conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "A store had 86 coloring books in stock. They ended up putting them on sale and getting rid of 37 of them. They put the remaining ones onto shelves with 7 on each shelf. How many shelves did they use?"
},
{
"from": "gpt",
"value": "To determine how many shelves the st... |
[
{
"from": "human",
"value": "Renu can do a piece of work in 8 days, but with the help of her friend Suma, they can do it in some days. Suma can do it alone in 8 days. In how many days can Renu and Suma do the work together?"
},
{
"from": "gpt",
"value": "To determine how many days Renu and Suma ... |
[
{
"from": "human",
"value": "Given proposition $p$: $\\alpha=\\beta$ is the necessary and sufficient condition for $\\tan \\alpha=\\tan \\beta$. Proposition $q$: $\\varnothing \\subseteq A$. Among the following propositions, the true ones are ______.\n\n$(1)$ $p$ or $q$ $(2)$ $p$ and $q$ $(3)$ $\\neg p$ $(4... |
[
{
"from": "human",
"value": "There are 200 female managers in a certain company. Given that 2/5 of all the employees are managers and 2/5 of all male employees are managers, find the total number of female employees in the company."
},
{
"from": "gpt",
"value": "Let's denote the total number of ... |
[
{
"from": "human",
"value": "Given that the polynomial in terms of $x$, $\\left(a-3\\right)x^{3}+4x^{2}+\\left(4-b\\right)x+3$, does not contain the cubic term and the linear term, then the value of $\\left(a-b\\right)^{2023}$ is ( )\n\nA: $1$\n\nB: $-1$\n\nC: $0$\n\nD: $-2$"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Let $p$ and $n$ be a natural numbers such that $p$ is a prime and $1+np$ is a perfect square. Prove that the number $n+1$ is sum of $p$ perfect squares."
},
{
"from": "gpt",
"value": "Given that \\( p \\) is a prime number and \\( 1 + np \\) is a perfe... |
[
{
"from": "human",
"value": "There is a reading room 4 floors higher than the academy which is on the 7th floor, and there is a bookstore that is 9 floors lower than the reading room. Find the floor where the bookstore is located."
},
{
"from": "gpt",
"value": "To determine the floor where the b... |
[
{
"from": "human",
"value": "Find the largest real number $x$ such that\n\\[\\frac{\\lfloor x \\rfloor}{x} = \\frac{11}{12}.\\]"
},
{
"from": "gpt",
"value": "To find the largest real number \\( x \\) such that\n\n\\[\n\\frac{\\lfloor x \\rfloor}{x} = \\frac{11}{12},\n\\]\n\nwe start by letting ... |
[
{
"from": "human",
"value": "Calculate how many squares, all sides parallel to the axes, can be formed within a region bounded by the equations \\(y = 2x\\), \\(y = 0\\), and \\(x = 6\\), where all vertices are integer coordinates.\n- **A)** 48\n- **B)** 51\n- **C)** 55\n- **D)** 57\n- **E)** 60"
},
{
... |
[
{
"from": "human",
"value": "Given proposition $p: \\forall x\\in R$, $x^{2}+4x-m\\geqslant 0$ always holds true; proposition $q: f\\left(x\\right)=-x^{2}+\\left(m-1\\right)x$ is monotonically increasing on $\\left(-\\infty ,-3\\right]$. If $p\\wedge q$ is a true proposition, then the range of real number $... |
[
{
"from": "human",
"value": "A retailer buys a radio for Rs 225. His overhead expenses are Rs 30. He sells the radio for Rs 300. What is the profit percent of the retailer?"
},
{
"from": "gpt",
"value": "To determine the profit percent of the retailer, we need to follow these steps:\n\n1. Calcul... |
[
{
"from": "human",
"value": "Given the function $f(x) = \\begin{cases} 2^{x}-1, & x\\leq 1 \\\\ f(x-2), & x>1 \\end{cases}$, find $f(4)$."
},
{
"from": "gpt",
"value": "To find \\( f(4) \\) for the given piecewise function \\( f(x) = \\begin{cases} 2^{x}-1, & x\\leq 1 \\\\ f(x-2), & x>1 \\end{ca... |
[
{
"from": "human",
"value": "\\[ 8.482 \\frac{3 \\operatorname{tg} x - \\operatorname{tg}^{3} x}{1 - \\operatorname{tg}^{2} x} \\cdot (\\cos 3x + \\cos x) = 2 \\sin 5x \\]"
},
{
"from": "gpt",
"value": "To solve the equation \\( 8.482 \\frac{3 \\operatorname{tg} x - \\operatorname{tg}^{3} x}{1 -... |
[
{
"from": "human",
"value": "The sequence $\\{a\\_n\\}$ satisfies $a\\_1=1$, $a\\_2= \\frac {2}{3}$, and $\\frac{1}{a_{n-1}} + \\frac{1}{a_{n+1}} = \\frac{2}{a\\_n} (n\\geqslant 2)$. Determine the value of $a\\_n$ equal to:\nA. $\\frac{2}{n+1}$\nB. $(\\frac{2}{3})^{n-1}$\nC. $(\\frac{2}{3})^{n}$\nD. $\\frac... |
[
{
"from": "human",
"value": "Write $3.68$ as a simplified fraction."
},
{
"from": "gpt",
"value": "To convert the decimal \\(3.68\\) into a simplified fraction, we can follow these steps:\n\n1. **Express the decimal as a fraction:**\n \\[\n 3.68 = \\frac{368}{100}\n \\]\n This is because... |
[
{
"from": "human",
"value": "A town is holding a fireworks display for New Year’s Eve. They light enough fireworks to show the full year then light enough fireworks to write “HAPPY NEW YEAR” in the sky. They then light another 50 boxes of fireworks as part of the display. Each box of fireworks contains 8 fi... |
[
{
"from": "human",
"value": "As shown in Figure 1.11.13, let $ABCDEF$ be a hexagon inscribed in a circle. Prove that its diagonals $AD, BE, CF$ intersect at a single point. (Brianchon's theorem)"
},
{
"from": "gpt",
"value": "To prove that the diagonals \\(AD\\), \\(BE\\), and \\(CF\\) of a hexa... |
[
{
"from": "human",
"value": "Let's modify the problem slightly to make it more challenging. Paco uses a spinner to select a number from 1 through 6, each with equal probability. Manu uses a different spinner to select a number from 1 through 12, each with equal probability. What is the probability that the ... |
[
{
"from": "human",
"value": "If $x$ is a real number and $k$ is a nonnegative integer, compute the value of\n\\[\n \\frac{\\binom{1/2}{2015} \\cdot 4^{2015}}{\\binom{4030}{2015}} \\, .\n\\]"
},
{
"from": "gpt",
"value": "To compute the value of the expression \\(\\frac{\\binom{1/2}{2015} \\cdot... |
[
{
"from": "human",
"value": "The device consists of a large number of independently operating elements with the same (very small) failure probability of each element over time $T$. Find the average number of elements that fail over time $T$ if the probability that at least one element fails during this time... |
[
{
"from": "human",
"value": "Given the set $M=\\{-1, 1, 2\\}$, and the set $N=\\{y|y=x^2, x\\in M\\}$, then $M\\cap N=$ ?"
},
{
"from": "gpt",
"value": "To determine the set \\( M \\cap N \\), we first need to find the set \\( N \\). The set \\( N \\) is defined as the set of all \\( y \\) such ... |
[
{
"from": "human",
"value": "Calculate the sum of the matrices $\\begin{pmatrix} 4 & 3 \\\\ -2 & 1 \\end{pmatrix}$ and $\\begin{pmatrix} -1 & 5 \\\\ 8 & -3 \\end{pmatrix}$."
},
{
"from": "gpt",
"value": "To find the sum of the matrices \\(\\begin{pmatrix} 4 & 3 \\\\ -2 & 1 \\end{pmatrix}\\) and ... |
[
{
"from": "human",
"value": "The second of two numbers is 4 more than the first. The sum of the two numbers is a certain value. The second number is 30. What is the sum of the two numbers?"
},
{
"from": "gpt",
"value": "To find the sum of the two numbers, we start by defining the first number. L... |
[
{
"from": "human",
"value": "Point $E$ is the midpoint of side $\\overline{CD}$ in square $ABCD,$ and $\\overline{BE}$ meets diagonal $\\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$\n\n$\\textbf{(A) } 100 \\qquad \\textbf{(B) } 108 \\qquad \\textbf{(C) } 120 \\... |
[
{
"from": "human",
"value": "Medians $\\overline{AD}$ and $\\overline{BE}$ of $\\triangle ABC$ form a $45^\\circ$ angle. If $AD= 18$ and $BE = 24$, find the area of $\\triangle ABC$."
},
{
"from": "gpt",
"value": "To find the area of \\(\\triangle ABC\\) given that the medians \\(\\overline{AD}\... |
[
{
"from": "human",
"value": "Suppose \\(a\\) and \\(b\\) are positive integers such that \\(a^{b}=2^{2023}\\). Compute the smallest possible value of \\(b^{a}\\)."
},
{
"from": "gpt",
"value": "To find the smallest possible value of \\(b^a\\) given that \\(a^b = 2^{2023}\\), we start by expressi... |
[
{
"from": "human",
"value": "Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$ . Prove that exists a subset $B=\\{a,b,c,d\\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$ "
},
{
"from": "gpt",
"value": "To solve the problem, we need to show... |
[
{
"from": "human",
"value": "Compute $\\binom{2024}{1}$."
},
{
"from": "gpt",
"value": "To compute \\(\\binom{2024}{1}\\), we start by recalling the definition of the binomial coefficient. The binomial coefficient \\(\\binom{n}{k}\\) is given by the formula:\n\n\\[\n\\binom{n}{k} = \\frac{n!}{k!... |
[
{
"from": "human",
"value": "Mr. Matheson has four cards, numbered 1, 2, 3, 4. He gives one card each to Ben, Wendy, Riley, and Sara. Ben is not given the number 1. Wendy's number is 1 greater than Riley's number. Which number could Sara not have been given?"
},
{
"from": "gpt",
"value": "To det... |
[
{
"from": "human",
"value": "Given the function $f(x)=\\sqrt{3}\\sin x-\\cos x$, where $x\\in R$, determine the range of $x$ such that $f(x)\\geqslant 1$."
},
{
"from": "gpt",
"value": "To determine the range of \\( x \\) such that \\( f(x) \\geq 1 \\) for the function \\( f(x) = \\sqrt{3} \\sin... |
[
{
"from": "human",
"value": "In a right triangle with integer length sides, the hypotenuse has length 39 units. How many units is the length of the shorter leg?"
},
{
"from": "gpt",
"value": "To find the length of the shorter leg in a right triangle with integer length sides and a hypotenuse of ... |
[
{
"from": "human",
"value": "A gardener plants two maple trees, six oaks, and six birch trees in a row. The gardener wants at least one oak at each end of the row. In random order, with each arrangement equally likely, find the probability that no two birch trees are adjacent. Express your answer as $\\frac... |
[
{
"from": "human",
"value": "If there exists a real number $x$ such that the inequality $ax^{2}-4x+a-3 \\lt 0$ holds, then the range of values for $a$ is ____."
},
{
"from": "gpt",
"value": "To determine the range of values for \\(a\\) such that the inequality \\(ax^2 - 4x + a - 3 < 0\\) holds f... |
[
{
"from": "human",
"value": "Let $\\mathbf{p}$ and $\\mathbf{q}$ be three-dimensional unit vectors with an angle of $45^\\circ$ between them. Find the area of the parallelogram whose diagonals are represented by $\\mathbf{p} + 3 \\mathbf{q}$ and $3 \\mathbf{p} + \\mathbf{q}$."
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Given the function $y=\\log_{a}(x+1)+2$, $(a > 0, a\\neq 1)$, the graph always passes through a fixed point. Determine this fixed point."
},
{
"from": "gpt",
"value": "To determine the fixed point through which the graph of the function \\( y = \\log_a(x+1) + 2 \\) a... |
[
{
"from": "human",
"value": "\nGiven circles $\\odot O_{1}$ and $\\odot O_{2}$ intersect at points $X$ and $Y$. A line $AB$ is a common tangent to $\\odot O_{1}$ and $\\odot O_{2}$, with point $A$ on $\\odot O_{1}$ and point $B$ on $\\odot O_{2}$. The tangents at point $X$ of $\\odot O_{1}$ and $\\odot O_{2... |
[
{
"from": "human",
"value": "A man is thrice as old as his son. After 12 years, he will be twice as old as his son. What is the son's current age?"
},
{
"from": "gpt",
"value": "Let's denote the son's current age as \\( x \\) years. According to the problem, the man is thrice as old as his son, ... |
[
{
"from": "human",
"value": "Given that the function $f(x)$ is an even function and is decreasing on $(0, +\\infty)$, prove that the function $f(x)$ is increasing on $(-\\infty, 0)$."
},
{
"from": "gpt",
"value": "To prove that the function \\( f(x) \\) is increasing on \\( (-\\infty, 0) \\) giv... |
[
{
"from": "human",
"value": "Let \\( P(x)=x^4 + 2x^3 - 13x^2 - 14x + 24 \\) be a polynomial with roots \\( r_1, r_2, r_3, r_4 \\).\nLet \\( Q \\) be the quartic polynomial with roots \\( r_1^2, r_2^2, r_3^2, r_4^2 \\), such that the coefficient of the \\( x^4 \\) term of \\( Q \\) is 1.\nSimplify the quotie... |
[
{
"from": "human",
"value": "Let $D$ be a point inside acute $\\triangle ABC$ such that the following condition is satisfied: \n\\[ DA \\cdot DB \\cdot AB + DB \\cdot DC \\cdot BC + DC \\cdot DA \\cdot CA = AB \\cdot BC \\cdot CA. \\]\nDetermine the geometric location of point $D$ and prove your conclusion.... |
[
{
"from": "human",
"value": "The average weight of a group of people increases by 1.5 kg when a person weighing 65 kg is replaced by a new person weighing 72.5 kg. How many people are in the group?"
},
{
"from": "gpt",
"value": "Let's denote the number of people in the group as \\( n \\) and the... |
[
{
"from": "human",
"value": "Given the parametric equation of line $l$ as $$\\begin{cases} x=t\\cos\\phi \\\\ y=-2+t\\sin\\phi \\end{cases}$$ (where $t$ is a parameter, $0 \\leq \\phi < \\pi$), and establishing a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as th... |
[
{
"from": "human",
"value": "Given three vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, and $\\overrightarrow{c}$ in the same plane, where $\\overrightarrow{a} = (2, 1)$.\n(1) If $|\\overrightarrow{c}| = 2\\sqrt{5}$ and $\\overrightarrow{c}$ is parallel to $\\overrightarrow{a}$, find the coordinates ... |
[
{
"from": "human",
"value": "Marge had a winning lottery ticket for $20000. She paid 55% of her winnings towards taxes. She then used a half of the remaining amount to pay off a part of her mortgage. Wanting to secure her future, she put a third of what was left into a retirement fund. Afterwards, she decid... |
[
{
"from": "human",
"value": "Given that $e$ is the base of the natural logarithm, if for any $x \\in [0,1]$, there exists a unique $y \\in [-1,1]$ such that $x + y^2e^y - a = 0$ holds, then the range of the real number $a$ is _____."
},
{
"from": "gpt",
"value": "To determine the range of the re... |
[
{
"from": "human",
"value": "Let $\\triangle ABC$ be scalene, with $BC$ as the largest side. Let $D$ be the foot of the perpendicular from $A$ on side $BC$ . Let points $K,L$ be chosen on the lines $AB$ and $AC$ respectively, such that $D$ is the midpoint of segment $KL$ . Prove that the p... |
[
{
"from": "human",
"value": "Given the general term formula of the sequence $\\{a_{n}\\}$ as $a_{n}= \\frac {1}{2n-1}$, where $n \\in \\mathbb{N}^{*}$\n(1) Find the sum of the first $n$ terms, $S_{n}$, of the sequence $\\{ \\frac {a_{n}+2}{a_{n}}\\}$\n(2) Let $b_{n}=a_{n}a_{n+1}$, find the sum of the first ... |
[
{
"from": "human",
"value": "Given $\\cos\\alpha = -\\frac{4}{5}$, and $\\alpha \\in \\left(\\frac{\\pi}{2}, \\pi\\right)$.\n\n(1) Find the value of $\\tan 2\\alpha$.\n\n(2) Find the value of $\\cos\\left(\\alpha + \\frac{\\pi}{3}\\right)$."
},
{
"from": "gpt",
"value": "Given \\(\\cos\\alpha = ... |
[
{
"from": "human",
"value": "A box contains 6 ballpoint pens, including 3 first-grade pens, 2 second-grade pens, and 1 third-grade pen. If 2 pens are randomly selected from the box, the probability that both are first-grade pens is ____."
},
{
"from": "gpt",
"value": "To determine the probabilit... |
[
{
"from": "human",
"value": "Given a line $l: kx-y-k+1=0$ and the graph of the function $f(x)= \\begin{cases} x^{3}-3x^{2}+2x+1, & x\\leqslant 2\\\\ ax-2a+1, & x > 2 \\end{cases}$ intersect at three points, with their $x$-coordinates being $x_{1}$, $x_{2}$, and $x_{3}$. If for any $0 < k < 3$, $x_{1}+x_{2}+... |
[
{
"from": "human",
"value": "If 3 letters are to be mailed into 4 mailboxes, then the total number of different ways to do this is ( )\nA: 24 ways\nB: 4 ways\nC: $4^3$ ways\nD: $3^4$ ways"
},
{
"from": "gpt",
"value": "To determine the total number of different ways to mail 3 letters into 4 mail... |
[
{
"from": "human",
"value": "Given a parabola symmetric with respect to the $x$-axis, its vertex is at the origin $O$, and it passes through the point $M(2,y_0)$. If the distance from point $M$ to the focus of the parabola is $3$, then $|OM|=\\boxed{?}$\n\nA: $2\\sqrt{2}$\nB: $2\\sqrt{3}$\nC: $4$\nD: $2\\sq... |
[
{
"from": "human",
"value": "A guy goes to the tool shop and buys 5 sets of drill bits. Each set cost 6 dollars. He then pays 10% tax on the order. What was the total amount he paid?"
},
{
"from": "gpt",
"value": "To determine the total amount the guy paid, we need to follow these steps:\n\n1... |
[
{
"from": "human",
"value": "Let the equation of the hyperbola C be $\\frac{{x}^{2}}{{a}^{2}}$-$\\frac{{y}^{2}}{{b}^{2}}$=1 (a>0, b>0), and the line passing through the focus of the parabola $y^2=4x$ and the point (0, b) is denoted as l. If one asymptote of C is parallel to l and the other asymptote is perp... |
[
{
"from": "human",
"value": "Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $E: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$, and point $M$ is on $E$ with $MF_1$ perpendicular to the x-axis and $\\sin \\angle MF_2F_1 = \\frac{1}{3}$, find the eccentricity of $E$.\n\nA: $\\sqrt{2}$\nB: $... |
[
{
"from": "human",
"value": "In the arithmetic sequence $\\{a_n\\}$, $d=-2$, $a_1+a_4+a_7+\\ldots+a_{31}=50$. What is the value of $a_2+a_6+a_{10}+\\ldots+a_{42}$?\n\nA: $60$\n\nB: $-82$\n\nC: $182$\n\nD: $-96$"
},
{
"from": "gpt",
"value": "**Analysis**\n\nThis problem examines the properties o... |
[
{
"from": "human",
"value": "The derivative of the function \\\\(y = \\cos 2x\\\\) is \\\\(({ })\\\\)\nA: \\\\(-\\sin 2x\\\\)\nB: \\\\(\\sin 2x\\\\)\nC: \\\\(-2\\sin 2x\\\\)\nD: \\\\(2\\sin 2x\\\\)"
},
{
"from": "gpt",
"value": "To find the derivative of the function \\( y = \\cos 2x \\), we ne... |
[
{
"from": "human",
"value": "Suppose four aluminum cans are needed to make a new can. Starting with 256 aluminum cans, how many new cans can eventually be made from these aluminum cans? (Remember that the first new cans that are made can then be recycled into even newer cans!) Do not include the original 25... |
[
{
"from": "human",
"value": "Given an arithmetic sequence $\\{a_n\\}$, if $a_4=4$ and $a_3 + a_8 = 5$, find the value of $a_7$.\nA: 1\nB: 2\nC: 3\nD: 4"
},
{
"from": "gpt",
"value": "To find the value of \\(a_7\\) in the arithmetic sequence \\(\\{a_n\\}\\), we start by using the general formula ... |
[
{
"from": "human",
"value": "Calculate the sum of the digits in the product of $\\underbrace{1111\\cdots 11}_{84\\text{ ones}} \\times \\underbrace{8888\\cdots 88}_{84\\text{ eights}}$, where a string of $84$ ones is multiplied by a string of $84$ eights.\n$\\text{(A)}\\ 672 \\qquad \\text{(B)}\\ 756 \\qqua... |
[
{
"from": "human",
"value": "A positive integer \\( A \\) divided by \\( 3! \\) gives a result where the number of factors is \\(\\frac{1}{3}\\) of the original number of factors. What is the smallest such \\( A \\)?"
},
{
"from": "gpt",
"value": "To solve for the smallest positive integer \\( A... |
[
{
"from": "human",
"value": "what is the perimeter , in meters , of a rectangular garden 16 meters wide that has the same area as a rectangular playground 16 meters long and 12 meters wide ?"
},
{
"from": "gpt",
"value": "To find the perimeter of the rectangular garden, we need to follow these s... |
[
{
"from": "human",
"value": "Find the greatest common divisor of 12,012 and 21,021."
},
{
"from": "gpt",
"value": "To find the greatest common divisor (GCD) of 12,012 and 21,021, we can use the Euclidean algorithm. The Euclidean algorithm involves repeated division and taking remainders until we... |
[
{
"from": "human",
"value": "When solving the linear equation $\\frac{1}{2}(x+1)=1-\\frac{1}{3}x$, the correct expression after eliminating the denominators is:\n\nA: $3\\left(x+1\\right)=1-2x$\n\nB: $3\\left(x+1\\right)=6-2x$\n\nC: $2\\left(x+1\\right)=6-3x$\n\nD: $2\\left(x+1\\right)=1-3x$"
},
{
"... |
[
{
"from": "human",
"value": "Solve the system of equations:\n\\[\n\\left\\{\n\\begin{array}{l}\nx-y+z-w=2, \\\\\nx^{2}-y^{2}+z^{2}-w^{2}=6, \\\\\nx^{3}-y^{3}+z^{3}-w^{3}=20, \\\\\nx^{4}-y^{4}+z^{4}-w^{4}=60.\n\\end{array}\n\\right.\n\\]"
},
{
"from": "gpt",
"value": "To solve the system of equat... |
[
{
"from": "human",
"value": "Solve the system of equations: \\(x_{1} x_{2}=x_{2} x_{3}=\\ldots=x_{n-1} x_{n}=x_{n} x_{1}=1\\)."
},
{
"from": "gpt",
"value": "To solve the system of equations \\(x_{1} x_{2}=x_{2} x_{3}=\\ldots=x_{n-1} x_{n}=x_{n} x_{1}=1\\), we start by examining the equations an... |
[
{
"from": "human",
"value": "As \\(t\\) takes on all real values, the set of points \\((x,y)\\) defined by\n\\[\nx = t^2 - 4,\n\\]\n\\[\ny = t^3 - 6t + 4\n\\]\nforms a curve. Compute the ordered pair \\((x,y)\\) where this curve crosses itself."
},
{
"from": "gpt",
"value": "To find the ordered ... |
[
{
"from": "human",
"value": "The length of a rectangle is increased by 20% and the breadth is reduced by some percentage. The effect on its area is 4.0000000000000036%. By what percentage is the breadth reduced?"
},
{
"from": "gpt",
"value": "Let the original length of the rectangle be \\( l \\)... |
[
{
"from": "human",
"value": "A man is 32 years older than his son. In two years, his age will be a certain multiple of the age of his son. The present age of the son is 30 years. What is the ratio of the man's age to his son's age in two years?"
},
{
"from": "gpt",
"value": "To determine the rat... |
[
{
"from": "human",
"value": "Minjun grew 0.2 meters (m) last year and 1/10 meter (m) this year. If Minjun's height was 1.1 meters (m) on January 1 last year, how tall is Minjun's height in meters (m) now?"
},
{
"from": "gpt",
"value": "To determine Minjun's current height, we need to add the gro... |
[
{
"from": "human",
"value": "Consider n, m, and p are positive integers, and let g(n, m, p) be a function defined as:\n\ng(n, m, p) = g(n - 1, m, p) - n + p * (n - m)\n\nwith the initial condition: g(4, 2, 3) = 20.\n\nWhat is the value of g(6, 2, 3)?"
},
{
"from": "gpt",
"value": "To find the va... |
[
{
"from": "human",
"value": "Suppose there are three coins - Coin A, Coin B, and Coin C. Each coin has a different probability of landing tails up: Coin A has a 3/4 probability, Coin B has a 1/2 probability, and Coin C has a 1/4 probability. If all three coins are flipped simultaneously, what is the probabi... |
[
{
"from": "human",
"value": "Given the function $f\\left(x\\right)=2^{x}-1$, a real number $x$ is randomly chosen in the interval $\\left[0,6\\right]$. The probability that $3\\leqslant f\\left(x\\right)\\leqslant 15$ is ( ).\n\nA: $\\frac{1}{3}$\n\nB: $\\frac{1}{4}$\n\nC: $\\frac{1}{2}$\n\nD: $\\frac{2}{3}... |
[
{
"from": "human",
"value": "Let point $P$ be the intersection point in the first quadrant of the hyperbola $\\frac{x^{2}}{a^{2}}- \\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and the circle $x^{2}+y^{2}=a^{2}+b^{2}$. Let $F\\_1$ and $F\\_2$ be the left and right foci of the hyperbola, respectively, and $|PF\\_1|... |
[
{
"from": "human",
"value": "If \\( x \\div y = 3 \\) and \\( y \\div z = \\frac{2}{5} \\), what is the value of \\( z \\div x \\)?"
},
{
"from": "gpt",
"value": "To find the value of \\( z \\div x \\), we start with the given equations:\n\n1. \\( x \\div y = 3 \\)\n2. \\( y \\div z = \\frac{2}{... |
[
{
"from": "human",
"value": "Given $f(x)=a\\sin x+bx+c(a,b,c\\in R)$, if $f(0)=-2$, $f(\\frac{\\pi}{2})=1$, find $f(-\\frac{\\pi}{2})$ \\_\\_\\_\\_\\_\\_."
},
{
"from": "gpt",
"value": "To find \\( f\\left(-\\frac{\\pi}{2}\\right) \\) for the function \\( f(x) = a \\sin x + bx + c \\), we start ... |
[
{
"from": "human",
"value": "Mr Castiel prepared some boar sausages on the weekend. He ate 2/5 of them on Monday, half of the remaining on Tuesday and saved the rest for Friday. He ate 3/4 of the remaining boar sausages on Friday, and 45 sausages are left. How many boar sausages did Mr Castiel prepare on th... |
[
{
"from": "human",
"value": "Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 65 islanders gathered for a meeting. Each of them made the following statement in turn: \"Among the statements made earlier, the true ones are exactly 20 less than the false ones.\" How ma... |
[
{
"from": "human",
"value": "Let \\( S \\) be the sum of all positive integers \\( n \\) such that \\( n^2 + 12n - 507 \\) is a perfect square. Find the remainder when \\( S \\) is divided by 1000."
},
{
"from": "gpt",
"value": "To determine the sum of all positive integers \\( n \\) such that \... |
[
{
"from": "human",
"value": "the regular price per can of a certain brand of soda is $ 0.15 . if the regular price per can is discounted 10 percent when the soda is purchased in 24 - can cases , what is the price of 75 cans of this brand of soda purchased in 24 - can cases ?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Given that the graph of the power function $y=f(x)$ passes through the point $(4,2)$, find the value of $f(\\frac{1}{4})$."
},
{
"from": "gpt",
"value": "Given that the graph of the power function \\( y = f(x) \\) passes through the point \\( (4,2) \\), we can expres... |
[
{
"from": "human",
"value": "Dave had some apps on his phone. After adding 1 app, he had 18 apps left. How many apps did he have initially?"
},
{
"from": "gpt",
"value": "To determine how many apps Dave initially had on his phone, we can set up a simple equation based on the information given. L... |
[
{
"from": "human",
"value": "Given that $g(x) = x^{-2} + \\frac{x^{-2}}{1+x^{-2}}$, find $g(g(-3))$ and express your answer as a common fraction."
},
{
"from": "gpt",
"value": "To find \\( g(g(-3)) \\), we first need to evaluate \\( g(-3) \\) and then use that result to evaluate \\( g \\) again.... |
[
{
"from": "human",
"value": "Compute $\\sin 240^\\circ$."
},
{
"from": "gpt",
"value": "To compute \\(\\sin 240^\\circ\\), we can follow these steps:\n\n1. **Determine the reference angle:**\n The angle \\(240^\\circ\\) is in the third quadrant. To find the reference angle, we subtract \\(180^... |
[
{
"from": "human",
"value": "Find the last two digits of the following sum: $$5! + 10! + 15! + \\cdots + 100!$$"
},
{
"from": "gpt",
"value": "To find the last two digits of the sum \\(5! + 10! + 15! + \\cdots + 100!\\), we need to determine the value of this sum modulo 100. We start by calculat... |
[
{
"from": "human",
"value": "Tammy wants to run 3500 meters per week. She knows her track at school is 50 meters around. If she wants to run the same amount each day, how many loops should she make per day?"
},
{
"from": "gpt",
"value": "To determine how many loops Tammy should make per day, we ... |
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