Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
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| solved_percentage
float64 0
100
|
|---|---|---|---|
18,000 |
If $8^{2x} = 11$, evaluate $2^{x + 1.5}$.
|
11^{1/6} \cdot 2 \sqrt{2}
| 0 |
18,001 |
Given the equation of the parabola $y^{2}=-4x$, and the equation of the line $l$ as $2x+y-4=0$. There is a moving point $A$ on the parabola. The distance from point $A$ to the $y$-axis is $m$, and the distance from point $A$ to the line $l$ is $n$. Find the minimum value of $m+n$.
|
\frac{6 \sqrt{5}}{5}-1
| 0 |
18,002 |
For an arithmetic sequence \(a_1, a_2, a_3, \dots\), let
\[ S_n = a_1 + a_2 + a_3 + \dots + a_n, \]
and let
\[ T_n = S_1 + S_2 + S_3 + \dots + S_n. \]
If you know the value of \( S_{2023}, \) then you can uniquely determine the value of \( T_n \) for some integer \( n \). What is this integer \( n \)?
|
3034
| 8.59375 |
18,003 |
Coach Randall is preparing a 6-person starting lineup for her soccer team, the Rangers, which has 15 players. Among the players, three are league All-Stars (Tom, Jerry, and Spike), and they are guaranteed to be in the starting lineup. Additionally, the lineup must include at least one goalkeeper, and there is only one goalkeeper available among the remaining players. How many different starting lineups are possible?
|
55
| 53.125 |
18,004 |
Given a cluster of circles ${C_n}:{({x-n})^2}+{({y-2n})^2}={n^2}(n≠0)$, the line $l:y=kx+b$ is a common tangent to them, then $k+b=$____.
|
\frac{3}{4}
| 77.34375 |
18,005 |
The non-negative numbers \(a, b, c\) sum up to 1. Find the maximum possible value of the expression
$$
(a + 3b + 5c) \cdot \left(a + \frac{b}{3} + \frac{c}{5}\right)
$$
|
\frac{9}{5}
| 11.71875 |
18,006 |
If A, B, and C stand in a row, calculate the probability that A and B are adjacent.
|
\frac{2}{3}
| 51.5625 |
18,007 |
Bob chooses a $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$ . For example, if Bob’s string is $0000$ and the stream of bits begins $101000001 \dots$ , then $N = 7$ .
|
30
| 11.71875 |
18,008 |
If a number $x$ is randomly taken from the interval $[0,2π]$, the probability that the value of $\sin x$ is between $0$ and $\frac{\sqrt{3}}{2}$ is _______.
|
\frac{1}{3}
| 65.625 |
18,009 |
If the complex number $a^2 - 1 + (a - 1)i$ (where $i$ is the imaginary unit) is purely imaginary, calculate the real number $a$.
|
-1
| 18.75 |
18,010 |
What is the value of the expression $2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2}}}$?
|
\frac{29}{12}
| 37.5 |
18,011 |
Each of the twelve letters in ``MATHEMATICS'' is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word ``CALM''? Express your answer as a common fraction.
|
\frac{5}{12}
| 33.59375 |
18,012 |
Given points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC$, $AB>r$, and the length of minor arc $BC$ is $r$, calculate the ratio of the length of $AB$ to the length of $BC$.
|
\frac{1}{2}\csc(\frac{1}{4})
| 0 |
18,013 |
A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the smaller square.
|
50\%
| 91.40625 |
18,014 |
Given that 2 students exercised for 0 days, 4 students exercised for 1 day, 2 students exercised for 2 days, 5 students exercised for 3 days, 4 students exercised for 4 days, 7 students exercised for 5 days, 3 students exercised for 6 days, and 2 students exercised for 7 days, find the mean number of days of exercise, rounded to the nearest hundredth.
|
3.66
| 0.78125 |
18,015 |
Select a number from \\(1\\), \\(2\\), \\(3\\), \\(4\\), \\(5\\), \\(6\\), \\(7\\), and calculate the probability of the following events:
\\((1)\\) The selected number is greater than \\(3\\);
\\((2)\\) The selected number is divisible by \\(3\\);
\\((3)\\) The selected number is greater than \\(3\\) or divisible by \\(3\\).
|
\dfrac{5}{7}
| 92.96875 |
18,016 |
Among all the five-digit numbers formed without repeating any of the digits 0, 1, 2, 3, 4, if they are arranged in ascending order, determine the position of the number 12340.
|
10
| 17.1875 |
18,017 |
What is the greatest 4-digit base 7 positive integer that is divisible by 7? (Express your answer in base 7.)
|
6660_7
| 39.84375 |
18,018 |
Given the ellipse E: $\\frac{x^{2}}{4} + \\frac{y^{2}}{2} = 1$, O is the coordinate origin, and a line with slope k intersects ellipse E at points A and B. The midpoint of segment AB is M, and the angle between line OM and AB is θ, with tanθ = 2 √2. Find the value of k.
|
\frac{\sqrt{2}}{2}
| 14.0625 |
18,019 |
A rectangular yard contains three flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, with the parallel sides measuring $10$ meters and $20$ meters. What fraction of the yard is occupied by the flower beds?
A) $\frac{1}{4}$
B) $\frac{1}{6}$
C) $\frac{1}{8}$
D) $\frac{1}{10}$
E) $\frac{1}{3}$
|
\frac{1}{6}
| 15.625 |
18,020 |
Let $p(n)$ denote the product of decimal digits of a positive integer $n$ . Computer the sum $p(1)+p(2)+\ldots+p(2001)$ .
|
184320
| 98.4375 |
18,021 |
In the plane rectangular coordinate system xOy, the origin is taken as the pole, the positive semi-axis of the x-axis is taken as the polar axis, and the polar coordinate system is established. The same unit length is adopted in both coordinate systems. The polar coordinate equation of the curve C is ρ = 4$\sqrt {2}$cos(θ - $\frac {π}{4}$), and the parameter equation of the line is $\begin{cases} x=2+ \frac {1}{2}t \\ y=1+ \frac { \sqrt {3}}{2}t\end{cases}$ (t is the parameter).
1. Find the rectangular coordinate equation of curve C and the general equation of line l.
2. Suppose point P(2, 1), if line l intersects curve C at points A and B, find the value of $| \frac {1}{|PA|} - \frac {1}{|PB|} |$.
|
\frac { \sqrt {31}}{7}
| 0 |
18,022 |
Given two spherical balls of different sizes placed in two corners of a rectangular room, where each ball touches two walls and the floor, and there is a point on each ball such that the distance from the two walls it touches to that point is 5 inches and the distance from the floor to that point is 10 inches, find the sum of the diameters of the two balls.
|
40
| 63.28125 |
18,023 |
Find the shortest distance from a point on the curve $y=x^{2}-\ln x$ to the line $x-y-2=0$.
|
\sqrt{2}
| 82.03125 |
18,024 |
The volume of a certain rectangular solid is $8 \text{ cm}^3$, its total surface area is $32 \text{ cm}^2$, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
|
32
| 82.8125 |
18,025 |
Given the system of equations $\begin{cases} x - 2y = z - 2u \\ 2yz = ux \end{cases}$, for every set of positive real number solutions $\{x, y, z, u\}$ where $z \geq y$, there exists a positive real number $M$ such that $M \leq \frac{z}{y}$. Find the maximum value of $M$.
|
6 + 4\sqrt{2}
| 22.65625 |
18,026 |
Let $ n$ be a positive integer and $ [ \ n ] = a.$ Find the largest integer $ n$ such that the following two conditions are satisfied:
$ (1)$ $ n$ is not a perfect square;
$ (2)$ $ a^{3}$ divides $ n^{2}$ .
|
24
| 66.40625 |
18,027 |
The perimeter of triangle \( \mathrm{ABC} \) is 1. A circle touches side \( \mathrm{AB} \) at point \( P \) and the extension of side \( \mathrm{AC} \) at point \( Q \). A line passing through the midpoints of sides \( \mathrm{AB} \) and \( \mathrm{AC} \) intersects the circumcircle of triangle \( \mathrm{APQ} \) at points \( X \) and \( Y \). Find the length of segment \( X Y \).
|
\frac{1}{2}
| 43.75 |
18,028 |
Given that the function $f(x) = e^x + \frac{a}{e^x}$ has a derivative $y = f'(x)$ that is an odd function and the slope of a tangent line to the curve $y = f(x)$ is $\frac{3}{2}$, determine the abscissa of the tangent point.
|
\ln 2
| 1.5625 |
18,029 |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?
|
2\sqrt{10}
| 30.46875 |
18,030 |
Steve guesses randomly on a 20-question multiple-choice test where each question has two choices. What is the probability that he gets at least half of the questions correct? Express your answer as a common fraction.
|
\frac{1}{2}
| 22.65625 |
18,031 |
A school plans to purchase volleyball and basketball in one go. The price of each basketball is $30$ yuan more expensive than a volleyball. It costs a total of $340$ yuan to buy $2$ volleyballs and $3$ basketballs.<br/>$(1)$ Find the price of each volleyball and basketball.<br/>$(2)$ If the school purchases a total of $60$ volleyballs and basketballs in one go, with a total cost not exceeding $3800$ yuan, and the number of volleyballs purchased is less than $39. Let the number of volleyballs be $m$ and the total cost be $y$ yuan.<br/>① Find the functional relationship of $y$ with respect to $m$ and find all possible values of $m$;<br/>② In what way should the school purchase to minimize the cost? What is the minimum cost?
|
3660
| 78.125 |
18,032 |
Given a tetrahedron O-ABC, where $\angle BOC=90^\circ$, $OA \perpendicular$ plane BOC, and $AB= \sqrt{10}$, $BC= \sqrt{13}$, $AC= \sqrt{5}$. Points O, A, B, and C are all on the surface of sphere S. Find the surface area of sphere S.
|
14\pi
| 33.59375 |
18,033 |
Given the common difference of the arithmetic sequence $\{a_n\}$ is $d$, and the sum of the first $n$ terms is $S_n$, with $a_1=d=1$, find the minimum value of $\frac{S_n+8}{a_n}$.
|
\frac{9}{2}
| 56.25 |
18,034 |
How many integer solutions \((x, y, z)\) are there to the equation \(xyz = 2008\)?
|
120
| 18.75 |
18,035 |
Four points are on a line segment.
If \( A B : B C = 1 : 2 \) and \( B C : C D = 8 : 5 \), then \( A B : B D \) equals
|
4 : 13
| 85.9375 |
18,036 |
Points A and B are on a circle of radius 7 and AB = 8. Point C is the midpoint of the minor arc AB. What is the length of the line segment AC?
|
\sqrt{98 - 14\sqrt{33}}
| 28.90625 |
18,037 |
Given the function f(x) = 3x^2 + 5x - 2, find the minimum value of the function f(x) in the interval [-2, -1].
|
-4
| 45.3125 |
18,038 |
Square $EFGH$ has sides of length 4. A point $P$ on $EH$ is such that line segments $FP$ and $GP$ divide the square’s area into four equal parts. Find the length of segment $FP$.
A) $2\sqrt{3}$
B) $3$
C) $2\sqrt{5}$
D) $4$
E) $2\sqrt{7}$
|
2\sqrt{5}
| 96.09375 |
18,039 |
Given an ellipse $C$ with its left and right foci at $F_{1}(-\sqrt{3},0)$ and $F_{2}(\sqrt{3},0)$, respectively, and the ellipse passes through the point $(-1, \frac{\sqrt{3}}{2})$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) Given a fixed point $A(1, \frac{1}{2})$, a line $l$ passing through the origin $O$ intersects the curve $C$ at points $M$ and $N$. Find the maximum area of $\triangle MAN$.
|
\sqrt{2}
| 10.9375 |
18,040 |
The least positive integer with exactly \(2023\) distinct positive divisors can be written in the form \(m \cdot 10^k\), where \(m\) and \(k\) are integers and \(10\) is not a divisor of \(m\). What is \(m+k?\)
A) 999846
B) 999847
C) 999848
D) 999849
E) 999850
|
999846
| 10.9375 |
18,041 |
A $50$-gon \(Q_1\) is drawn in the Cartesian plane where the sum of the \(x\)-coordinates of the \(50\) vertices equals \(150\). A constant scaling factor \(k = 1.5\) applies only to the \(x\)-coordinates of \(Q_1\). The midpoints of the sides of \(Q_1\) form a second $50$-gon, \(Q_2\), and the midpoints of the sides of \(Q_2\) form a third $50$-gon, \(Q_3\). Find the sum of the \(x\)-coordinates of the vertices of \(Q_3\).
|
225
| 50.78125 |
18,042 |
On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights always told the truth, liars always lied. Each follower answered the same as the majority of the preceding respondents, and if the "Yes" and "No" answers were split equally, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island residents?
|
1009
| 49.21875 |
18,043 |
Find the absolute value of the difference of single-digit integers $C$ and $D$ such that in base 8:
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & D & D & C_8 \\
-& & & \mathbf{6} & \mathbf{3} & D_8 \\
\cline{2-6}
& & C & \mathbf{3} & \mathbf{1} & \mathbf{5_8}
\end{array} $$
Express your answer in base $8$.
|
5_8
| 3.125 |
18,044 |
In the polar coordinate system, the polar equation of curve $C$ is $\rho =6\sin \theta$, and the polar coordinates of point $P$ are $(\sqrt{2},\frac{\pi }{4})$. Taking the pole as the origin of coordinates and the positive half-axis of the $x$-axis as the polar axis, a plane rectangular coordinate system is established.
(1) Find the rectangular coordinate equation of curve $C$ and the rectangular coordinates of point $P$;
(2) A line $l$ passing through point $P$ intersects curve $C$ at points $A$ and $B$. If $|PA|=2|PB|$, find the value of $|AB|$.
|
3 \sqrt{2}
| 17.1875 |
18,045 |
Given that the wavelength of each light quantum is approximately $688$ nanometers and $1$ nanometer is equal to $0.000000001$ meters, express the wavelength of each light quantum in scientific notation.
|
6.88\times 10^{-7}
| 69.53125 |
18,046 |
Determine how many triangles can be formed using the vertices of a regular hexadecagon (a 16-sided polygon).
|
560
| 82.03125 |
18,047 |
Given a hyperbola with eccentricity $e$ and an ellipse with eccentricity $\frac{\sqrt{2}}{2}$ share the same foci $F_{1}$ and $F_{2}$. If $P$ is a common point of the two curves and $\angle F_{1}PF_{2}=60^{\circ}$, then $e=$ ______.
|
\frac{\sqrt{6}}{2}
| 46.09375 |
18,048 |
Monica decides to tile the floor of her 15-foot by 20-foot dining room. She plans to create a two-foot-wide border using one-foot by one-foot square tiles around the edges of the room and fill in the rest of the floor with three-foot by three-foot square tiles. Calculate the total number of tiles she will use.
|
144
| 10.9375 |
18,049 |
Given the function $f(x)=e^{x}+ \frac {2x-5}{x^{2}+1}$, determine the value of the real number $m$ such that the tangent line to the graph of the function at the point $(0,f(0))$ is perpendicular to the line $x-my+4=0$.
|
-3
| 70.3125 |
18,050 |
The entire contents of the jug can exactly fill 9 small glasses and 4 large glasses of juice, and also fill 6 small glasses and 6 large glasses. If the entire contents of the jug is used to fill only large glasses, determine the maximum number of large glasses that can be filled.
|
10
| 99.21875 |
18,051 |
Given that point P is on the left branch of the hyperbola $x^2-y^2=4$, and $F_1$, $F_2$ are the left and right foci of the hyperbola, respectively, then $|PF_1|-|PF_2|$ equals to __.
|
-4
| 66.40625 |
18,052 |
Given the function $y=\cos(2x- \frac{\pi}{6})$, find the horizontal shift required to transform the graph of $y=\sin 2x$ into the graph of $y=\cos(2x- \frac{\pi}{6})$.
|
\frac{\pi}{6}
| 41.40625 |
18,053 |
On a rectangular sheet of paper, a picture is drawn in the shape of a "cross" formed by two rectangles $ABCD$ and $EFGH$, where the sides are parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.
|
52.5
| 64.84375 |
18,054 |
When randomly selecting three line segments from the lengths 1, 3, 5, 7, and 9, calculate the probability that the selected line segments cannot form a triangle.
|
\frac{7}{10}
| 30.46875 |
18,055 |
Given that $a > 0$, if the area enclosed by the curve $y=\sqrt{x}$ and the lines $x=a$, $y=0$ is equal to $a$, find the value of $a$.
|
\frac{9}{4}
| 84.375 |
18,056 |
Extend the definition of the binomial coefficient to $C_x^m = \frac{x(x-1)\dots(x-m+1)}{m!}$ where $x\in\mathbb{R}$ and $m$ is a positive integer, with $C_x^0=1$. This is a generalization of the binomial coefficient $C_n^m$ (where $n$ and $m$ are positive integers and $m\leq n$).
1. Calculate the value of $C_{-15}^3$.
2. Let $x > 0$. For which value of $x$ does $\frac{C_x^3}{(C_x^1)^2}$ attain its minimum value?
3. Can the two properties of binomial coefficients $C_n^m = C_n^{n-m}$ (Property 1) and $C_n^m + C_n^{m-1} = C_{n+1}^m$ (Property 2) be extended to $C_x^m$ where $x\in\mathbb{R}$ and $m$ is a positive integer? If so, write the extended form and provide a proof. If not, explain why.
|
\sqrt{2}
| 66.40625 |
18,057 |
Let $\lfloor x \rfloor$ represent the integer part of the real number $x$, and $\{x\}$ represent the fractional part of the real number $x$, e.g., $\lfloor 3.1 \rfloor = 3, \{3.1\} = 0.1$. It is known that all terms of the sequence $\{a\_n\}$ are positive, $a\_1 = \sqrt{2}$, and $a\_{n+1} = \lfloor a\_n \rfloor + \frac{1}{\{a\_n\}}$. Find $a\_{2017}$.
|
4032 + \sqrt{2}
| 15.625 |
18,058 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a \neq b$, $c= \sqrt{3}$, and $\cos^2A - \cos^2B = \sqrt{3}\sin A\cos A - \sqrt{3}\sin B\cos B$.
$(I)$ Find the magnitude of angle $C$.
$(II)$ If $\sin A= \frac{4}{5}$, find the area of $\triangle ABC$.
|
\frac{8\sqrt{3}+18}{25}
| 51.5625 |
18,059 |
Given the function $f(x) = \frac {1}{2}x^{2} + x - 2\ln{x}$ ($x > 0$):
(1) Find the intervals of monotonicity for $f(x)$.
(2) Find the extreme values of the function $f(x)$.
|
\frac {3}{2}
| 53.125 |
18,060 |
Five friends — Sarah, Lily, Emma, Nora, and Kate — performed in a theater as quartets, with one friend sitting out each time. Nora performed in 10 performances, which was the most among all, and Sarah performed in 6 performances, which was the fewest among all. Calculate the total number of performances.
|
10
| 23.4375 |
18,061 |
Given that the line $l$ passes through the point $P(3,4)$<br/>(1) Its intercept on the $y$-axis is twice the intercept on the $x$-axis, find the equation of the line $l$.<br/>(2) If the line $l$ intersects the positive $x$-axis and positive $y$-axis at points $A$ and $B$ respectively, find the minimum area of $\triangle AOB$.
|
24
| 68.75 |
18,062 |
A beam of light is emitted from point $P(1,2,3)$, reflected by the $Oxy$ plane, and then absorbed at point $Q(4,4,4)$. The distance traveled by the light beam is ______.
|
\sqrt{62}
| 86.71875 |
18,063 |
Remove all perfect squares from the sequence of positive integers $1, 2, 3, \ldots$ to obtain a new sequence. What is the 2003rd term of this new sequence?
|
2048
| 91.40625 |
18,064 |
9 seeds are divided among three pits, labeled A, B, and C, with each pit containing 3 seeds. Each seed has a 0.5 probability of germinating. If at least one seed in a pit germinates, then that pit does not need to be replanted; if no seeds in a pit germinate, then that pit needs to be replanted.
(Ⅰ) Calculate the probability that pit A does not need to be replanted;
(Ⅱ) Calculate the probability that exactly one of the three pits does not need to be replanted;
(Ⅲ) Calculate the probability that at least one pit needs to be replanted. (Round to three decimal places).
|
0.330
| 71.875 |
18,065 |
Given that $x$ and $y$ are positive numbers satisfying the equation $xy = \frac{x-y}{x+3y}$, find the maximum value of $y$.
|
\frac{1}{3}
| 36.71875 |
18,066 |
Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there?
|
112
| 31.25 |
18,067 |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\]
|
1001
| 65.625 |
18,068 |
Four equal circles with diameter $6$ are arranged such that three circles are tangent to one side of a rectangle and the fourth circle is tangent to the opposite side. All circles are tangent to at least one other circle with their centers forming a straight line that is parallel to the sides of the rectangle they touch. The length of the rectangle is twice its width. Calculate the area of the rectangle.
|
648
| 10.15625 |
18,069 |
What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$ , $a$ , and $a + 1$ has integer area?
|
52
| 70.3125 |
18,070 |
The number $1025$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$?
|
27
| 0 |
18,071 |
(1) Simplify: $\dfrac{\sin(\pi -\alpha)\cos(\pi +\alpha)\sin(\dfrac{\pi}{2}+\alpha)}{\sin(-\alpha)\sin(\dfrac{3\pi}{2}+\alpha)}$.
(2) Given $\alpha \in (\dfrac{\pi}{2}, \pi)$, and $\sin(\pi -\alpha) + \cos \alpha = \dfrac{7}{13}$, find $\tan \alpha$.
|
-\dfrac{12}{5}
| 69.53125 |
18,072 |
Sequentially throw a die and record the numbers obtained as $a$ and $b$. Perform the following operations:
```
INPUT "a, b="; a, b
IF a >= b THEN
y = a - b
ELSE
y = b - a
END IF
PRINT y
END
```
(1) If $a=3$, $b=6$, find the value of $y$ output by the computer after running the program.
(2) If "the output value of $y$ is 2" is event A, calculate the probability of event A occurring.
|
\frac{2}{9}
| 87.5 |
18,073 |
Given: The curve $C$ has the polar coordinate equation: $ρ=a\cos θ (a>0)$, and the line $l$ has the parametric equations: $\begin{cases}x=1+\frac{\sqrt{2}}{2}t\\y=\frac{\sqrt{2}}{2}t\end{cases}$ ($t$ is the parameter)
1. Find the Cartesian equation of the curve $C$ and the line $l$;
2. If the line $l$ is tangent to the curve $C$, find the value of $a$.
|
a=2(\sqrt{2}-1)
| 50 |
18,074 |
Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \]
|
14
| 71.875 |
18,075 |
Given $\sin(\alpha - \beta) = \frac{1}{3}$ and $\cos \alpha \sin \beta = \frac{1}{6}$, calculate the value of $\cos(2\alpha + 2\beta)$.
|
\frac{1}{9}
| 16.40625 |
18,076 |
Given the vectors $\overrightarrow{m}=(\cos x,\sin x)$ and $\overrightarrow{n}=(2 \sqrt {2}+\sin x,2 \sqrt {2}-\cos x)$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$, where $x\in R$.
(I) Find the maximum value of the function $f(x)$;
(II) If $x\in(-\frac {3π}{2},-π)$ and $f(x)=1$, find the value of $\cos (x+\frac {5π}{12})$.
|
-\frac {3 \sqrt {5}+1}{8}
| 0 |
18,077 |
A conservatory houses five pairs of different animals, one male and one female of each type. The feeder must alternate between feeding a male and a female each time. If the feeder begins by feeding a female hippopotamus, how many ways can the feeder complete feeding all the animals?
|
2880
| 17.96875 |
18,078 |
Given that $a$ and $b$ are positive real numbers satisfying $9a^{2}+b^{2}=1$, find the maximum value of $\frac{ab}{3a+b}$.
|
\frac{\sqrt{2}}{12}
| 42.96875 |
18,079 |
A workshop produces a type of instrument with a fixed cost of $7500. Each additional unit of the instrument requires an additional investment of $100. The total revenue function is given by: $H(x) = \begin{cases} 400x - x^2, (0 \leq x \leq 200) \\ 40000, (x > 200) \end{cases}$, where $x$ is the monthly production volume of the instrument. (Note: Profit = Total Revenue - Total Cost)
(I) Express the profit as a function of the monthly production volume $x$;
(II) What monthly production volume will yield the maximum profit for the workshop? What is the maximum profit?
|
15000
| 88.28125 |
18,080 |
Wang Lei and her older sister walk from home to the gym to play badminton. It is known that the older sister walks 20 meters more per minute than Wang Lei. After 25 minutes, the older sister reaches the gym, and then realizes she forgot the racket. She immediately returns along the same route to get the racket and meets Wang Lei at a point 300 meters away from the gym. Determine the distance between Wang Lei's home and the gym in meters.
|
1500
| 19.53125 |
18,081 |
Let $\alpha$ be a root of $x^6-x-1$ , and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$ . It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$ . Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$ .
|
727
| 19.53125 |
18,082 |
Determine the area of the circle described by the equation \(3x^2 + 3y^2 - 15x + 9y + 27 = 0\) in terms of \(\pi\).
|
\frac{\pi}{2}
| 53.125 |
18,083 |
Find the last two digits of the sum $$6! + 1 + 12! + 1 + 18! + 1 + \cdots + 96! + 1.$$
|
36
| 32.03125 |
18,084 |
Three boys and two girls are to stand in a row according to the following requirements. How many different arrangements are there? (Answer with numbers)
(Ⅰ) The two girls stand next to each other;
(Ⅱ) Girls cannot stand at the ends;
(Ⅲ) Girls are arranged from left to right from tallest to shortest;
(Ⅳ) Girl A cannot stand at the left end, and Girl B cannot stand at the right end.
|
78
| 98.4375 |
18,085 |
Given triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, $c\cos A= \frac{4}{b}$, and the area of $\triangle ABC$, $S \geq 2$.
(1) Determine the range of possible values for angle $A$.
(2) Find the maximum value of the function $f(x) = \cos^2 A + \sqrt{3}\sin^2\left(\frac{\pi}{2}+ \frac{A}{2}\right) - \frac{\sqrt{3}}{2}$.
|
\frac{1}{2} + \frac{\sqrt{6}}{4}
| 3.125 |
18,086 |
Express $537_8 + 1C2E_{16}$ as a base 10 integer, where $C$ and $E$ denote the hexadecimal digits with values 12 and 14 respectively.
|
7565
| 53.90625 |
18,087 |
A line $l$ is drawn through the fixed point $M(4,0)$, intersecting the parabola $y^{2}=4x$ at points $A$ and $B$. $F$ is the focus of the parabola. Calculate the minimum area of the triangle $\triangle AFB$.
|
12
| 14.84375 |
18,088 |
A digital watch now displays time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits when it displays time in this format, where the hour ranges from 00 to 23 and the minutes range from 00 to 59.
|
24
| 2.34375 |
18,089 |
In a new diagram below, we have $\cos \angle XPY = \frac{3}{5}$. A point Z is placed such that $\angle XPZ$ is a right angle. What is $\sin \angle YPZ$?
[asy]
pair X, P, Y, Z;
P = (0,0);
X = Rotate(-aCos(3/5))*(-2,0);
Y = (2,0);
Z = Rotate(-90)*(2,0);
dot("$Z$", Z, S);
dot("$Y$", Y, S);
dot("$X$", X, W);
dot("$P$", P, S);
draw(X--P--Y--Z--cycle);
[/asy]
|
\frac{3}{5}
| 53.125 |
18,090 |
Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of school A winning in the three events are 0.5, 0.4, and 0.8, respectively, and the results of each event are independent.<br/>$(1)$ Find the probability of school A winning the championship;<br/>$(2)$ Let $X$ represent the total score of school B, find the distribution table and expectation of $X$.
|
13
| 17.1875 |
18,091 |
Using the Horner's method, calculate the value of the polynomial $f(x)=2x^{4}-x^{3}+3x^{2}+7$ at $x=3$, and find the corresponding value of $v_{3}$.
|
54
| 78.125 |
18,092 |
Quadrilateral $ALEX,$ pictured below (but not necessarily to scale!)
can be inscribed in a circle; with $\angle LAX = 20^{\circ}$ and $\angle AXE = 100^{\circ}:$
|
80
| 35.9375 |
18,093 |
Given the set $A=\{x|0<x+a\leq5\}$, and the set $B=\{x|-\frac{1}{2}\leq x<6\}$
(Ⅰ) If $A\subseteq B$, find the range of the real number $a$;
(Ⅱ) If $A\cap B$ is a singleton set, find the value of the real number $a$.
|
\frac {11}{2}
| 36.71875 |
18,094 |
A set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a+b$.
|
-1
| 66.40625 |
18,095 |
Let $S = \{r_1, r_2, \ldots, r_n\} \subseteq \{1, 2, 3, \ldots, 50\}$, and the sum of any two numbers in $S$ cannot be divisible by 7. The maximum value of $n$ is ____.
|
23
| 85.9375 |
18,096 |
To obtain the graph of the function $$y=2\sin(x+ \frac {\pi}{6})\cos(x+ \frac {\pi}{6})$$, determine the horizontal shift required to transform the graph of the function $y=\sin 2x$.
|
\frac {\pi}{6}
| 29.6875 |
18,097 |
March 12th is Tree Planting Day. A school organizes 65 high school students and their parents to participate in the "Plant a tree, green a land" tree planting activity as a family unit. The activity divides the 65 families into two groups, A and B. Group A is responsible for planting 150 silver poplar seedlings, while group B is responsible for planting 160 purple plum seedlings. According to previous statistics, it takes $\frac{2}{5}h$ for a family to plant a silver poplar seedling and $\frac{3}{5}h$ to plant a purple plum seedling. Assuming that groups A and B start planting at the same time, in order to minimize the duration of the tree planting activity, the number of families in group A is \_\_\_\_\_\_, and the duration of the activity is \_\_\_\_\_\_ $(h)$.
|
\frac{12}{5}
| 0.78125 |
18,098 |
Given that the vertices of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ lie on the surface of a sphere $O$, the lateral area of the regular triangular prism $ABC-A_{1}B_{1}C_{1}$ is $6$, and the base area is $\sqrt{3}$, calculate the surface area of the sphere $O$.
|
\frac{19\pi}{3}
| 88.28125 |
18,099 |
Four planes divide space into $n$ parts at most. Calculate $n$.
|
15
| 57.8125 |
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