conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Calculate the value of $\\left(\\sqrt{5!\\cdot 4!}\\right)^2$."
},
{
"from": "gpt",
"value": "To calculate the value of \\(\\left(\\sqrt{5! \\cdot 4!}\\right)^2\\), we start by simplifying the expression inside the square root and then squaring it. The square root an... |
[
{
"from": "human",
"value": "Find the value of $x + x(x^{x^2})$ when $x = 3$.\n$\\textbf{(A)}\\ 59052 \\qquad\\textbf{(B)}\\ 59053 \\qquad\\textbf{(C)}\\ 59055 \\qquad\\textbf{(D)}\\ 59060 \\qquad\\textbf{(E)}\\ 19686$"
},
{
"from": "gpt",
"value": "To find the value of \\( x + x(x^{x^2}) \\) wh... |
[
{
"from": "human",
"value": "Randy has $200 in his piggy bank. He spends 2 dollars every time he goes to the store. He makes 4 trips to the store every month. How much money, in dollars, will be in his piggy bank after a year?"
},
{
"from": "gpt",
"value": "To determine how much money Randy will... |
[
{
"from": "human",
"value": "Two diesel trains of length 120 m and 280 m are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. In what time will they be clear of each other from the moment they meet?"
},
{
"from": "gpt",
"value": "To determine the time it takes fo... |
[
{
"from": "human",
"value": "Side $\\overline{AB}$ of $\\triangle ABC$ is length $8$. The bisector of angle $A$ meets $\\overline{BC}$ at $D$, and $CD = 2$. The set of all possible values of $AC$ is an open interval $(m, n)$. What is $m+n$?\nA) 9\nB) 10\nC) 11\nD) 12\nE) 13"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "The market value of a 10.5% stock, in which an income of Rs. 756 is derived by investing a certain amount, brokerage being 1/4%, is 90.02777777777779. How much was invested in the stock?"
},
{
"from": "gpt",
"value": "To determine the amount invested in the stock, we... |
[
{
"from": "human",
"value": "Miss Walter has 50 gold stickers. She also has twice as many silver stickers as gold stickers, and 20 fewer bronze stickers than silver stickers. She wants to give the same number of stickers to each of her 5 students. How many stickers will each student receive?"
},
{
... |
[
{
"from": "human",
"value": "Given that the discrete random variable $X$ follows a binomial distribution $B(n,p)$, where $n \\in \\mathbb{N}^{*}$, $0 \\lt p \\lt 1$, and the probability of $X$ being odd is $a$, and the probability of $X$ being even is $b$, which of the following statements are correct?\n\nA... |
[
{
"from": "human",
"value": "Two trains leave a station traveling in the same direction. Train A leaves traveling at a constant speed of 50 mph, while Train B leaves traveling at a constant speed of 80 mph. Train B left the station 30 minutes after Train A left. In how many minutes will Train B overtake Tra... |
[
{
"from": "human",
"value": "Given \\( n \\in \\mathbf{Z}, f: \\mathbf{Z} \\rightarrow \\mathbf{R} \\) such that \\( f(n) = \\begin{cases} \nn-10, & n>100 \\\\\nf(f(n+11)), & n \\leq 100 \n\\end{cases} \\), which of the following statements is correct?\n\nA. When \\( n \\leq 100 \\), the value of \\( f(n) \... |
[
{
"from": "human",
"value": "A homeowner needs to clear snow from a driveway that is 30 feet long and 3 feet wide. The snow has accumulated to a depth of 0.75 feet. How many cubic feet of snow must be removed?"
},
{
"from": "gpt",
"value": "To determine the volume of snow that needs to be remove... |
[
{
"from": "human",
"value": "Given that $\\{a\\_n\\}$ is an arithmetic sequence, $S\\_n$ is the sum of its first $n$ terms. If $a\\_1= \\frac {1}{2}$, $S\\_2=a\\_3$, find the expression for $S\\_n$."
},
{
"from": "gpt",
"value": "Given that \\(\\{a_n\\}\\) is an arithmetic sequence with the firs... |
[
{
"from": "human",
"value": "Let $P$ be the point on line segment $\\overline{AB}$ such that $AP:PB = 4:1.$ Determine $\\overrightarrow{P}$ in terms of $\\overrightarrow{A}$ and $\\overrightarrow{B}$, and find the ordered pair $(t, u)$ such that \n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrig... |
[
{
"from": "human",
"value": "Define an even function $f(x)$ on $\\mathbb{R}$ that satisfies: for any $x_1, x_2 \\in [0, +\\infty)$ ($x_1 \\neq x_2$), it holds that $$\\frac {f(x_{2})-f(x_{1})}{x_{2}-x_{1}} < 0$$, then ( )\n\nA: $f(-2) < f(1) < f(3)$\n\nB: $f(1) < f(-2) < f(3)$\n\nC: $f(3) < f(-2) < f(1)$\n... |
[
{
"from": "human",
"value": "Given an ellipse $C$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1\\left(a \\gt b \\gt 0\\right)$ with left and right foci $F_{1}$, $F_{2}$. If there exists a point $M$ on the ellipse $C$ such that $|F_{1}F_{2}|$ is the geometric mean of $|MF_{1}|$ and $|MF_{2}|$, then the range of ecce... |
[
{
"from": "human",
"value": "Moving horizontally and vertically from point to point along the lines in the diagram below, how many routes are there from point $A$ to point $B$ which consist of six horizontal moves and six vertical moves?\n\n[asy]\nfor(int i=0; i<=6;++i)\n{\ndraw((i,i)--(6,i),black);\ndr... |
[
{
"from": "human",
"value": "Prove the following inequality if we know that $a$ and $b$ are the legs of a right triangle , and $c$ is the length of the hypotenuse of this triangle: $$ 3a + 4b \\le 5c. $$ When does equality holds?"
},
{
"from": "gpt",
"value": "To prove the inequality \... |
[
{
"from": "human",
"value": "Let $\\{b_k\\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+mn$ for all positive integers $m$ and $n$. Find $b_{15}$."
},
{
"from": "gpt",
"value": "To find \\( b_{15} \\) for the sequence \\(\\{b_k\\}\\) defined by \\( b_1 = 2 \\) and \\( b_{m+n... |
[
{
"from": "human",
"value": "What is the value of ( p + q ) / ( p - q ) if p / q = 7?"
},
{
"from": "gpt",
"value": "Given the equation \\( \\frac{p}{q} = 7 \\), we need to find the value of \\( \\frac{p + q}{p - q} \\).\n\nFirst, we can express \\( p \\) in terms of \\( q \\) using the given eq... |
[
{
"from": "human",
"value": "The volume of a rectangular solid whose side face, front face, and bottom face have areas of $18\\text{ in}^2$, $15\\text{ in}^2$, and $10\\text{ in}^2$ respectively is:\n$\\textbf{(A)}\\ 20\\sqrt{3}\\text{ in}^3\\qquad\\textbf{(B)}\\ 25\\sqrt{3}\\text{ in}^3\\qquad\\textbf{(C)}... |
[
{
"from": "human",
"value": "An article is sold at a certain price, making a profit of 60%. The cost price of the article is Rs. 375. What is the selling price of the article?"
},
{
"from": "gpt",
"value": "To determine the selling price of the article, we start by understanding the relationship... |
[
{
"from": "human",
"value": "There are total 821 numbers of chicks and chickens in the coop. If there are 267 chicks, find how many more chickens there are than chicks."
},
{
"from": "gpt",
"value": "To determine how many more chickens there are than chicks, we need to follow these steps:\n\n1. ... |
[
{
"from": "human",
"value": "In the sequence $\\{a_n\\}$, $a_1=1$, $a_{n+2}+(-1)^n a_n=2$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\\{a_n\\}$. Then, $S_{100}=$ ______."
},
{
"from": "gpt",
"value": "To solve for \\( S_{100} \\) in the sequence \\(\\{a_n\\}\\) where \\(a_1 = ... |
[
{
"from": "human",
"value": "According to census statistics, the probability of a woman of childbearing age giving birth to a boy or a girl is equal. If a second child is allowed, the probability that a woman of childbearing age has two girls is ( ).\n\nA: $ \\dfrac {1}{2}$ \nB: $ \\dfrac {1}{3}$ \nC: $ \... |
[
{
"from": "human",
"value": "The negation of the proposition \"There exists $x\\_0 > 1$ such that $x\\_0^2 - x\\_0 + 2016 > 0$\" is $\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_$."
},
{
"from": "gpt",
"value": "To find the negation of the proposition \"There exists \\( x_0 > 1 \\) such that \\( x_0^2 - x_0 + ... |
[
{
"from": "human",
"value": "The championship finals of the Chinese Volleyball Super League adopts a seven-game four-win system, which means that if one team wins four games first, that team will be the overall champion, and the competition will end. Now team A and team B are in the finals. Since the two te... |
[
{
"from": "human",
"value": "In a right triangle, the lengths of the two legs are $x$ and $y$, and the hypotenuse is $z$. A perpendicular dropped from the right angle vertex divides $z$ into segments $u$ and $v$, adjacent respectively to $x$ and $y$. If $x : y = 2 : 5$, what is the ratio of $u$ to $v$?\n$\\... |
[
{
"from": "human",
"value": "Tom, Dorothy, Sammy, and Mark went on a vacation and agreed to split the costs evenly. During their trip, Tom paid $150, Dorothy paid $180, Sammy paid $240, and Mark paid $130. In order to share costs equally, Tom gave Sammy $t$ dollars, Dorothy gave Sammy $d$ dollars, and Mark ... |
[
{
"from": "human",
"value": "A rectangular field has a length of 23 m and one of its diagonals measures 33 m. The field is on a sloped area, losing a uniform 3m of its height on every 5m of its length. A circular pond of radius 3 m is also located in the field. What is the area of the habitable land left in... |
[
{
"from": "human",
"value": "The domain of the function $f(x) = \\begin{cases} 2x - x^2 & (0 \\leqslant x \\leqslant 3) \\\\ x^2 + 6x & (-2 \\leqslant x < 0) \\end{cases}$ is _______."
},
{
"from": "gpt",
"value": "To determine the domain of the function \\( f(x) \\), we need to consider the int... |
[
{
"from": "human",
"value": "A rectangle has a perimeter of 40 units, and one side is at least 3 units longer than the other side. Both dimensions are whole numbers. What is the maximum possible area of this rectangle in square units?"
},
{
"from": "gpt",
"value": "To determine the maximum possi... |
[
{
"from": "human",
"value": "In an acute triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. If $\\frac{b}{a} + \\frac{a}{b} = 6\\cos C$, then $\\frac{\\tan C}{\\tan A} + \\frac{\\tan C}{\\tan B} =$ \\_\\_\\_\\_\\_\\_."
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "An electric pump can fill a tank in 5 hours. Because of a leak in the tank, it took 10 hours to fill the tank. If the tank is full, how much time will the leak take to empty it?"
},
{
"from": "gpt",
"value": "To determine how long it will take for the leak to empty t... |
[
{
"from": "human",
"value": "There are five numbers 10, 11, 12, 13, and 14. What is the sum of the second largest number and the smallest number?"
},
{
"from": "gpt",
"value": "To find the sum of the second largest number and the smallest number in the set {10, 11, 12, 13, 14}, we need to identi... |
[
{
"from": "human",
"value": "I made the mistake of adding this number to 63, when I should have divided 36 by this number. The result of this miscalculation was 69. How much is it if I calculate it right?"
},
{
"from": "gpt",
"value": "Let's denote the unknown number by \\( x \\). According to t... |
[
{
"from": "human",
"value": "A triangle is formed with one vertex at the vertex of the parabola $y = 2x^2 - 2$ and the other two vertices at the intersections of the line $y = r$ and the parabola. If the area of the triangle is between $32$ and $128$ inclusive, find all possible values of $r$. Express your ... |
[
{
"from": "human",
"value": "Jane Doe invested a certain amount of money in mutual funds and real estate. The total amount she invested was $250,000. If she invested 3 times as much in real estate as she did in mutual funds, what was her total investment in real estate?"
},
{
"from": "gpt",
"val... |
[
{
"from": "human",
"value": "In a city of $500$ adults, every adult owns a car, a motorcycle, or both. If $480$ adults own cars and $120$ adults own motorcycles, how many of the car owners do not own a motorcycle?\n$\\textbf{(A)}\\ 100 \\qquad \\textbf{(B)}\\ 280 \\qquad \\textbf{(C)}\\ 380 \\qquad \\textbf... |
[
{
"from": "human",
"value": "Emily is making bead necklaces for her 44 friends. She made some necklaces and each necklace takes 2 beads. Emily had 52 beads. How many necklaces did Emily make?"
},
{
"from": "gpt",
"value": "To determine how many necklaces Emily made, we need to follow these steps... |
[
{
"from": "human",
"value": "Consider the polynomial $$Q(x) = \\left(\\frac{x^{20} - 1}{x-1}\\right)^2 - x^{19}.$$ Find the sum of the smallest five angles, \\( \\beta_1 + \\beta_2 + \\beta_3 + \\beta_4 + \\beta_5 \\), where the complex zeros of \\( Q(x) \\) can be expressed in the form \\( z_k = s_k \\left... |
[
{
"from": "human",
"value": "Calculate the area, in square units, of the triangle formed by the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2) (x-1)$."
},
{
"from": "gpt",
"value": "To find the area of the triangle formed by the \\(x\\)- and \\(y\\)-intercepts of the curve \\(y = (x-3)^2... |
[
{
"from": "human",
"value": "Given an arithmetic sequence \\(\\{a_n\\}\\) where \\(a_2=6\\) and \\(a_5=15\\), let \\(b_n=a_{2n}\\). Find the sum of the first 5 terms of the sequence \\(\\{b_n\\}\\). \nA: \\(30\\)\nB: \\(45\\)\nC: \\(90\\)\nD: \\(186\\)"
},
{
"from": "gpt",
"value": "To find the ... |
[
{
"from": "human",
"value": "Given the function \\( f(x) = \\ln(1 + ax) - x^2 \\) where \\( a > 0 \\) and \\( x \\in (0, 1] \\):\n1. Determine the intervals in which the function \\( f(x) \\) is monotonic.\n2. Given that the inequality \\( \\frac{1}{n^2} + \\lambda \\geqslant \\ln \\left(1 + \\frac{2}{n}\\r... |
[
{
"from": "human",
"value": "In how many ways can four people sit in a row of five vacant chairs if the first chair must always be occupied?"
},
{
"from": "gpt",
"value": "To determine the number of ways four people can sit in a row of five vacant chairs with the condition that the first chair m... |
[
{
"from": "human",
"value": "Consider the piecewise function \\(f(x)\\) defined as follows on the interval \\([-4, 4]\\):\n\\[\nf(x) = \\begin{cases} \nx + 2 & \\text{for } -4 \\leq x \\leq -1 \\\\\n-x & \\text{for } -1 < x < 2 \\\\\nx - 1 & \\text{for } 2 \\leq x \\leq 4\n\\end{cases}\n\\]\nDetermine the r... |
[
{
"from": "human",
"value": "Given the sets $P=\\{x|x^2-2x\\geq0\\}$, $Q=\\{x|1<x\\leq2\\}$, then $(\\complement_R P) \\cap Q=$ ( )\nA: $[0,1)$\nB: $(0,2]$\nC: $(1,2)$\nD: $[1,2]$"
},
{
"from": "gpt",
"value": "To solve the problem, we need to determine the set \\((\\complement_R P) \\cap Q\\)... |
[
{
"from": "human",
"value": "Sides $\\overline{AF}$ and $\\overline{CD}$ of regular hexagon $ABCDEF$ are extended to meet at point $Q$. What is the degree measure of angle $Q$?"
},
{
"from": "gpt",
"value": "To determine the degree measure of angle \\( Q \\) formed by the extensions of sides \\(... |
[
{
"from": "human",
"value": "Let \\( ABC \\) be a triangle and \\( I \\) the center of its inscribed circle. The angle bisectors \\( (AI), (BI), (CI) \\) intersect the sides \\( [BC], [CA], [AB] \\) at \\( D, E, F \\) respectively. The perpendicular bisector of the segment \\( [AD] \\) intersects the lines ... |
[
{
"from": "human",
"value": "A printing company operates a machine that can print 600 flyers in 10 minutes. They are planning to install an additional machine so that both machines working together can print 1000 flyers in 4 minutes. What equation would you use to determine how many minutes \\( x \\) it wou... |
[
{
"from": "human",
"value": "Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\\triangle ABC$, respectively, and $ac\\sin A + 4\\sin C = 4c\\sin A$.\n$(1)$ Find the value of $a$;\n$(2)$ Circle $O$ is the circumcircle of $\\triangle ABC$ (with $O$ inside $\\triangle ABC$), and the ... |
[
{
"from": "human",
"value": "Let \\( \\triangle ABC \\) be an acute-angled triangle with orthocenter \\( H \\). Let \\( E \\) and \\( F \\) be the feet of the altitudes from \\( B \\) and \\( C \\), respectively. Let \\( J \\) be the midpoint of \\( [HB] \\) and \\( K \\) the midpoint of \\( [HC] \\).\n\nSh... |
[
{
"from": "human",
"value": "A club consists of boys and girls, initially with $30\\%$ of the members being girls. After three girls leave and three boys join, girls make up $25\\%$ of the club. How many girls were originally in the club?\n$\\textbf{(A) } 12 \\qquad\\textbf{(B) } 15 \\qquad\\textbf{(C) } 18... |
[
{
"from": "human",
"value": "Consider the statements:\n$\\textbf{(1)}\\ \\text{p or q is true}\\qquad\\textbf{(2)}\\ \\text{p is true and q is false}\\qquad\\textbf{(3)}\\ \\text{p is false and q is true}\\qquad\\textbf{(4)}\\ \\text{p and q are both false.}$\nHow many of these imply the negation of the sta... |
[
{
"from": "human",
"value": "if the space diagonal of cube c is 5 inches long , what is the length , in inches , of the diagonal of the base of cube c ?"
},
{
"from": "gpt",
"value": "To find the length of the diagonal of the base of a cube given the length of its space diagonal, we can follow t... |
[
{
"from": "human",
"value": "In the figure, $ABCD$ is an isosceles trapezoid with $AD = BC = 7$, $AB = 6$, and $DC = 14$. The point $C$ is on $\\overline{DF}$. In right triangle $DEF$, point $B$ is now the midpoint of side $\\overline{DE}$. Find the length of segment $CF$.\nA) 0\nB) 0.5\nC) 1\nD) 1.5\nE) 2"... |
[
{
"from": "human",
"value": "Let \\\\(p: x < 3\\\\), \\\\(q: -1 < x < 3\\\\). Then \\\\(p\\\\) is a \\_\\_\\_\\_\\_\\_ condition for \\\\(q\\\\) to hold. (Fill in the blank with \"sufficient but not necessary\", \"necessary but not sufficient\", \"necessary and sufficient\", or \"neither sufficient nor nece... |
[
{
"from": "human",
"value": "A child builds towers using identically shaped cubes of different colors. How many different towers with a height of $7$ cubes can the child build with $3$ red cubes, $4$ blue cubes, and $2$ yellow cubes?\n$\\textbf{(A)}\\ 5040 \\qquad$\n$\\textbf{(B)}\\ 6720 \\qquad$\n$\\textbf... |
[
{
"from": "human",
"value": "Let the moving point $M(x, y)$ have a distance ratio of 2 to point $F(4,0)$ and the line $x=3$. Find the equation of the trajectory of $M(x, y)$:\n\n(A) $\\frac{x^{2}}{12}-\\frac{y^{2}}{4}=1$\n\n(B) $\\frac{x^{2}}{4}-\\frac{y^{2}}{12}=1$\n\n(C) $3 x^{2}-y^{2}-16 x+20=0$\n\n(D) $... |
[
{
"from": "human",
"value": "Given the parabola $y^{2}=2px$ and a point $M(1,m)$ on it, the distance from $M$ to its focus is $5$. Then, the equation of the directrix of this parabola is\nA: $x=8$\nB: $x=-8$\nC: $x=4$\nD: $x=-4$"
},
{
"from": "gpt",
"value": "To determine the equation of the dir... |
[
{
"from": "human",
"value": "Evaluate $\\left( \\lfloor -6.5 \\rfloor \\cdot \\lceil 6.5 \\rceil \\right)^2 \\cdot \\left( \\lfloor -5.5 \\rfloor \\cdot \\lceil 5.5 \\rceil \\right)^2 \\cdot \\left( \\lfloor -4.5 \\rfloor \\cdot \\lceil 4.5 \\rceil \\right)^2 \\cdot \\left( \\lfloor -3.5 \\rfloor \\cdot \\l... |
[
{
"from": "human",
"value": "Let $m$ and $n$ be constants. Consider that the equation \\[\\frac{(x+m)(x+n)(x+8)}{(x+2)^2} = 0\\] has exactly $3$ distinct roots. While the equation \\[\\frac{(x+2m)(x+4)(x+10)}{(x+n)(x+8)} = 0\\] has exactly $1$ distinct root. Compute $50m + n.$"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "If a sum of money doubles itself in 10 years at compound interest, compounded semiannually, what is the annual interest rate?"
},
{
"from": "gpt",
"value": "To find the annual interest rate when a sum of money doubles itself in 10 years with compound interest compoun... |
[
{
"from": "human",
"value": "When rolling a fair 6-sided die, what is the probability of a 2 or 4 being rolled?"
},
{
"from": "gpt",
"value": "To determine the probability of rolling a 2 or a 4 on a fair 6-sided die, we can follow these steps:\n\n1. Identify the total number of possible outcomes... |
[
{
"from": "human",
"value": "The base of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ has a side length of $2$ and the side length of the prism is $\\sqrt{3}$. $D$ is the midpoint of $BC$. What is the volume of the triangular pyramid $A-B_{1}DC_{1}$?\n\nA: $3$\n\nB: $\\frac{3}{2}$\n\nC: $1$\n\nD: $\\f... |
[
{
"from": "human",
"value": "What is the smallest integer value of $x$ that satisfies the equation $10x^2 - 40x + 36 = 0$?"
},
{
"from": "gpt",
"value": "To find the smallest integer value of \\( x \\) that satisfies the equation \\( 10x^2 - 40x + 36 = 0 \\), we start by solving the quadratic eq... |
[
{
"from": "human",
"value": "Given $x \\in \\mathbb{R}$, identify the necessary but not sufficient condition that satisfies the inequality $\\lg(x+1) < 1$.\nA: $-1 < x < 9$\nB: $x > -1$\nC: $x > 1$\nD: $1 < x < 9$"
},
{
"from": "gpt",
"value": "To solve the inequality \\(\\lg(x+1) < 1\\), we nee... |
[
{
"from": "human",
"value": "In the complex plane, the point corresponding to the complex number $ \\dfrac{2}{1-i} $ is located in\nA: The first quadrant\nB: The second quadrant\nC: The third quadrant\nD: The fourth quadrant"
},
{
"from": "gpt",
"value": "To determine the quadrant in which the p... |
[
{
"from": "human",
"value": "Let a right triangle have its longer leg of length 1 and its shorter leg of length $t$. Let another right triangle have its legs of lengths $2t$ and $1-t^{2}$. Show that the angle opposite to the leg of length $2t$ in the second triangle is twice the smallest angle of the origin... |
[
{
"from": "human",
"value": "In $\\triangle ABC$, $D$, $E$, and $F$ are points on $BC$, $CA$, and $AB$ respectively, such that $CD=\\frac{3}{5}BC$, $EC=\\frac{1}{2}AC$, $AF=\\frac{1}{3}AB$. Let $P$ be a point inside the quadrilateral $AEDF$ (not on the boundary). Given that $\\overrightarrow{DP}=-\\frac{1}{... |
[
{
"from": "human",
"value": "Do there exist strictly positive integers $n$ and $m$ such that $5^{n}$ and $6^{m}$ end with the same four digits?"
},
{
"from": "gpt",
"value": "To determine if there exist strictly positive integers \\( n \\) and \\( m \\) such that \\( 5^n \\) and \\( 6^m \\) end ... |
[
{
"from": "human",
"value": "Comets are celestial bodies in the solar system with bright tails, and their orbits are ellipses with the Sun as one focus. A comet is measured to have its perihelion (the point closest to the Sun) at a distance of about 2 astronomical units from the center of the Sun, and its a... |
[
{
"from": "human",
"value": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a, b, c$ (with $b \\neq 1$). If both $\\frac{C}{A}$ and $\\frac{\\sin B}{\\sin A}$ are roots of the equation $\\log_{\\sqrt{5}} x=\\log_{b}(4x-4)$, then $\\triangle ABC$:\nA. is an isosceles tria... |
[
{
"from": "human",
"value": "How many positive integers $n$ satisfy $400 < n^2 < 1600$?"
},
{
"from": "gpt",
"value": "To determine how many positive integers \\( n \\) satisfy the inequality \\( 400 < n^2 < 1600 \\), we start by finding the range of \\( n \\) by taking the square root of each p... |
[
{
"from": "human",
"value": "A honeybee is freely flying inside a cube with an edge length of $3$. If during its flight, the bee always maintains a distance greater than $1$ from all $6$ surfaces of the cube, this is referred to as \"safe flight\". What is the probability of the bee flying safely?"
},
{... |
[
{
"from": "human",
"value": "When the vectors $\\begin{pmatrix} -3 \\\\ 2 \\end{pmatrix}$ and $\\begin{pmatrix} 4 \\\\ -1 \\end{pmatrix}$ are both projected onto the same vector $\\mathbf{v},$ the result is $\\mathbf{p}$ in both cases. Find $\\mathbf{p}.$"
},
{
"from": "gpt",
"value": "To find ... |
[
{
"from": "human",
"value": "Two factors. Find two integers whose difference is minimal and whose product is 1234567890."
},
{
"from": "gpt",
"value": "To find two integers whose difference is minimal and whose product is 1234567890, we need to find two factors of 1234567890 that are as close to... |
[
{
"from": "human",
"value": "On curve $C_{1}$, any point $M$ satisfies $|MF_{1}|+|MF_{2}|=4$, where $F_{1}(-\\sqrt{3},0)$, $F_{2}(\\sqrt{3},0)$. The focus of parabola $C_{2}$ is the intersection point of the line $y=x-1$ and the $x$-axis, and its vertex is the origin $O$.\n$(1)$ Find the standard equations ... |
[
{
"from": "human",
"value": "a train 110 m long is running with a speed of 24 km / hr . in what time will it pass a man who is running at 6 km / hr in the direction opposite to that in which the train is going ?"
},
{
"from": "gpt",
"value": "To determine the time it takes for the train to pass ... |
[
{
"from": "human",
"value": "Suppose that $x$ and $y$ are nonzero real numbers such that $\\frac{4x - 2y}{3x + y} = 3$. What is the value of $\\frac{5x - y}{2x + 4y}$?\n$\\textbf{(A)}\\ -4\\qquad\\textbf{(B)}\\ -3\\qquad\\textbf{(C)}\\ 0\\qquad\\textbf{(D)}\\ 2\\qquad\\textbf{(E)}\\ 3$"
},
{
"from":... |
[
{
"from": "human",
"value": "Find the number of square units in the area of the shaded region in the following figure: [asy]\nsize(200);\ndraw((0,0)--(40,0)--(40,40)--(0, 40)--cycle);\ndot((40,0));dot((0,0));dot((15,0));dot((40,30));dot((40,40)); dot((25,40));dot((0,40));dot((0,20));\nfilldraw((0,0)--(15,0)... |
[
{
"from": "human",
"value": "Ralph is going to practice playing tennis with a tennis ball machine that shoots out tennis balls for Ralph to hit. He loads up the machine with 175 tennis balls to start with. Out of the first set of balls, he manages to hit 2/5 of them. Of the next 75 tennis balls, he manages ... |
[
{
"from": "human",
"value": "What is the remainder of $9^6 + 8^8 + 7^9$ divided by 7?"
},
{
"from": "gpt",
"value": "To find the remainder of \\(9^6 + 8^8 + 7^9\\) when divided by 7, we can simplify each term modulo 7.\n\nFirst, consider \\(9^6 \\mod 7\\):\n\\[9 \\equiv 2 \\pmod{7}\\]\nSo,\n\\[9... |
[
{
"from": "human",
"value": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $|\\overrightarrow{b}|=3$, and the projection of $\\overrightarrow{a}$ on the direction of $\\overrightarrow{b}$ is $\\frac{3}{2}$, then $\\overrightarrow{a} \\cdot \\overrightarrow{b}=$ ______."
},
{
... |
[
{
"from": "human",
"value": "Let $f(x)=(1-x^2)^{\\frac 32}$ . Denote $M$ the maximum value of $|f'(x)|$ in $(-1,\\ 1)$ . Prove that $\\int_{-1}^1 f(x)\\ dx\\leq M.$ *1981 Musashi Institute of Technology entrance exam/Electro Telecommunication, Civil Enginerring*"
},
{
"from": "gpt",
"val... |
[
{
"from": "human",
"value": "Given the function $f(x) = |x-1| - 2|x+1|$, the maximum value of $f(x)$ is $a$.\n(1) Find the value of $a$.\n(2) If $\\frac{1}{m} + \\frac{1}{2n} = a$ where $m>0$ and $n>0$, compare the magnitude of $2m+n$ with $2$."
},
{
"from": "gpt",
"value": "To find the maximum ... |
[
{
"from": "human",
"value": "What is the maximum number of balls of clay of radius $3$ that can completely fit inside a cube of side length $10$ assuming the balls can be reshaped but not compressed before they are packed in the cube?\n$\\textbf{(A) }7\\qquad\\textbf{(B) }8\\qquad\\textbf{(C) }9\\qquad\\tex... |
[
{
"from": "human",
"value": "Find the greatest common divisor of 111 and 148 using the method of successive subtraction (更相减损术).\n\nA: 1 \nB: 17 \nC: 23 \nD: 37"
},
{
"from": "gpt",
"value": "To find the greatest common divisor (GCD) of 111 and 148 using the method of successive subtraction (... |
[
{
"from": "human",
"value": "The number $121_b$, written in the integral base $b$, is the square of an integer, for \n$\\textbf{(A)}\\ b = 10,\\text{ only}\\qquad\\textbf{(B)}\\ b = 10\\text{ and }b = 5,\\text{ only}\\qquad$\n$\\textbf{(C)}\\ 2\\leq b\\leq 10\\qquad\\textbf{(D)}\\ b > 2\\qquad\\textbf{(E)}\... |
[
{
"from": "human",
"value": "A construction company is building some apartment buildings with 12 floors each. The apartments are almost ready to sell but all of them need doors to be completed. Each floor has 6 apartments, and each apartment needs 7 doors in total. The company needs to buy 1008 doors. How m... |
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