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40.3k
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5.15k
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100
|
|---|---|---|---|
18,200 |
Given an arithmetic sequence $\{a_n\}$, it is known that $\frac {a_{11}}{a_{10}} + 1 < 0$, and the sum of the first $n$ terms of the sequence, $S_n$, has a maximum value. Find the maximum value of $n$ for which $S_n > 0$.
|
19
| 71.09375 |
18,201 |
How many pages does the book "Folk Tales" have if from the first page to the last page, a total of 357 digits were used to number the pages?
|
155
| 72.65625 |
18,202 |
Given $$\overrightarrow {m} = (\sin \omega x + \cos \omega x, \sqrt {3} \cos \omega x)$$, $$\overrightarrow {n} = (\cos \omega x - \sin \omega x, 2\sin \omega x)$$ ($\omega > 0$), and the function $f(x) = \overrightarrow {m} \cdot \overrightarrow {n}$, if the distance between two adjacent axes of symmetry of $f(x)$ is not less than $\frac {\pi}{2}$.
(1) Find the range of values for $\omega$;
(2) In $\triangle ABC$, where $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ respectively, and $a=2$, when $\omega$ is at its maximum, $f(A) = 1$, find the maximum area of $\triangle ABC$.
|
\sqrt {3}
| 0 |
18,203 |
Given that $[x]$ represents the greatest integer less than or equal to $x$, if
$$
[x+0.1]+[x+0.2]+[x+0.3]+[x+0.4]+[x+0.5]+[x+0.6]+[x+0.7]+[x+0.8]+[x+0.9]=104
$$
then the smallest value of $x$ is ( ).
|
11.5
| 42.96875 |
18,204 |
A certain machine has a display showing an integer $x$, and two buttons, $\mathrm{A}$ and $\mathrm{B}$. When button $\mathrm{A}$ is pressed, the number $x$ on the display is replaced by $2x + 1$. When button $\mathrm{B}$ is pressed, the number $x$ on the display is replaced by $3x - 1$. What is the largest two-digit number that can be obtained by pressing some sequence of buttons $\mathrm{A}$ and $\mathrm{B}$ starting from the number 5 on the display?
|
95
| 3.125 |
18,205 |
Given $m=(\sqrt{3}\sin \omega x,\cos \omega x)$, $n=(\cos \omega x,-\cos \omega x)$ ($\omega > 0$, $x\in\mathbb{R}$), $f(x)=m\cdot n-\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\frac{\pi}{2}$.
$(1)$ Find the intervals of monotonic increase for the function $f(x)$;
$(2)$ If in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b=\sqrt{7}$, $f(B)=0$, $\sin A=3\sin C$, find the values of $a$, $c$ and the area of $\triangle ABC$.
|
\frac{3\sqrt{3}}{4}
| 22.65625 |
18,206 |
Given the parametric equation of line $l$ as $\begin{cases} x=m+\frac{\sqrt{2}}{2}t \\ y=\frac{\sqrt{2}}{2}t \end{cases} (t \text{ is the parameter})$, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C$ is $\rho^{2}\cos ^{2}\theta+3\rho^{2}\sin ^{2}\theta=12$, and the left focus $F$ of the curve $C$ is on line $l$.
(1) If line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $|FA| \cdot |FB|$;
(2) Find the maximum value of the perimeter of the inscribed rectangle of curve $C$.
|
16
| 71.09375 |
18,207 |
Add $A85_{12}$ and $2B4_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary.
|
1179_{12}
| 79.6875 |
18,208 |
Expand and find the sum of the coefficients of the expression $-(2x - 5)(4x + 3(2x - 5))$.
|
-15
| 70.3125 |
18,209 |
Given that $S_{n}$ is the sum of the first $n$ terms of a positive geometric sequence $\{a_{n}\}$, and $S_{3}=10$, find the minimum value of $2S_{9}-3S_{6}+S_{3}$.
|
-\frac{5}{4}
| 4.6875 |
18,210 |
For all positive integers $n$ greater than 2, the greatest common divisor of $n^5 - 5n^3 + 4n$ is.
|
120
| 58.59375 |
18,211 |
Consider a triangle $DEF$ where the angles of the triangle satisfy
\[ \cos 3D + \cos 3E + \cos 3F = 1. \]
Two sides of this triangle have lengths 12 and 14. Find the maximum possible length of the third side.
|
2\sqrt{127}
| 1.5625 |
18,212 |
The sum of three numbers \(x, y,\) and \(z\) is 120. If we decrease \(x\) by 10, we get the value \(M\). If we increase \(y\) by 10, we also get the value \(M\). If we multiply \(z\) by 10, we also get the value \(M\). What is the value of \(M\)?
|
\frac{400}{7}
| 98.4375 |
18,213 |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect cube factor other than one?
|
15
| 10.15625 |
18,214 |
Given the origin $O$ of a Cartesian coordinate system as the pole and the non-negative half-axis of the $x$-axis as the initial line, a polar coordinate system is established. The polar equation of curve $C$ is $\rho\sin^2\theta=4\cos\theta$.
$(1)$ Find the Cartesian equation of curve $C$;
$(2)$ The parametric equation of line $l$ is $\begin{cases} x=1+ \frac{2\sqrt{5}}{5}t \\ y=1+ \frac{\sqrt{5}}{5}t \end{cases}$ ($t$ is the parameter), let point $P(1,1)$, and line $l$ intersects with curve $C$ at points $A$, $B$. Calculate the value of $|PA|+|PB|$.
|
4\sqrt{15}
| 56.25 |
18,215 |
In a 6 by 6 grid, each of the 36 small squares measures 1 cm by 1 cm and is shaded. Six unshaded circles are placed on top of the grid. One large circle is centered at the center of the grid with a radius equal to 1.5 cm, and five smaller circles each with a radius of 0.5 cm are placed at the center of the outer border of the grid. The area of the visible shaded region can be written in the form $C-D\pi$ square cm. What is the value of $C+D$?
|
39.5
| 89.84375 |
18,216 |
In product inspection, the method of sampling inspection is often used. Now, 4 products are randomly selected from 100 products (among which there are 3 defective products) for inspection. The number of ways to exactly select 2 defective products is ____. (Answer with a number)
|
13968
| 81.25 |
18,217 |
Among 5 table tennis players, there are 2 veteran players and 3 new players. Now, 3 players are to be selected to form a team and arranged in positions 1, 2, and 3 to participate in a team competition. The arrangement must include at least 1 veteran player, and among players 1 and 2, there must be at least 1 new player. How many such arrangements are there? (Answer with a number).
|
48
| 55.46875 |
18,218 |
Given 3 zeros and 2 ones are randomly arranged in a row, calculate the probability that the 2 ones are not adjacent.
|
\frac{3}{5}
| 82.03125 |
18,219 |
A cone and a sphere are placed inside a cylinder without overlapping. All bases are aligned and rest on the same flat surface. The cylinder has a radius of 6 cm and a height of 15 cm. The cone has a radius of 6 cm and a height of 7.5 cm. The sphere has a radius of 6 cm. Find the ratio of the volume of the cone to the total volume of the cylinder and the sphere.
|
\frac{15}{138}
| 0 |
18,220 |
A $4 \times 4$ square is divided into $16$ unit squares. Each unit square is painted either white or black, each with a probability of $\frac{1}{2}$, independently. The square is then rotated $180^\circ$ about its center. After rotation, any white square that occupies a position previously held by a black square is repainted black; other squares retain their original color. Determine the probability that the entire grid is black after this process.
|
\frac{6561}{65536}
| 21.09375 |
18,221 |
A certain school conducted physical fitness tests on the freshmen to understand their physical health conditions. Now, $20$ students are randomly selected from both male and female students as samples. Their test data is organized in the table below. It is defined that data $\geqslant 60$ indicates a qualified physical health condition.
| Level | Data Range | Number of Male Students | Number of Female Students |
|---------|--------------|-------------------------|---------------------------|
| Excellent | $[90,100]$ | $4$ | $6$ |
| Good | $[80,90)$ | $6$ | $6$ |
| Pass | $[60,80)$ | $7$ | $6$ |
| Fail | Below $60$ | $3$ | $2$ |
$(Ⅰ)$ Estimate the probability that the physical health level of the freshmen in this school is qualified.
$(Ⅱ)$ From the students in the sample with an excellent level, $3$ students are randomly selected for retesting. Let the number of female students selected be $X$. Find the distribution table and the expected value of $X$.
$(Ⅲ)$ Randomly select $2$ male students and $1$ female student from all male and female students in the school, respectively. Estimate the probability that exactly $2$ of these $3$ students have an excellent health level.
|
\frac{31}{250}
| 5.46875 |
18,222 |
Given real numbers $a_1$, $a_2$, $a_3$ are not all zero, and positive numbers $x$, $y$ satisfy $x+y=2$. Let the maximum value of $$\frac {xa_{1}a_{2}+ya_{2}a_{3}}{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$$ be $M=f(x,y)$, then the minimum value of $M$ is \_\_\_\_\_\_.
|
\frac { \sqrt {2}}{2}
| 0 |
18,223 |
Let $\triangle PQR$ be a right triangle with $Q$ as the right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If the area of $\triangle PQR$ is $200$ and $PR = 40$, find the length of $QS$.
|
10
| 50 |
18,224 |
In $\triangle ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. The lengths of sides $AC$ and $BC$ are 6 and 7 respectively. Calculate the length of side $AB$.
|
\sqrt{17}
| 33.59375 |
18,225 |
Calculate the value of $V_3$ for the polynomial $f(x) = 2x^6 + 5x^5 + 6x^4 + 23x^3 - 8x^2 + 10x - 3$ at $x = -4$ using the Horner's method.
|
-49
| 50.78125 |
18,226 |
In triangle \( \triangle ABC \), the sides opposite to the angles \( \angle A \), \( \angle B \), and \( \angle C \) are denoted as \( a \), \( b \), and \( c \) respectively. If \( b^{2}=a^{2}+c^{2}-ac \), and \( c-a \) is equal to the height \( h \) from vertex \( A \) to side \( AC \), then find \( \sin \frac{C-A}{2} \).
|
\frac{1}{2}
| 50 |
18,227 |
Convert $1729_{10}$ to base 6.
|
12001_6
| 11.71875 |
18,228 |
Given the points $(7, -9)$ and $(1, 7)$ as the endpoints of a diameter of a circle, calculate the sum of the coordinates of the center of the circle, and also determine the radius of the circle.
|
\sqrt{73}
| 58.59375 |
18,229 |
Machines A, B, and C operate independently and are supervised by a single worker, who cannot attend to two or more machines simultaneously. Given that the probabilities of these machines operating without needing supervision are 0.9, 0.8, and 0.85 respectively, calculate the probability that during a certain period,
(1) all three machines operate without needing supervision;
(2) the worker is unable to supervise effectively, leading to a shutdown.
|
0.059
| 7.8125 |
18,230 |
How many different ways can 6 different books be distributed according to the following requirements?
(1) Among three people, A, B, and C, one person gets 1 book, another gets 2 books, and the last one gets 3 books;
(2) The books are evenly distributed to A, B, and C, with each person getting 2 books;
(3) The books are divided into three parts, with one part getting 4 books and the other two parts getting 1 book each;
(4) A gets 1 book, B gets 1 book, and C gets 4 books.
|
30
| 85.15625 |
18,231 |
Liam builds two snowmen using snowballs of radii 4 inches, 6 inches, and 8 inches for the first snowman. For the second snowman, he uses snowballs that are 75% of the size of each corresponding ball in the first snowman. Assuming all snowballs are perfectly spherical, what is the total volume of snow used in cubic inches? Express your answer in terms of $\pi$.
|
\frac{4504.5}{3}\pi
| 0 |
18,232 |
Two people are selected from each of the two groups. Group A has 5 boys and 3 girls, and Group B has 6 boys and 2 girls. Calculate the number of ways to select 4 people such that exactly 1 girl is included.
|
345
| 5.46875 |
18,233 |
Among all triangles $ABC$, find the maximum value of $\cos A + \cos B \cos C$.
|
\frac{3}{4}
| 78.90625 |
18,234 |
Using the digits $0$, $1$, $2$, $3$, $4$, $5$, how many different five-digit even numbers greater than $20000$ can be formed without repetition?
|
240
| 57.8125 |
18,235 |
Given that $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$ and $0 < \alpha < \pi$, find the value of $\tan(\alpha - \frac{\pi}{4})$.
|
2\sqrt{2}
| 74.21875 |
18,236 |
Translate the graph of the function y = 2sin( $$\frac {π}{3}$$ - x) - cos( $$\frac {π}{6}$$ + x) by shifting it to the right by $$\frac {π}{4}$$ units. Determine the minimum value of the corresponding function.
|
-1
| 70.3125 |
18,237 |
Let $S$ be the set of all real numbers $x$ such that $0 \le x \le 2016 \pi$ and $\sin x < 3 \sin(x/3)$ . The set $S$ is the union of a finite number of disjoint intervals. Compute the total length of all these intervals.
|
1008\pi
| 27.34375 |
18,238 |
Given an angle $a$ and a point $P(-4,3)$ on its terminal side, find the value of $\dfrac{\cos \left( \dfrac{\pi}{2}+a\right)\sin \left(-\pi-a\right)}{\cos \left( \dfrac{11\pi}{2}-a\right)\sin \left( \dfrac{9\pi}{2}+a\right)}$.
|
- \dfrac{3}{4}
| 70.3125 |
18,239 |
The lottery now consists of two drawings. First, a PowerBall is picked from among 30 numbered balls. Second, six LuckyBalls are picked from among 49 numbered balls. To win the lottery, you must pick the PowerBall number correctly and also correctly pick the numbers on all six LuckyBalls (order does not matter for the LuckyBalls). What is the probability that the ticket I hold has the winning numbers?
|
\frac{1}{419,512,480}
| 0 |
18,240 |
There are 7 volunteers to be arranged for community service activities on Saturday and Sunday, with 3 people arranged for each day, calculate the total number of different arrangements.
|
140
| 2.34375 |
18,241 |
Given the numbers 252 and 630, find the ratio of the least common multiple to the greatest common factor.
|
10
| 52.34375 |
18,242 |
Given $\sin \alpha + \cos \alpha = - \frac{\sqrt{5}}{2}$, and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, calculate the value of $\cos \alpha - \sin \alpha$.
|
\frac{\sqrt{3}}{2}
| 73.4375 |
18,243 |
A triangular pyramid \( S-ABC \) has a base in the shape of an equilateral triangle with side lengths of 4. It is known that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \).
|
\frac{268\pi}{11}
| 0 |
18,244 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$ with $a > b$, $a > c$. The radius of the circumcircle of $\triangle ABC$ is $1$, and the area of $\triangle ABC$ is $S= \sqrt{3} \sin B\sin C$.
$(1)$ Find the size of angle $A$;
$(2)$ If a point $D$ on side $BC$ satisfies $BD=2DC$, and $AB\perp AD$, find the area of $\triangle ABC$.
|
\dfrac{ \sqrt{3}}{4}
| 11.71875 |
18,245 |
Let $a_1=24$ and form the sequence $a_n$ , $n\geq 2$ by $a_n=100a_{n-1}+134$ . The first few terms are $$ 24,2534,253534,25353534,\ldots $$ What is the least value of $n$ for which $a_n$ is divisible by $99$ ?
|
88
| 29.6875 |
18,246 |
Bill draws two circles which intersect at $X,Y$ . Let $P$ be the intersection of the common tangents to the two circles and let $Q$ be a point on the line segment connecting the centers of the two circles such that lines $PX$ and $QX$ are perpendicular. Given that the radii of the two circles are $3,4$ and the distance between the centers of these two circles is $5$ , then the largest distance from $Q$ to any point on either of the circles can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $100m+n$ .
*Proposed by Tristan Shin*
|
4807
| 18.75 |
18,247 |
In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find the area of quadrilateral WXYZ.
|
200
| 7.8125 |
18,248 |
For how many integers $n$ with $1 \le n \le 2023$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]equal to zero, where $n$ needs to be an even multiple of $5$?
|
202
| 75 |
18,249 |
Let the function $y=f(x)$ have the domain $D$. If for any $x_{1}, x_{2} \in D$, when $x_{1}+x_{2}=2a$, it always holds that $f(x_{1})+f(x_{2})=2b$, then the point $(a,b)$ is called the symmetry center of the graph of the function $y=f(x)$. Study a symmetry point of the graph of the function $f(x)=x^{3}+\sin x+2$, and using the above definition of the symmetry center, we can obtain $f(-1)+f(- \frac {9}{10})+\ldots+f(0)+\ldots+f( \frac {9}{10})+f(1)=$ \_\_\_\_\_\_.
|
42
| 95.3125 |
18,250 |
Given a geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, satisfying $S_{n} = 2^{n} + r$ (where $r$ is a constant). Define $b_{n} = 2\left(1 + \log_{2} a_{n}\right)$ for $n \in \mathbf{N}^{*}$.
(1) Find the sum of the first $n$ terms of the sequence $\{a_{n} b_{n}\}$, denoted as $T_{n}$.
(2) If for any positive integer $n$, the inequality $\frac{1 + b_{1}}{b_{1}} \cdot \frac{1 + b_{2}}{b_{2}} \cdots \frac{1 + b_{n}}{b_{n}} \geq k \sqrt{n + 1}$ holds, determine the maximum value of the real number $k$.
|
\frac{3 \sqrt{2}}{4}
| 7.8125 |
18,251 |
Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one?
|
20
| 37.5 |
18,252 |
For how many integers $n$ between 1 and 150 is the greatest common divisor of 21 and $n$ equal to 3?
|
43
| 69.53125 |
18,253 |
In $\triangle ABC$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, and angles $A, B, C$ form an arithmetic sequence.
(1) Find the measure of angle $B$.
(2) If $a=4$ and the area of $\triangle ABC$ is $S=5\sqrt{3}$, find the value of $b$.
|
\sqrt{21}
| 87.5 |
18,254 |
The left and right foci of the hyperbola $C$ are respectively $F_1$ and $F_2$, and $F_2$ coincides with the focus of the parabola $y^2=4x$. Let the point $A$ be an intersection of the hyperbola $C$ with the parabola, and suppose that $\triangle AF_{1}F_{2}$ is an isosceles triangle with $AF_{1}$ as its base. Then, the eccentricity of the hyperbola $C$ is _______.
|
\sqrt{2} + 1
| 38.28125 |
18,255 |
Ben rolls six fair 12-sided dice, and each of the dice has faces numbered from 1 to 12. What is the probability that exactly three of the dice show a prime number?
|
\frac{857500}{2985984}
| 0.78125 |
18,256 |
Let \( q(x) = 2x - 5 \) and \( r(q(x)) = x^3 + 2x^2 - x - 4 \). Find \( r(3) \).
|
88
| 46.875 |
18,257 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m} = (\cos B, \cos C)$ and $\overrightarrow{n} = (2a + c, b)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(I) Find the measure of angle $B$ and the range of $y = \sin 2A + \sin 2C$;
(II) If $b = \sqrt{13}$ and $a + c = 4$, find the area of triangle $ABC$.
|
\frac{3\sqrt{3}}{4}
| 88.28125 |
18,258 |
Given that in the polar coordinate system, the equation of the curve $\Omega$ is $\rho=6\cos \theta$. With the pole as the origin of the rectangular coordinate system, the polar axis as the positive semi-axis of the $x$-axis, and the same length unit in both coordinate systems, establish a rectangular coordinate system. The parametric equations of line $l$ are $\begin{cases} x=4+t\cos \theta \\ y=-1+t\sin \theta \end{cases} (t \text{ is a parameter, } \theta \in \mathbb{R})$.
(I) Find the rectangular coordinate equation of the curve $\Omega$ and the general equation of line $l$.
(II) Suppose line $l$ intersects curve $\Omega$ at points $A$ and $C$. Line $l_0$, which passes through point $(4,-1)$ and is perpendicular to line $l$, intersects curve $\Omega$ at points $B$ and $D$. Find the maximum area of quadrilateral $ABCD$.
|
16
| 7.8125 |
18,259 |
Given an arithmetic sequence $\{a_{n}\}$, where $a_{1}+a_{8}=2a_{5}-2$ and $a_{3}+a_{11}=26$, calculate the sum of the first 2022 terms of the sequence $\{a_{n} \cdot \cos n\pi\}$.
|
2022
| 84.375 |
18,260 |
In $\triangle ABC$, it is known that $BC=1$, $B= \frac{\pi}{3}$, and the area of $\triangle ABC$ is $\sqrt{3}$. Determine the length of $AC$.
|
\sqrt{13}
| 75.78125 |
18,261 |
In \(\triangle ABC\), \(D\) is the midpoint of \(BC\), \(E\) is the foot of the perpendicular from \(A\) to \(BC\), and \(F\) is the foot of the perpendicular from \(D\) to \(AC\). Given that \(BE = 5\), \(EC = 9\), and the area of \(\triangle ABC\) is 84, compute \(|EF|\).
|
\frac{6 \sqrt{37}}{5}
| 39.0625 |
18,262 |
How many distinct sequences of five letters can be made from the letters in "PROBLEMS" if each letter can be used only once and each sequence must begin with "S" and not end with "M"?
|
720
| 59.375 |
18,263 |
Given that $|\overrightarrow{a}|=5$, $|\overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -12$, find the projection of vector $\overrightarrow{a}$ on vector $\overrightarrow{b}$.
|
-4
| 35.9375 |
18,264 |
When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product $abc = 72$.
|
\frac{1}{24}
| 35.15625 |
18,265 |
Given the digits $0$, $1$, $2$, $3$, $4$, $5$, find the number of six-digit numbers that can be formed without repetition and with odd and even digits alternating.
|
60
| 69.53125 |
18,266 |
The sum other than $11$ which occurs with the same probability when all 8 dice are rolled is equal to what value.
|
45
| 50 |
18,267 |
Given that $\sqrt{x}+\frac{1}{\sqrt{x}}=3$, determine the value of $\frac{x}{x^{2}+2018 x+1}$.
|
$\frac{1}{2025}$
| 0 |
18,268 |
Simplify first, then choose a suitable value for $x$ from $2$, $-2$, and $-6$ to substitute and evaluate.<br/>$\frac{{x}^{3}+2x^{2}}{{x}^{2}-4x+4}÷\frac{4x+8}{x-2}-\frac{1}{x-2}$.
|
-1
| 36.71875 |
18,269 |
Find the percentage of people with a grade of "excellent" among the selected individuals.
|
20\%
| 6.25 |
18,270 |
There are four tiles marked X, and three tiles marked O. The seven tiles are randomly arranged in a row. What is the probability that the two outermost positions in the arrangement are occupied by X and the middle one by O?
|
\frac{6}{35}
| 30.46875 |
18,271 |
A number is formed using the digits 1, 2, ..., 9. Any digit can be used more than once, but adjacent digits cannot be the same. Once a pair of adjacent digits has occurred, that pair, in that order, cannot be used again. How many digits are in the largest such number?
|
73
| 2.34375 |
18,272 |
When two fair dice are thrown once each, what is the probability that the upward-facing numbers are different and that one of them shows a 3?
|
\frac{5}{18}
| 92.96875 |
18,273 |
In triangle $ABC$, $AB = 7$, $BC = 24$, and the area of triangle $ABC$ is 84 square units. Given that the length of median $AM$ from $A$ to $BC$ is 12.5, find $AC$.
|
25
| 14.84375 |
18,274 |
Given the limit of the ratio of an infinite decreasing geometric series \(\{a_{n}\}\) satisfies \(\lim _{n \rightarrow \infty} \frac{a_{1} + a_{4} + a_{7} + \cdots + a_{3n-2}}{a_{1} + a_{2} + \cdots + a_{n}} = \frac{3}{4}\), find the common ratio of the series.
|
\frac{\sqrt{21} - 3}{6}
| 0 |
18,275 |
1. The focal distance of the parabola $4x^{2}=y$ is \_\_\_\_\_\_\_\_\_\_\_\_
2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \_\_\_\_\_\_\_\_\_\_\_\_
3. In the plane, the distance formula between a point $(x_{0},y_{0})$ and a line $Ax+By+C=0$ is $d= \frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}$. By analogy, the distance between the point $(0,1,3)$ and the plane $x+2y+3z+3=0$ is \_\_\_\_\_\_\_\_\_\_\_\_
4. If point $A$ has coordinates $(1,1)$, $F_{1}$ is the lower focus of the ellipse $5y^{2}+9x^{2}=45$, and $P$ is a moving point on the ellipse, then the maximum value of $|PA|+|PF_{1}|$ is $M$, the minimum value is $N$, so $M-N=$ \_\_\_\_\_\_\_\_\_\_\_\_
|
2\sqrt{2}
| 3.90625 |
18,276 |
Given the function $f(x)=a\cos (x+\frac{\pi }{6})$, its graph passes through the point $(\frac{\pi }{2}, -\frac{1}{2})$.
(1) Find the value of $a$;
(2) If $\sin \theta =\frac{1}{3}, 0 < \theta < \frac{\pi }{2}$, find $f(\theta ).$
|
\frac{2\sqrt{6}-1}{6}
| 81.25 |
18,277 |
The reciprocal of $-2$ is equal to $\frac{1}{-2}$.
|
-\frac{1}{2}
| 89.84375 |
18,278 |
$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is
|
186
| 86.71875 |
18,279 |
In a polar coordinate system with the pole at point $O$, the curve $C\_1$: $ρ=6\sin θ$ intersects with the curve $C\_2$: $ρ\sin (θ+ \frac {π}{4})= \sqrt {2}$. Determine the maximum distance from a point on curve $C\_1$ to curve $C\_2$.
|
3+\frac{\sqrt{2}}{2}
| 76.5625 |
18,280 |
What is the largest value of $n$ less than 100,000 for which the expression $9(n-3)^5 - 2n^3 + 17n - 33$ is a multiple of 7?
|
99999
| 1.5625 |
18,281 |
Point A is a fixed point on a circle with a circumference of 3. If a point B is randomly selected on the circumference of the circle, the probability that the length of the minor arc is less than 1 is ______.
|
\frac{2}{3}
| 30.46875 |
18,282 |
Given the function $f(x)=\sin x\cos x-\cos ^{2}x$.
$(1)$ Find the interval where $f(x)$ is decreasing.
$(2)$ Let the zeros of $f(x)$ on $(0,+\infty)$ be arranged in ascending order to form a sequence $\{a_{n}\}$. Find the sum of the first $10$ terms of $\{a_{n}\}$.
|
\frac{95\pi}{4}
| 34.375 |
18,283 |
If $2^n$ divides $5^{256} - 1$ , what is the largest possible value of $n$ ? $
\textbf{a)}\ 8
\qquad\textbf{b)}\ 10
\qquad\textbf{c)}\ 11
\qquad\textbf{d)}\ 12
\qquad\textbf{e)}\ \text{None of above}
$
|
10
| 90.625 |
18,284 |
Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., find the value of the 8th term.
|
84
| 11.71875 |
18,285 |
A box contains seven cards, each with a different integer from 1 to 7 written on it. Avani takes three cards from the box and then Niamh takes two cards, leaving two cards in the box. Avani looks at her cards and then tells Niamh "I know the sum of the numbers on your cards is even." What is the sum of the numbers on Avani's cards?
|
12
| 52.34375 |
18,286 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $2b\cos C=a\cos C+c\cos A$.
$(I)$ Find the magnitude of angle $C$;
$(II)$ If $b=2$ and $c= \sqrt {7}$, find $a$ and the area of $\triangle ABC$.
|
\dfrac {3 \sqrt {3}}{2}
| 0 |
18,287 |
Convert the binary number $101110_2$ to an octal number.
|
56_8
| 85.9375 |
18,288 |
For how many values of $c$ in the interval $[0, 1000]$ does the equation \[5 \lfloor x \rfloor + 3 \lceil x \rceil = c\]have a solution for $x$?
|
251
| 56.25 |
18,289 |
Let $ABC$ be a triangle whose angles measure $A$ , $B$ , $C$ , respectively. Suppose $\tan A$ , $\tan B$ , $\tan C$ form a geometric sequence in that order. If $1\le \tan A+\tan B+\tan C\le 2015$ , find the number of possible integer values for $\tan B$ . (The values of $\tan A$ and $\tan C$ need not be integers.)
*Proposed by Justin Stevens*
|
11
| 3.125 |
18,290 |
If $x$ is a real number and $\lceil x \rceil = 9,$ how many possible values are there for $\lceil x^2 \rceil$?
|
17
| 89.0625 |
18,291 |
The prime numbers 2, 3, 5, 7, 11, 13, 17 are arranged in a multiplication table, with four along the top and the other three down the left. The multiplication table is completed and the sum of the twelve entries is tabulated. What is the largest possible sum of the twelve entries?
\[
\begin{array}{c||c|c|c|c|}
\times & a & b & c & d \\ \hline \hline
e & & & & \\ \hline
f & & & & \\ \hline
g & & & & \\ \hline
\end{array}
\]
|
841
| 3.90625 |
18,292 |
Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length.
|
13
| 28.125 |
18,293 |
Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass.
|
42
| 14.84375 |
18,294 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $a$, $b$, and $c$ form a geometric progression, and $\cos B = \frac{3}{4}$.
(1) If $\overrightarrow{BA} \cdot \overrightarrow{BC} = \frac{3}{2}$, find the value of $a+c$;
(2) Find the value of $\frac{\cos A}{\sin A} + \frac{\cos C}{\sin C}$.
|
\frac{4\sqrt{7}}{7}
| 47.65625 |
18,295 |
What is the area of the smallest square that can contain a circle of radius 6?
|
144
| 98.4375 |
18,296 |
How many three-digit whole numbers have at least one 8 or at least one 9 as digits?
|
452
| 90.625 |
18,297 |
When two fair dice are rolled once each, what is the probability that one of the upward-facing numbers is 2, given that the two numbers are not the same?
|
\frac{1}{3}
| 93.75 |
18,298 |
If the two roots of the quadratic $5x^2 + 4x + k$ are $\frac{-4 \pm i \sqrt{379}}{10}$, what is $k$?
|
19.75
| 2.34375 |
18,299 |
Two congruent cones, each with a radius of 15 cm and a height of 10 cm, are enclosed within a cylinder. The bases of the cones are the bases of the cylinder, and the height of the cylinder is 30 cm. Determine the volume in cubic centimeters of the space inside the cylinder that is not occupied by the cones. Express your answer in terms of $\pi$.
|
5250\pi
| 89.0625 |
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