number $n$ is said to be *literally 1434* if, when every digit is replaced by its remainder when divided by $5$ , the result is $1434$ . For example, $1984$ is *literally 1434* because $1$ mod $5$ is $1$ , $9$ mod $5$ is $4$ , $8$ mod $5$ is $3$... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$ , $\\angle{BAD}\\cong\\angle{ADC}$ and $\\angle{ABD}\\cong\\angle{BCD}$ , $AB=8$ , $BD=10$ , and $BC=6$ . The length $CD$ may be written in the form $\\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ ."
... |
[
{
"from": "human",
"value": "Given that $i$ is the imaginary unit, the point corresponding to the complex number $z= \\frac {3+i}{1+i}$ lies in the $($ $)$th quadrant.\n\nA: first\nB: second\nC: third\nD: fourth"
},
{
"from": "gpt",
"value": "To determine the quadrant in which the point corresp... |
[
{
"from": "human",
"value": "Given the function $f\\left(x\\right)=\\left\\{\\begin{array}{l}x-1,x≤0\\\\{x^2},x＞0\\end{array}\\right.$, the range of real numbers $x$ that satisfies $f\\left(x+1\\right) \\lt 4$ is ( ).\n\nA: $(-1,0)$\n\nB: $(-\\infty ,4)$\n\nC: $(-\\infty ,0)\\cup \\left(0,1\\right)$\n\nD: $... |
[
{
"from": "human",
"value": "Hunter needs to make a square patch of land whose perimeter is twice as large as a rectangular patch of land. The rectangular patch of land has a certain length and a width of 300 feet. The length of one side of the square patch of land is 700 feet. What is the length of the rec... |
[
{
"from": "human",
"value": "Given the function $f(x)=x^{3}+ax^{2}+bx+c$ has extreme values at $x=-2$ and $x=1$ \n(Ⅰ) Find the values of $a$ and $b$, and the intervals of monotonicity for the function $f(x)$. \n(Ⅱ) If for $x∈[-1,2]$, the inequality $f(x) < c^{2}$ always holds, find the range of values for... |
[
{
"from": "human",
"value": "Point \\( P \\) lies on side \\( A B \\) of a convex quadrilateral \\( A B C D \\). Let \\( \\omega \\) be the incircle of triangle \\( C P D \\), and let \\( I \\) be its incenter. Suppose that \\( \\omega \\) is tangent to the incircles of triangles \\( A P D \\) and \\( B P C... |
[
{
"from": "human",
"value": "Determine the necessary and sufficient conditions for lines $l_1: ax+2y-1=0$ and $l_2: x+(a+1)y+4=0$ to be parallel. Proposition p states: the condition is $a=-2$. Proposition q states: If there exist three non-collinear points in plane $\\alpha$ that are equidistant from plane ... |
[
{
"from": "human",
"value": "I looked under ___bed and found ___book I lost last week.\n\nA. the; a B. the; the C. blank; the D. the; blank"
},
{
"from": "gpt",
"value": "To determine the correct answer, let's analyze the sentence step by step:\n\n1. **Identify the context*... |
[
{
"from": "human",
"value": "In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have a radius of $20$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ ... |
[
{
"from": "human",
"value": "Macey saves to buy herself a shirt that costs $3. She was able to save $1.50 already. How many weeks does she need to save for the remaining amount if she saves $0.50 per week?"
},
{
"from": "gpt",
"value": "To determine how many weeks Macey needs to save for the rem... |
[
{
"from": "human",
"value": "In the countries of Dillie and Dallie, the currencies are dillers and dallers respectively, with the exchange rate in Dillie being 1 diller for 10 dallers and in Dallie being 1 daller for 10 dillers. A novice financier has 1 diller and can freely travel between the two countries... |
[
{
"from": "human",
"value": "A board is cut into two pieces, one twice as long as the other. The length of the shorter piece is 2 ft. What is the total length of the board?"
},
{
"from": "gpt",
"value": "To determine the total length of the board, we start by identifying the lengths of the two p... |
[
{
"from": "human",
"value": "Each month, after Jill pays for rent, utilities, food, and other necessary expenses, she has some of her net monthly salary left as discretionary income. Of this discretionary income, she puts 30% into a vacation fund, 20% into savings, and spends 35% on eating out and socializi... |
[
{
"from": "human",
"value": "The general term formula of the sequence $\\left\\{ a_n \\right\\}$ is $a_n=3n-23$. When $S_n$ is minimized, $n=$ ."
},
{
"from": "gpt",
"value": "To determine the value of \\( n \\) that minimizes the sum \\( S_n \\) of the sequence \\( \\{a_n\\} \\) where... |
[
{
"from": "human",
"value": "a man buys 50 pens at marked price of 46 pens from a whole seller . if he sells these pens giving a discount of 1 % , what is the profit percent ?"
},
{
"from": "gpt",
"value": "Let's denote the marked price of one pen as \\( P \\).\n\nFirst, we calculate the total c... |
[
{
"from": "human",
"value": "How many positive factors does 210 have?"
},
{
"from": "gpt",
"value": "To determine the number of positive factors of 210, we start by finding its prime factorization. We do this by dividing 210 by the smallest prime numbers until we are left with 1.\n\n1. Divide 21... |
[
{
"from": "human",
"value": "Given \\( n \\geq 4 \\) is an integer, prove that in any \\( n \\)-element subset \\( S \\) of the set \\( \\{1, 2, \\cdots, 2n-1\\} \\), there exists a subset of \\( S \\) whose sum of all elements is a multiple of \\( 2n \\)."
},
{
"from": "gpt",
"value": "To prove... |
[
{
"from": "human",
"value": "X and Y started a business by investing Rs. 36000 and Rs. 42000 respectively. After 4 months, Z joined in the business with an investment of Rs. 48000. Then, what was the share of Z in the profit of Rs. 13970?"
},
{
"from": "gpt",
"value": "To determine the share of ... |
[
{
"from": "human",
"value": "Given the function $f(x)=\\frac{x^2}{1+x^2}$,\n(1) Find the value of $f(2)+f(\\frac{1}{2})$ and $f(3)+f(\\frac{1}{3})$;\n(2) Prove that $f(x)+f(\\frac{1}{x})$ is a constant."
},
{
"from": "gpt",
"value": "To solve the given problem, we start by analyzing the function... |
[
{
"from": "human",
"value": "There are 38 books on the shelf. Marta put some more books on the shelf. There are 48 books on the shelf now. How many books did Marta add to the shelf?"
},
{
"from": "gpt",
"value": "To determine how many books Marta added to the shelf, we can follow these steps:\n\... |
[
{
"from": "human",
"value": "Given a rhombus with diagonals of 14 cm and 20 cm, and an angle θ between the diagonals, find its area."
},
{
"from": "gpt",
"value": "To find the area of a rhombus given its diagonals, we can use the formula for the area of a rhombus in terms of its diagonals. The a... |
[
{
"from": "human",
"value": "Ten friends eat at Wendy's and ordered the following: a platter of Taco Salad that cost $10, 6 orders of Dave's Single hamburger that cost $5 each, 5 sets of french fries that cost $2.50, 7 cups of peach lemonade that cost $2 each, 4 orders of apple pecan salad that cost $6 each... |
[
{
"from": "human",
"value": "Evaluate the product \\[ (b-5) \\cdot (b-4) \\cdot (b-3) \\cdot (b-2) \\cdot (b-1) \\cdot b, \\] where $b=3$."
},
{
"from": "gpt",
"value": "To evaluate the product \\((b-5) \\cdot (b-4) \\cdot (b-3) \\cdot (b-2) \\cdot (b-1) \\cdot b\\) where \\(b=3\\), we substitut... |
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