task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | Given that the random variable $X$ follows a normal distribution $N(3, \frac{1}{4})$, and $P(X > \frac{7}{2})=0.1587$, determine the value of $P(\frac{5}{2} \leqslant X \leqslant \frac{7}{2})$. | 0.6826 | 77 | 6 |
math | Let $ABCD$ be a convex quadrilateral with $AC=7$ and $BD=17$. Let $M, P, N, Q$ be the midpoints of sides $AB, BC, CD, DA$ respectively. Compute $MN^{2}+PQ^{2}$. | 169 | 63 | 3 |
math | Given that $a$ is an odd multiple of $17$, find the greatest common divisor of $2a^2 + 33a + 85$ and $a + 17$. | 34 | 43 | 2 |
math | A play includes three different male roles, five women audition, and seven men audition. Additionally, there are two roles open to either gender. Determine how many ways there are to cast these five roles if roles can only be filled as per gender specifications. | 15120 | 50 | 5 |
math | The Grand Old Duke of York had 10,000 men. He lost 10% of them on the way to the top of the hill, and he lost 15% of the rest as he marched them back down the hill. What percentage of the 10,000 men were still there when they reached the bottom of the hill?
A) \(76 \frac{1}{2} \%\)
B) \(75 \%\)
C) \(73 \frac{1}{2} ... | 76 \frac{1}{2} \% | 138 | 10 |
math | Given a positive integer \(n\), find the number of positive integer solutions \((x, y)\) to the equation
$$
\frac{1}{n}=\frac{1}{x}+\frac{1}{y} \quad(x \neq y)
$$ | d(n^2) - 1 | 58 | 8 |
math | Let \( n \geqslant 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n} \) be two sets of real numbers, satisfying \( \sum_{j=1}^{n} \alpha_{j}^{2}<1 \) and \( \sum_{j=1}^{n} \beta_{j}^{2}<1 \). Define
\[ A^{2}=1-\sum_{j=1}^{n} \alpha_{j}^{2}, \, B^{2}=1-\sum_{j=1}^{n} \beta_{j}^{2}, ... | 0 | 254 | 1 |
math | A natural number \( A \) is written twice in a row (for example, writing 12 as 1212) to form a new number \( B \). If \( B \) is a multiple of 2016, what is the smallest \( A \)? | 288 | 60 | 3 |
math | A classroom has enough chairs to seat $312_5$ students. If $3$ students are supposed to sit on one bench, how many benches does the classroom have? | 28 | 37 | 2 |
math | Given the fractional equation about $x$: $\frac{m-1}{x+1}=1$, and the solution is a negative number, then the range of $m$ is ______. | m < 2 \text{ and } m \neq 1 | 39 | 15 |
math | What integer value of $x$ will satisfy $x + 8 > 9$ and $-3x > -15$? | 2, 3, 4 | 29 | 7 |
math | Given the hyperbola $C$: $\frac{y^{2}}{2} - \frac{x^{2}}{b} = 1 (b > 0)$ with an eccentricity of $2$, find the product of the distances from any point on $C$ to its asymptotes. | \frac{3}{2} | 63 | 7 |
math | Handy Fiona helped her neighbor, spending 1 hour 30 minutes on Monday, 1 hour 15 minutes on Tuesday, from 9:10 AM to 12:20 PM on Wednesday, and 45 minutes on Thursday. She is paid $\textdollar4$ per hour. Calculate the total amount Handy Fiona earned for the week. | 26.67 | 77 | 5 |
math | Find the angle of inclination for the line \(l\) with the parametric equations
$$ \begin{cases}
x = -\sqrt{3}t, \\
y = 1 + 3t
\end{cases}
(t \text{ is the parameter}).$$ | \frac{2\pi}{3} | 59 | 9 |
math | Given that among any group of people, there must be at least two people who share the same birthday, determine the minimum number of people. | 367 | 28 | 3 |
math | The value range of the function $y=\sin x-\sin |x|$ can be determined. | [-2,2] | 20 | 5 |
math | Let proposition \\(p:\\) be the inequality about \\(a\\) that \\(\forall {x} \in \mathbb{R}, {x}^{2} - 4x + {a}^{2} > 0\\); proposition \\(q:\\) be that the quadratic equation in \\(x\\), \\({x}^{2} + (a+1)x + a - 1 = 0\\), has one root greater than zero and the other less than zero; proposition \\(r:\\) be the solutio... | (3, +\infty) | 233 | 8 |
math | Write down two decimals that are greater than 3.5 and less than 3.6: \_\_\_\_\_\_, \_\_\_\_\_\_;
Write down three decimals that are between 0 and 0.1: \_\_\_\_\_\_, \_\_\_\_\_\_, \_\_\_\_\_\_. | 3.51, 3.52; 0.01, 0.02, 0.03 | 71 | 28 |
math | Suppose \( x, y, z \) and \( \lambda \) are positive real numbers such that
\[
\begin{aligned}
yz & = 6\lambda x \\
xz & = 6\lambda y \\
xy & = 6\lambda z \\
x^2 + y^2 + z^2 & = 1
\end{aligned}
\]
Find the value of \( (xyz\lambda)^{-1} \). | 54 | 95 | 2 |
math | The positive five-digit integers that use each of the five digits $1, 2, 3, 4, 5$ exactly once are ordered from least to greatest. What is the $44^{\text{th}}$ integer in the list? | 25143 | 54 | 5 |
math | Given two points $M(m+3,3m+5)$ and $N(2m-1,1)$ in the coordinate plane.<br/>$(1)$ Find the range of $m$ when the slope of the line $MN$ is acute angle;<br/>$(2)$ If the direction vector of the line $MN$ is $\overrightarrow{a}=(1,-2023)$, find the value of $m$. | \frac{2024}{505} | 93 | 12 |
math | From the digits $1$, $2$, $3$, $4$, $5$, randomly select $3$ digits to form a three-digit number without repetition. Find:
$(1)$ The probability that this three-digit number is a multiple of $5$;
$(2)$ The probability that this three-digit number is an odd number. | \frac{3}{5} | 68 | 7 |
math | A district in a city is laid out in an $11 \times 11$ grid. Every day, a sprinkler truck departs from the bottom-left corner $A(0,0)$ and travels along the streets to reach the top-right corner $B(10,10)$. At each intersection, the driver randomly chooses a direction, as long as it does not deviate from the shortest pa... | 1 - \frac{\binom{18}{9}}{\binom{20}{10}} | 145 | 23 |
math | What is the least positive integer $n$ such that $4725$ is a factor of $n!$? | 15 | 26 | 2 |
math | Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $45^\circ$. The area of face $ABC$ is $150$, the area of face $BCD$ is $100$, and $BC=10$. Find the volume of the tetrahedron. | 500\sqrt{2} | 73 | 8 |
math | Let the function $f(x)$ ($x \in \mathbb{R}$) have a period of $3$. When $x \in [-2,1)$, $f(x)$ is defined as follows:
\[f(x)= \begin{cases} x+a, & -2\leqslant x < 0 \\ ( \dfrac {1}{2})^{x}, & 0\leqslant x < 1\end{cases}\]
1. Find $f( \dfrac {13}{2})=$ \_\_\_\_\_\_ ;
2. If $f(x)$ has a minimum value but no maximum v... | (1, \dfrac {5}{2}] | 161 | 11 |
math | For any \( x_1, x_2 \) \((x_1 \neq x_2)\) in the domain of the function \( f(x) \), the following statements hold:
(1) \( f(x_1 + x_2) = f(x_1) \cdot f(x_2) \)
(2) \( f(x_1 \cdot x_2) = f(x_1) + f(x_2) \)
(3) \( \frac{f(x_1) - f(x_2)}{x_1 - x_2} > 0 \)
(4) \( f\left(\frac{x_1 + x_2}{2}\right) > \frac{f(x_1) + f(x_2)}{2... | (1),(3) | 190 | 5 |
math | Given that $α∈\{-1,1,2,\frac{3}{5},\frac{7}{2}\}$, determine the set of all $α$ values for which the function $y=x^{α}$ has a domain of $R$ and is an odd function. | \{1,\frac{3}{5}\} | 60 | 11 |
math | Given a man with mass $m$ jumps off a high bridge with a bungee cord attached, falls through a maximum distance $H$ , and experiences a maximum tension in the bungee cord that is four times the man's weight, find the spring constant $k$ of the bungee cord. | \frac{8mg}{H} | 69 | 8 |
math | Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median. | 234 | 55 | 3 |
math | A square pattern is composed of 5 black and 20 white square tiles. Around the pattern, attach a border consisting of white tiles on all sides. What is the ratio of black tiles to white tiles in the extended pattern? | \frac{5}{44} | 47 | 8 |
math | A chicken farm plans to purchase a total of 2000 chicks of type A and type B for breeding. It is known that each chick of type A costs 2 yuan, and each chick of type B costs 3 yuan.
(1) If purchasing these chicks costs a total of 4500 yuan, how many chicks of type A and type B were purchased respectively?
(2) If the mo... | 1200 \text{ type A chicks, } 800 \text{ type B chicks, } 4800 \text{ yuan} | 192 | 34 |
math | Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$ , Anne and Carl in $5$ . How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back? | \frac{59}{20} | 63 | 9 |
math | Five times $A$'s money added to twice $B$'s money is more than $100$, and four times $A$'s money minus $B$'s money equals $40$. Given that $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, solve for the system of inequalities. | a>\frac{180}{13}, b>\frac{200}{13} | 81 | 22 |
math | The largest prime factor of 101101101101 is a four-digit number $N$. Compute $N$. | 9901 | 30 | 4 |
math | In a high school, there are 400 male students and 600 female students in the second year. To analyze the final physics survey test scores, a sample is obtained using stratified random sampling based on the male-to-female ratio. In the sample, the average score of male students is 80 points with a variance of 10, and th... | 112 | 121 | 3 |
math | Let \(a\), \(b\), and \(c\) be real numbers such that
\[
\cos a + \cos b + \cos c = \sin a + \sin b + \sin c = 1.
\]
Find the sum of all possible values of \(\cos 2a + \cos 2b + \cos 2c.\) | -2 | 78 | 2 |
math | For which integer values of $n$ is the number $n^{4} + 4$ composite? | n \neq \pm 1 | 22 | 8 |
math | Isosceles trapezoid \(ABCD\) with bases \(AB\) and \(CD\) has a point \(P\) on \(AB\) with \(AP=11, BP=27\), \(CD=34\), and \(\angle CPD=90^{\circ}\). Compute the height of isosceles trapezoid \(ABCD\). | 15 | 82 | 2 |
math | In a sequence, the arrangement of numbers in each row begins from 4 and increases by 2 in each subsequent row. Each number in a row appears as many times as its value divided by 2. The first three rows are shown as:
$\bullet$ Row 1: $4,$ $4$
$\bullet$ Row 2: $6,$ $6,$ $6,$ $6$
$\bullet$ Row 3: $8,$ $8,$ $8,$ $8,$ $8... | 18 | 128 | 2 |
math | Suppose the sequence $\{a\_n\}$ satisfies the following condition: for any $n\in\mathbb{N}^*$, there are only finitely many positive integers $m$ such that $a\_m < n$ holds. Let $b\_n$ denote the number of such $m$'s. This gives rise to a new sequence $\{b\_n\}$. For example, if the sequence $\{a\_n\}$ is $1$, $2$, $3$... | 2^{2015} | 214 | 7 |
math | Find all triples of natural numbers \(a, b\), and \(c\) for which the numbers \(a^{2}-23\) and \(b^{2}-23\) are prime, and \(\left(a^{2}-23\right)\left(b^{2}-23\right)=c^{2}-23\). | (5,6,7) \text{ and } (6,5,7) | 71 | 19 |
math | Calculate the slope angle of the line $x+2=0$. | \frac {π}{2} | 14 | 7 |
math | Find the number of distinct arrangements in a row of all natural numbers from 1 to 10 such that the sum of any three consecutive numbers is divisible by 3. | 1728 | 35 | 4 |
math | Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus ride. | 45 | 86 | 2 |
math | Given the equation about \( x \)
$$
\sqrt{m-x^{2}}=\log _{2}(x+n)
$$
When there is one positive and one negative real root, the range of \( n \) is \( 3 \leqslant n<4 \). Then the value of the constant \( m \) is $\qquad$ . | 4 | 77 | 1 |
math | The teacher of the summer math camp brought with him several shirts, several pairs of pants, several pairs of shoes, and two jackets for the entire summer. On each lesson, he wore pants, a shirt, and shoes, and wore a jacket for some lessons. On any two lessons, at least one element of his attire or shoes was different... | 216 | 147 | 3 |
math | The interval in which the root of the function $y = \ln x + 2x - 3$ must lie is to be determined. | (1, 2) | 30 | 6 |
math | Find the sum of the squares of all integer solutions of the equation \(x^4 - 49x^2 + 576 = 0\). | 104 | 34 | 3 |
math | In a class with 35 students, a survey was conducted on their preferences regarding mathematics and literature, and it was found that:
- 7 men like mathematics;
- 6 men like literature;
- 5 men and 8 women said they do not like either;
- there are 16 men in the class;
- 5 students like both;
- 11 students like only mat... | 6 | 103 | 1 |
math | The equation \( x^2 - (1B)x + B1 = 0 \) has positive integer solutions where \( B \) is a positive single digit. How many such \( B \)s exist? (Since \( B \) represents a digit, if \( B = 3 \) then \( B1 \) represents the integer 31.) | 2 | 75 | 1 |
math | Henry walked 15 meters due north from a tree. He then turned due east and walked 40 feet. He then turned due south and walked a distance of 15 meters plus 50 feet. Finally, he turned and walked 10 additional meters to the north. Determine how many feet away from his original starting point is Henry. | 43.54 \text{ feet} | 72 | 10 |
math | Evaluate the expression:
\[
\frac{3}{2-\frac{3}{4}}
\]
A) $\frac{8}{5}$
B) $\frac{12}{5}$
C) $\frac{5}{4}$
D) $\frac{4}{3}$
E) $3$ | \frac{12}{5} | 67 | 8 |
math | The diagonals of a convex quadrilateral are 12 and 18 and intersect at point \( O \).
Find the sides of the quadrilateral formed by the points of intersection of the medians of triangles \( AOB, BOC, COD \), and \( AOD \). | 4,6,4,6 | 59 | 7 |
math | Consider a function $f(x),$ defined for $0 \le x \le 1,$ with the following properties:
(i) $f(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $f(x) \le f(y).$
(iii) $f(1 - x) = 1 - f(x)$ for all $0 \le x \le 1.$
(iv) $f\left(\frac{x}{4}\right) = \frac{f(x)}{3}$ for $0 \le x \le 1.$
Find $f\left(\frac{3}{8}\right).$ | f\left(\frac{3}{8}\right) = \frac{2}{9} | 143 | 20 |
math | How many arithmetic sequences, where the common difference is a natural number greater than 2, satisfy the conditions that the first term is 1783, the last term is 1993, and the number of terms is at least 3? | 13 | 53 | 2 |
math | Find the minimum value of
\[ 3 \sqrt[3]{x} + \frac{1}{x^2} \]
for \( x > 0 \). | 4 | 36 | 1 |
math | A list of five positive integers has the following properties:
$\bullet$ The only integer in the list that occurs more than once is $7,$
$\bullet$ its median is $12,$ and
$\bullet$ its average (mean) is $13.$
What is the largest possible integer that could appear in the list? | 26 | 69 | 2 |
math | If $\log_8 (5x) = 3$, find $\log_x 125$. Express your answer in simplest fractional form. | \frac{3 \log 5}{9 \log 2 - \log 5} | 30 | 20 |
math | Given a sequence $\left\{a_n\right\}$ satisfying $a_1=1, a_2=3$, if $\left|a_{n+1}-a_n\right|=2^n$ $(n\in\mathbb{N}^*)$, and $\left\{a_{2n-1}\right\}$ is an increasing sequence, $\left\{a_{2n}\right\}$ is a decreasing sequence, then find $\lim\limits_{n\to\infty} \frac{a_{2n-1}}{a_{2n}}$. | -\frac{1}{2} | 126 | 7 |
math | Given the set $A=\{a+2,2a^{2}+a\}$, if $3 \in A$, find the value of $a$. | -\frac{3}{2} | 35 | 7 |
math | A pair of dice is rolled twice, and the coordinates of point $P(m,n)$ are determined by the sequence of the dots $m$ and $n$ obtained. The probability that point $P$ lies outside the circle $x^2+y^2=17$ is $\_\_\_\_\_\_\_.$ | \dfrac{13}{18} | 67 | 9 |
math | Given the function $f(x)=ax^2+bx+1$ with the conditions $-1\leq f(-1) \leq 1$ and $-2\leq f(2) \leq 2$, determine the range of $f(3)$. | [-7, 3] | 61 | 6 |
math | Given circle $M$: $x^{2}+y^{2}-2x+a=0$.
(1) If $a=-8$, find the equation of the tangent line to circle $M$ passing through point $P(4,5)$;
(2) If $AB$ is any diameter of circle $M$ and $\overrightarrow{OA} \cdot \overrightarrow{OB}=-6$ (where $O$ is the origin), find the radius of circle $M$. | \sqrt{7} | 104 | 5 |
math | The values of $p$, $q$, $r$ and $s$ are 3, 4, 5 and 6, but not necessarily in that order. What is the largest possible value of the sum of the four products $pq$, $qr$, $rs$, and $ps$? | 80 | 63 | 2 |
math | Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\cos x+\sin x)^{2}}
$$ | \ln 2 - \frac{1}{2} | 42 | 12 |
math | Determine the value of $x$ between 0 and 180 such that
\[\tan (150^\circ - x^\circ) = \frac{\sin 150^\circ - \sin x^\circ}{\cos 150^\circ - \cos x^\circ}.\] | 120 | 68 | 3 |
math | Construct a sequence of positive integers by applying three rules beginning at 65. The rules are:
Rule 1: If the integer is less than 15, multiply it by 7.
Rule 2: If the integer is even and greater than 14, divide it by 2.
Rule 3: If the integer is odd and greater than 14, subtract 5 from it.
Find the $50^\text{th}$ t... | 30 | 122 | 2 |
math | A triangle has an area of $40$, one side of length $12$, and the median to that side of length $10$. Let $\theta$ be the acute angle formed by that side and the median. What is $\cos{\theta}$?
A) $\frac{1}{3}$
B) $\frac{2}{3}$
C) $\frac{\sqrt{5}}{3}$
D) $\frac{3}{2}$
E) $\frac{\sqrt{2}}{2}$ | \frac{\sqrt{5}}{3} | 109 | 10 |
math | Given that $α \in (0,π)$, and $\cos α + \sin α = -\frac{1}{5}$, find the value of $\tan 2α$. | -\frac{24}{7} | 40 | 8 |
math | Let $z_1$ and $z_2$ be the complex roots of $z^2 + az + b = 0$, where $a$ and $b$ are complex numbers. In the complex plane, points 0, $z_1$, and $2z_2$ form an isosceles triangle with $0z_1 = 0z_2$. Find $\frac{a^2}{b}$. | 4.5 | 93 | 3 |
math | (1) The equation of the tangent line to the function $y=x^{2}+x-1$ at the point $(1,1)$ is ______________.
(2) Given the ellipse $\dfrac{x^{2}}{36}+ \dfrac{y^{2}}{9}=1$ and a point inside the ellipse $P(4,2)$, the slope of the line containing the chord with $P$ as its midpoint is ______________.
(3) Given the sets $A... | \dfrac{4}{3} | 279 | 7 |
math | The roots of the quadratic equation $x\left(x-2\right)=2-x$ can be found by solving the equation. | x_1=-1, x_2=2 | 27 | 11 |
math | A wooden cube $n$ units on each side is painted red on four of its six faces and then cut into $n^3$ unit cubes. Exactly one-third of the total number of faces of the unit cubes are red. What is $n$? | 2 | 53 | 1 |
math | Given sets $A=\{x|1\leq x<6\}$, $B=\{x|2<x<9\}$.
1. Find $A\cap B$;
2. Find $(C_{R}B) \cup A$. | (-\infty,6)\cup[9,+\infty) | 55 | 15 |
math | Given four positive integers \(a, b, c,\) and \(d\) satisfying the equations \(a^2 = c(d + 20)\) and \(b^2 = c(d - 18)\). Find the value of \(d\). | 180 | 54 | 3 |
math | If the one-variable quadratic equation $x^{2}+2x+m=0$ has two real roots with the same sign, then the range of $m$ is ____. | 0 < m \leq 1 | 37 | 8 |
math | In the coordinate plane, a point is called an integer point if both its x-coordinate and y-coordinate are integers. For any natural number \( n \), the point \( O \) (the origin) is connected to the point \( A_n(n, n+3) \). Let \( f(n) \) denote the number of integer points on the line segment \( OA_n \) excluding the ... | 1326 | 111 | 4 |
math | The Mad Hatter's clock gains 15 minutes per hour, and the March Hare's clock loses 10 minutes per hour. Once, they set their clocks by the Dormouse's clock (which stopped and always shows 12:00) and agreed to meet at 5 o'clock in the evening for a traditional five-o'clock tea. How long will the Mad Hatter wait for the ... | 2 \text{ hours} | 103 | 6 |
math | Let $h(x) = x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $h(x^{14})$ is divided by the polynomial $h(x)$? | 8 | 60 | 1 |
math | A cube with edge length 2 units has vertices $Q_1,Q_2,Q_3,Q_4,Q_1',Q_2',Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$. Vertices $Q_i$ and $Q_i'$ are opposite each other for $1 \leq i \leq 4$. A regular octahedron has one vertex at each of the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q_3}$... | 2\sqrt{2} | 197 | 6 |
math | Let $A$ , $B$ , $C$ , $D$ , $E$ , and $F$ be $6$ points around a circle, listed in clockwise order. We have $AB = 3\sqrt{2}$ , $BC = 3\sqrt{3}$ , $CD = 6\sqrt{6}$ , $DE = 4\sqrt{2}$ , and $EF = 5\sqrt{2}$ . Given that $\overline{AD}$ , $\overline{BE}$ , and $\overline{CF}$ are concurrent, determine ... | 225 | 152 | 3 |
math | In the set of three-digit positive integers, if the digit in the tens place is less than the sum of the digits in the units and hundreds places, then such a number is called a "hump number". For example, $102$ and $546$ are "hump numbers". Among the "hump numbers" formed without repeating digits from the five digits $1... | 30 | 105 | 2 |
math | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $$a^2 + b^2 + \sqrt{2}ab = c^2$$. Find the range of values for $\tan A \tan 2B$. | (0, \frac{1}{2}) | 70 | 10 |
math | Given that the right focus of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ is $F$, and the upper vertex is $A$. If the line $AF$ is tangent to the circle $O$: ${x^2} + {y^2} = \frac{{3{a^2}}}{16}$, calculate the eccentricity of the ellipse. | \frac{\sqrt{3}}{2} | 104 | 10 |
math | A nine-digit number 222222222 is written on a blackboard. The following operation is performed on this number: erase the last digit, multiply the remaining number by 4, then add the erased digit back to the result, and write the number on the blackboard. This process is repeated until a single digit number is obtained.... | 6 | 81 | 1 |
math | What are the coordinates of the point of tangency of the two circles defined by $x^2 - 2x + y^2 - 10y + 17 = 0$ and $x^2 - 8x + y^2 - 10y + 49 = 0$? | (2.5, 5) | 68 | 8 |
math | Taxi driver Li Shifu's operation on the first afternoon of National Day was conducted on a north-south road. If heading south is denoted as "$-$" and heading north is denoted as "$+$", his driving situation that afternoon was as follows: (unit: kilometers, each trip with passengers $)-2,-3,-6,+8,-9,-7,-5,+13.$
$(1)$ W... | 106.08 \text{ yuan} | 303 | 11 |
math | Let $a$ and $b$ be positive real numbers. Consider the following five statements:
1. $\frac{1}{a} < \frac{1}{b}$
2. $a^2 > b^2$
3. $a > b$
4. $a > 0$
5. $b > 0$
What is the maximum number of these statements that can be true for any values of $a$ and $b$? | 4 | 95 | 1 |
math | In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as the diameter, intersects side \( BC \) at point \( K \). It turns out that \( \angle BAK = \angle ACB \). Find the area of triangle \( BOC \). | 21 | 84 | 2 |
math | Given a sequence $\{a_{n}\}$ such that $a_{n}\cdot a_{n+1}\cdot a_{n+2}=-\frac{1}{2}$, $a_{1}=-2$, $a_{2}=\frac{1}{4}$, calculate the maximum value of the product of the first $n$ terms of $\{a_{n}\}$. | 1 | 84 | 1 |
math | The distance from a fixed point $P$ on a plane to two vertices $A$ and $B$ of an equilateral triangle $ABC$ are $AP=2$ and $BP=3$. Determine the maximum possible length of the line segment $PC$. | 5 | 54 | 1 |
math | Given the function $f(x) = \frac{e^x}{e^x - 1}$, where \( x > 0 \):
1. Find the equation of the tangent line to the graph of \( y = f(x) \) at the point \( (\ln 2, f(\ln 2)) \).
2. Let \( g(x) = \frac{k}{x + 1} \), where \( x > 0 \) and \( k \in \mathbb{N}^* \). If \( f(x) > g(x) \) holds true for all values within the... | k = 3 | 140 | 4 |
math | Determine the correct number(s) in the following statements.
\[①2+i > 1+i\]
\[②\] If a number is real, then its imaginary part does not exist.
\[③\] If $z= \frac {1}{i}$, then $z^{3}+1$ corresponds to the point in the first quadrant of the complex plane. | ③ | 81 | 2 |
math | Given the functions $f(x)=2(x+1)$ and $g(x)=x+ \ln x$, points $A$ and $B$ are located on the graphs of $f(x)$ and $g(x)$ respectively, and their y-coordinates are always equal. Calculate the minimum distance between points $A$ and $B$. | \frac{3}{2} | 70 | 7 |
math | The given arithmetic sequence $\{{a_{n}}\}$ satisfies ${a_{3}}+{{a}_{4}}=4$ and ${{a}_{5}}+{{a}_{7}}=6$.
(1) Find the general term formula for $\{{a_{n}}\}$.
(2) Let ${{b}_{n}}=[{{a}_{n}}]$, where $[x]$ denotes the greatest integer not exceeding $x$. Find the sum of the first 10 terms of the sequence $\{{{b}_{n}}\}$. | 24 | 116 | 2 |
math | Triangles $\triangle ABC$ and $\triangle PQR$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,10)$, $C(20,0)$, $P(30,0)$, $Q(30,20)$, $R(50,0)$. A rotation of $n$ degrees counterclockwise around the point $(x,y)$ where $0<n<180$, will transform $\triangle ABC$ to $\triangle PQR$. Find $n+x+y$. | 120 | 117 | 3 |
math | What is the largest prime factor of 4141? | 101 | 13 | 3 |
math | The parabola $y=ax^2+\frac{1}{2}x+c$ intersects the $x$-axis at points $A$ and $B$, and intersects the $y$-axis at point $C$. The line $y=-\frac{1}{2}x-2$ passes through points $A$ and $C$. <br/>$(1)$① Find the values of $a$ and $c$; <br/>② Let the vertex of the parabola $y=ax^2+\frac{1}{2}x+c$ be $D$, find the area of... | \frac{17}{4} | 217 | 8 |
math | Given the set $A=\{x|x=a_0+a_1\times2+a_2\times2^2+a_3\times2^3\}$, where $a_i\in\{0,1,2\}$ ($i=0,1,2,3$), and $a_0\neq0$, find the sum of all elements in set $A$. | 837 | 85 | 3 |
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