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23,500 | Given that $m$ is a positive integer, and given that $\mathop{\text{lcm}}[40, m] = 120$ and $\mathop{\text{lcm}}[m, 45] = 180$, what is $m$? | m = 36 | 3.125 |
23,501 | In the plane rectangular coordinate system $xOy$, the equation of the hyperbola $C$ is $x^{2}-y^{2}=1$. Find all real numbers $a$ greater than 1 that satisfy the following requirement: Through the point $(a, 0)$, draw any two mutually perpendicular lines $l_{1}$ and $l_{2}$. If $l_{1}$ intersects the hyperbola $C$ at points $P$ and $Q$, and $l_{2}$ intersects $C$ at points $R$ and $S$, then $|PQ| = |RS|$ always holds. | \sqrt{2} | 71.09375 |
23,502 | While hiking in a valley, a hiker first walks 15 miles north, then 8 miles east, then 9 miles south, and finally 2 miles east. How far is the hiker from the starting point after completing these movements? | 2\sqrt{34} | 79.6875 |
23,503 | What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls? | $\dfrac{10}{21}$ | 0 |
23,504 | Solve the equation $(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$ | -1 | 67.1875 |
23,505 | Given that $F\_1$ is the left focus of the hyperbola $C$: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, point $B$ has coordinates $(0, b)$, and the line $F\_1B$ intersects with the two asymptotes of hyperbola $C$ at points $P$ and $Q$. If $\overrightarrow{QP} = 4\overrightarrow{PF\_1}$, find the eccentricity of the hyperbola $C$. | \frac{3}{2} | 45.3125 |
23,506 | Price of some item has decreased by $5\%$ . Then price increased by $40\%$ and now it is $1352.06\$ $ cheaper than doubled original price. How much did the item originally cost? | 2018 | 82.03125 |
23,507 | Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:
(a) The novels "Beetle in the Anthill" and "Waves Extinguish the Wind" are next to each other (in any order).
(b) The stories "Restlessness" and "A Story About Friendship and Non-friendship" are next to each other (in any order).
In how many ways can Vasya do this?
Choose the correct answer:
a) \(4 \cdot 7!\);
b) \(9!\);
c) \(\frac{9!}{4!}\);
d) \(4! \cdot 7!\);
e) another answer. | 4 \cdot 7! | 99.21875 |
23,508 | Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\left( x + \frac{1}{y} \right) \left( x + \frac{1}{y} - 1024 \right) + \left( y + \frac{1}{x} \right) \left( y + \frac{1}{x} - 1024 \right).\] | -524288 | 60.15625 |
23,509 | A geometric progression with 10 terms starts with the first term as 2 and has a common ratio of 3. Calculate the sum of the new geometric progression formed by taking the reciprocal of each term in the original progression.
A) $\frac{29523}{59049}$
B) $\frac{29524}{59049}$
C) $\frac{29525}{59049}$
D) $\frac{29526}{59049}$ | \frac{29524}{59049} | 69.53125 |
23,510 | Let an integer be such that when divided by 20, the remainder is 11. What is the sum of the remainders when the same integer is divided by 4 and by 5? Additionally, find the smallest such integer greater than 50. | 51 | 91.40625 |
23,511 | What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 60? | 4087 | 3.125 |
23,512 | Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$ , denote $P(x)$ as the probability that the length of $AB$ is less than $x$ . Find the minimum possible integer value of $x$ such that $\text{P}(x) > \frac{2}{3}$ . | 174 | 91.40625 |
23,513 | Xinjiang region has a dry climate and is one of the three major cotton-producing areas in China, producing high-quality long-staple cotton. In an experiment on the germination rate of a certain variety of long-staple cotton seeds, research institute staff selected experimental fields with basically the same conditions, sowed seeds simultaneously, and determined the germination rate, obtaining the following data:
| Number of<br/>cotton seeds| $100$ | $200$ | $500$ | $1000$ | $2000$ | $5000$ | $10000$ |
|---|---|---|---|---|---|---|---|
| Number of<br/>germinated seeds| $98$ | $192$ | $478$ | $953$ | $1902$ | $4758$ | $9507$ |
Then the germination rate of this variety of long-staple cotton seeds is approximately ______ (rounded to $0.01$). | 0.95 | 89.0625 |
23,514 | $(Ⅰ){0.064^{-\frac{1}{3}}}+\sqrt{{{(-2)}^4}}-{(π+e)^0}-{9^{\frac{3}{2}}}×{({\frac{{\sqrt{3}}}{3}})^4}$；<br/>$(Ⅱ)\frac{{lg18+lg5-lg60}}{{{{log}_2}27×lg2-lg8}}$. | \frac{1}{3} | 60.15625 |
23,515 | Define a modified Ackermann function \( A(m, n) \) with the same recursive relationships as the original problem:
\[ A(m,n) = \left\{
\begin{aligned}
&n+1& \text{ if } m = 0 \\
&A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\
&A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0.
\end{aligned}
\right.\]
Compute \( A(3, 2) \). | 29 | 88.28125 |
23,516 | Given $\cos \left(\alpha- \frac{\pi}{6}\right) + \sin \alpha = \frac{4}{5} \sqrt{3}$, find the value of $\sin \left(\alpha+ \frac{7\pi}{6}\right)$. | -\frac{4}{5} | 85.15625 |
23,517 | Find the coefficient of the $x^{3}$ term in the expansion of $(x^{2}-x+1)^{10}$. | -210 | 29.6875 |
23,518 | Determine the minimum possible value of the sum
\[
\frac{a}{3b} + \frac{b}{6c} + \frac{c}{9a},
\]
where \( a, b, \) and \( c \) are positive real numbers. | \frac{1}{3\sqrt[3]{2}} | 0 |
23,519 | How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 | 100 |
23,520 | Given two lines $l_{1}$: $(m+1)x+2y+2m-2=0$ and $l_{2}$: $2x+(m-2)y+2=0$, if $l_{1} \parallel l_{2}$, then $m=$ ______. | -2 | 32.8125 |
23,521 | Two circles with centers \( M \) and \( N \), lying on the side \( AB \) of triangle \( ABC \), are tangent to each other and intersect the sides \( AC \) and \( BC \) at points \( A, P \) and \( B, Q \) respectively. Additionally, \( AM = PM = 2 \) and \( BN = QN = 5 \). Find the radius of the circumcircle of triangle \( ABC \), given that the ratio of the area of triangle \( AQn \) to the area of triangle \( MPB \) is \(\frac{15 \sqrt{2 + \sqrt{3}}}{5 \sqrt{3}}\). | 10 | 0.78125 |
23,522 | find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have $$ \{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\} $$ . | {0} | 0 |
23,523 | Let \( ABCDEF \) be a hexagon such that the diagonals \( AD, BE, \) and \( CF \) intersect at the point \( O \), and the area of the triangle formed by any three adjacent points is 2 (for example, the area of \(\triangle BCD\) is 2). Find the area of the hexagon. | 12 | 92.96875 |
23,524 | Simplify and find the value of: $a^{2}-\left(a-\dfrac{2a}{a+1}\right)\div \dfrac{a^{2}-2a+1}{a^{2}-1}$, where $a$ is a solution of the equation $x^{2}-x-\dfrac{7}{2}=0$. | \dfrac{7}{2} | 58.59375 |
23,525 | 18. Given the function $f(x)=x^3+ax^2+bx+5$, the equation of the tangent line to the curve $y=f(x)$ at the point $x=1$ is $3x-y+1=0$.
Ⅰ. Find the values of $a$ and $b$;
Ⅱ. Find the maximum and minimum values of $y=f(x)$ on the interval $[-3,1]$. | \frac{95}{27} | 0.78125 |
23,526 | The complex numbers corresponding to the vertices $O$, $A$, and $C$ of the parallelogram $OABC$ are $0$, $3+2i$, and $-2+4i$, respectively.<br/>$(1)$ Find the complex number corresponding to point $B$;<br/>$(2)$ In triangle $OAB$, find the height $h$ on side $OB$. | \frac{16\sqrt{37}}{37} | 39.0625 |
23,527 | If the graph of the function $f(x)=3\sin(2x+\varphi)$ is symmetric about the point $\left(\frac{\pi}{3},0\right)$ $(|\varphi| < \frac{\pi}{2})$, determine the equation of one of the axes of symmetry of the graph of $f(x)$. | \frac{\pi}{12} | 74.21875 |
23,528 | Determine the least positive period $q$ of the functions $g$ such that $g(x+2) + g(x-2) = g(x)$ for all real $x$. | 12 | 38.28125 |
23,529 | Given the function $f(x)=2\sqrt{3}\sin x\cos x-\cos (\pi +2x)$.
(1) Find the interval(s) where $f(x)$ is monotonically increasing.
(2) In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(C)=1,c=\sqrt{3},a+b=2\sqrt{3}$, find the area of $\Delta ABC$. | \frac{3\sqrt{3}}{4} | 92.96875 |
23,530 | The probability of event A occurring is $P (0<p<1)$. Find the maximum value of the variance of the number of occurrences of the event A, $X$. | \frac{1}{4} | 18.75 |
23,531 | In an International track meet, 256 sprinters participate in a 100-meter dash competition. If the track has 8 lanes, and only the winner of each race advances to the next round while the others are eliminated, how many total races are needed to determine the champion sprinter? | 37 | 46.09375 |
23,532 | In $\triangle ABC$, $a=3 \sqrt {3}$, $c=2$, $B=150^{\circ}$, find the length of side $b$ and the area of $\triangle ABC$. | \frac{3 \sqrt {3}}{2} | 0 |
23,533 | Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$ | \frac{1}{\sqrt{2}} | 0 |
23,534 | In $\triangle ABC$, the ratio $AC:CB$ is $2:3$. The bisector of the exterior angle at $C$ intersects $BA$ extended at point $Q$ ($A$ is between $Q$ and $B$). Find the ratio $QA:AB$. | 2:1 | 3.125 |
23,535 | The numbers from 1 to 9 are placed at the vertices of a cube such that the sum of the four numbers on each face is the same. Find the common sum. | 22.5 | 0.78125 |
23,536 | The expression $\cos 2x + \cos 6x + \cos 10x + \cos 14x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a$, $b$, $c$, and $d$. Find $a+b+c+d$. | 18 | 77.34375 |
23,537 | Let $C$ be a point on the parabola $y = x^2 - 4x + 7,$ and let $D$ be a point on the line $y = 3x - 5.$ Find the shortest distance $CD$ and also ensure that the projection of point $C$ over line $y = 3x - 5$ lands on point $D$. | \frac{0.25}{\sqrt{10}} | 0 |
23,538 | The amount of heat \( Q \) received by a certain substance when heated from 0 to \( T \) is determined by the formula \( Q = 0.1054t + 0.000002t^2 \) (\( Q \) is in joules, \( t \) is in kelvins). Find the heat capacity of this substance at \( 100 \) K. | 0.1058 | 97.65625 |
23,539 | A deck of fifty-two cards consists of four $1$'s, four $2$'s,..., and four $13$'s. A matching pair (two cards with the same number) is removed from the deck. Determine the probability that two randomly selected cards from the remaining deck also form a pair, and express the result as the sum of the numerator and denominator of the simplified fraction. | 1298 | 49.21875 |
23,540 | In $\triangle ABC$, if $a=3$, $b= \sqrt {3}$, $\angle A= \dfrac {\pi}{3}$, then the size of $\angle C$ is \_\_\_\_\_\_. | \dfrac {\pi}{2} | 98.4375 |
23,541 | In a group of 10 basketball teams which includes 2 strong teams, the teams are randomly split into two equal groups for a competition. What is the probability that the 2 strong teams do not end up in the same group? | \frac{5}{9} | 58.59375 |
23,542 | Find the number of ordered pairs \((a, b)\) of positive integers such that \(a\) and \(b\) both divide \(20^{19}\), but \(ab\) does not. | 444600 | 49.21875 |
23,543 | The probability that students A and B stand together among three students A, B, and C lined up in a row is what? | \frac{2}{3} | 88.28125 |
23,544 | Given the polar equation of curve $C_1$ is $\rho^2=\frac {2}{3+\cos2\theta}$, establish a rectangular coordinate system with the pole O as the origin and the polar axis as the positive direction of the x-axis. After stretching all the x-coordinates of points on curve $C_1$ to twice their original values and shortening all the y-coordinates to half of their original values, we obtain curve $C_2$.
1. Write down the rectangular coordinate equation of curve $C_1$.
2. Take any point R on curve $C_2$ and find the maximum distance from point R to the line $l: x + y - 5 = 0$. | \frac{13\sqrt{2}}{4} | 35.9375 |
23,545 | There are ten digits: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$.
$(1)$ How many unique three-digit numbers can be formed without repetition?
$(2)$ How many unique four-digit even numbers can be formed without repetition? | 2296 | 64.84375 |
23,546 | A coordinate system and parametric equations problem (4-4):
In the rectangular coordinate system $xoy$, a curve $C_1$ is defined by the parametric equations $\begin{cases} x = 1 + \cos \alpha \\ y = \sin^2 \alpha - \frac{9}{4} \end{cases}$, where $\alpha$ is the parameter and $\alpha \in \mathbb{R}$. In the polar coordinate system with the origin $O$ as the pole and the nonnegative $x$-axis as the polar axis (using the same length units), there are two other curves: $C_2: \rho \sin \left( \theta + \frac{\pi}{4} \right) = -\frac{\sqrt{2}}{2}$ and $C_3: \rho = 2 \cos \theta$.
(I) Find the rectangular coordinates of the intersection point $M$ of the curves $C_1$ and $C_2$.
(II) Let $A$ and $B$ be moving points on the curves $C_2$ and $C_3$, respectively. Find the minimum value of the distance $|AB|$. | \sqrt{2} - 1 | 66.40625 |
23,547 | If a polygon is formed by connecting a point on one side of the polygon to all its vertices, forming $2023$ triangles, determine the number of sides of this polygon. | 2024 | 15.625 |
23,548 | Evaluate the expression $1 - \frac{1}{1 + \sqrt{5}} + \frac{1}{1 - \sqrt{5}}$. | 1 - \frac{\sqrt{5}}{2} | 74.21875 |
23,549 | In the arithmetic sequence $\{a\_n\}$, it is known that $a\_1 - a\_4 - a\_8 - a\_{12} + a\_{15} = 2$. Find the value of $S\_{15}$. | -30 | 60.9375 |
23,550 | Given a function $f(x)$ that satisfies $f(x+3)=-f(x)$, when $x\in \left[-3,0\right)$, $f(x)=2^{x}+\sin \frac{πx}{3}$, determine the value of $f(2023)$. | -\frac{1}{4} + \frac{\sqrt{3}}{2} | 54.6875 |
23,551 | Find the integer $n,$ $-180 < n < 180,$ such that $\tan n^\circ = \tan 345^\circ.$ | -15 | 91.40625 |
23,552 | Given that Anne, Cindy, and Ben repeatedly take turns tossing a die in the order Anne, Cindy, Ben, find the probability that Cindy will be the first one to toss a five. | \frac{30}{91} | 10.15625 |
23,553 | Arrange the 7 numbers $39, 41, 44, 45, 47, 52, 55$ in a sequence such that the sum of any three consecutive numbers is a multiple of 3. What is the maximum value of the fourth number in all such arrangements? | 47 | 31.25 |
23,554 | 2 diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon? | \dfrac{42}{119} | 0 |
23,555 | Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 | 100 |
23,556 | Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$. | 350 | 0 |
23,557 | The members of a club are arranged in a rectangular formation. When they are arranged in 10 rows, there are 4 positions unoccupied in the formation. When they are arranged in 11 rows, there are 5 positions unoccupied. How many members are in the club if the membership is between 150 and 300? | 226 | 15.625 |
23,558 | Arrange the natural numbers according to the following triangular pattern. What is the sum of the numbers in the 10th row?
```
1
2 3 4
5 6 7 8 9
10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25
...
``` | 1729 | 66.40625 |
23,559 | To calculate $41^2$, Tom mentally computes $40^2$ and adds a number. To find $39^2$, Tom subtracts a number from $40^2$. What number does he add to calculate $41^2$ and subtract to calculate $39^2$? | 79 | 64.84375 |
23,560 | Given the function $f(x)=- \sqrt {3}\sin ^{2}x+\sin x\cos x$.
(1) Find the value of $f( \frac {25π}{6})$.
(2) Find the smallest positive period of the function $f(x)$ and its maximum and minimum values in the interval $[0, \frac {π}{2}]$. | -\sqrt{3} | 38.28125 |
23,561 | Given 18 parking spaces in a row, 14 cars arrive and occupy spaces at random, followed by Auntie Em, who requires 2 adjacent spaces, determine the probability that the remaining spaces are sufficient for her to park. | \frac{113}{204} | 3.125 |
23,562 | What is the area, in square units, of a triangle that has sides of $5, 3,$ and $3$ units? Express your answer in simplest radical form. | \frac{5\sqrt{11}}{4} | 92.96875 |
23,563 | Find the maximum value of $$ \int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx $$ over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$ \int^1_0|f'(x)|^2dx\le1. $$ | \frac{2}{3} | 63.28125 |
23,564 | Thirty clever students from 6th, 7th, 8th, 9th, and 10th grades were tasked with creating forty problems for an olympiad. Any two students from the same grade came up with the same number of problems, while any two students from different grades came up with a different number of problems. How many students came up with one problem each? | 26 | 3.125 |
23,565 | Given that bricklayer Alice can build a wall alone in 8 hours, and bricklayer Bob can build it alone in 12 hours, and they complete the wall in 6 hours when working together with a 15-brick-per-hour decrease in productivity, determine the number of bricks in the wall. | 360 | 87.5 |
23,566 | Given the sequence $1990-1980+1970-1960+\cdots -20+10$, calculate the sum. | 1000 | 50.78125 |
23,567 | In triangle $XYZ$, $\angle Z=90^\circ$, $XZ=3$ and $YZ=4$. Points $W$ and $V$ are on $\overline{XY}$ and $\overline{YZ}$, respectively, and $\angle WVZ=90^\circ$. If $WV=2$, then what is the length of $WY$? | \frac{10}{3} | 38.28125 |
23,568 | Given that $f(x)$ is an odd function defined on $\mathbb{R}$, when $x > 0$, $f(x)=2^{x}+ \ln \frac{x}{4}$. Let $a_{n}=f(n-5)$, then the sum of the first $8$ terms of the sequence $\{a_{n}\}$ is $\_\_\_\_\_\_\_\_\_.$ | -16 | 81.25 |
23,569 | The line $y = 2$ intersects the graph of $y = 3x^2 + 2x - 5$ at the points $C$ and $D$. Determine the distance between $C$ and $D$ and express this distance in the form $\frac{\sqrt{p}}{q}$, where $p$ and $q$ are coprime positive integers. | \frac{2\sqrt{22}}{3} | 68.75 |
23,570 | What is the product of all real numbers that are tripled when added to their reciprocals? | -\frac{1}{2} | 85.15625 |
23,571 | What is the minimum number of equilateral triangles, of side length 1 unit, needed to cover an equilateral triangle of side length 15 units? | 225 | 100 |
23,572 | Given the parabola C: y^2 = 4x, point B(3,0), and the focus F, point A lies on C. If |AF| = |BF|, calculate the length of |AB|. | 2\sqrt{2} | 63.28125 |
23,573 | It is known that the variance of each of the given independent random variables does not exceed 4. Determine the number of such variables for which the probability that the deviation of the arithmetic mean of the random variable from the arithmetic mean of their mathematical expectations by no more than 0.25 exceeds 0.99. | 6400 | 55.46875 |
23,574 | Find the distance between the foci of the hyperbola $x^2 - 4x - 9y^2 - 18y = 45.$ | \frac{40}{3} | 89.0625 |
23,575 | A checker can move in one direction on a divided strip into cells, moving either to the adjacent cell or skipping one cell in one move. In how many ways can it move 10 cells? 11 cells? | 144 | 83.59375 |
23,576 | Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 | 99.21875 |
23,577 | An engineer invested $\$15,\!000$ in a nine-month savings certificate that paid a simple annual interest rate of $9\%$. After nine months, she invested the total value of her investment in another nine-month certificate. After another nine months, the investment was worth $\$17,\!218.50$. If the annual interest rate of the second certificate is $s\%,$ what is $s$? | 10 | 36.71875 |
23,578 | Given a sequence $\{a_n\}$ that satisfies $a_1=33$ and $a_{n+1}-a_n=2n$, find the minimum value of $\frac {a_{n}}{n}$. | \frac {21}{2} | 14.84375 |
23,579 | The sum of the first 3 terms of a geometric sequence $\{a_n\}$ is 13, and the sum of the first 6 terms is 65. Find $S_{12}$. | 1105 | 55.46875 |
23,580 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $\overrightarrow{a}=(\cos A,\cos B)$, $\overrightarrow{b}=(a,2c-b)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$.
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) If $b=3$ and the area of $\triangle ABC$, $S_{\triangle ABC}=3 \sqrt {3}$, find the value of $a$. | \sqrt {13} | 0 |
23,581 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 69.53125 |
23,582 | A sector with a central angle of 135° has an area of $S_1$, and the total surface area of the cone formed by it is $S_2$. Find the value of $\frac{S_{1}}{S_{2}}$. | \frac{8}{11} | 50.78125 |
23,583 | Let $x_0$ be a zero of the function $f(x)=\sin \pi x$, and it satisfies $|x_0|+f(x_0+ \frac{1}{2}) < 33$. Calculate the number of such zeros. | 65 | 19.53125 |
23,584 | The coefficient of $x^2$ in the expansion of $(x-1) - (x-1)^2 + (x-1)^3 - (x-1)^4 + (x-1)^5$ is ____. | -20 | 39.0625 |
23,585 | When the municipal government investigated the relationship between changes in citizens' income and tourism demand, a sample of 5000 people was randomly selected using the independence test method. The calculation showed that $K^{2}=6.109$. Based on this data, the municipal government asserts that the credibility of the relationship between changes in citizens' income and tourism demand is ______ $\%.$
Attached: Common small probability values and critical values table:
| $P(K^{2}\geqslant k_{0})$ | $0.15$ | $0.10$ | $0.05$ | $0.025$ | $0.010$ | $0.001$ |
|---------------------------|--------|--------|--------|---------|---------|---------|
| $k_{0}$ | $2.072$| $2.706$| $3.841$| $5.024$ | $6.635$ | $10.828$| | 97.5\% | 21.875 |
23,586 | In the Cartesian coordinate system $xOy$, establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis, using the same unit of length in both coordinate systems. Given that circle $C$ has a center at point ($2$, $\frac{7π}{6}$) in the polar coordinate system and a radius of $\sqrt{3}$, and line $l$ has parametric equations $\begin{cases} x=- \frac{1}{2}t \\ y=-2+ \frac{\sqrt{3}}{2}t\end{cases}$.
(1) Find the Cartesian coordinate equations for circle $C$ and line $l$.
(2) If line $l$ intersects circle $C$ at points $M$ and $N$, find the area of triangle $MON$. | \sqrt{2} | 8.59375 |
23,587 | Given 2009 points, where no three points are collinear, along with three vertices, making a total of 2012 points, and connecting these 2012 points to form non-overlapping small triangles, calculate the total number of small triangles that can be formed. | 4019 | 18.75 |
23,588 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 64.84375 |
23,589 | Evaluate \(\left(d^d - d(d-2)^d\right)^d\) when \( d = 4 \). | 1358954496 | 13.28125 |
23,590 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $\frac{b}{c} = \frac{2\sqrt{3}}{3}$ and $A + 3C = \pi$.
$(1)$ Find the value of $\cos C$;
$(2)$ Find the value of $\sin B$;
$(3)$ If $b = 3\sqrt{3}$, find the area of $\triangle ABC$. | \frac{9\sqrt{2}}{4} | 63.28125 |
23,591 | Determine the number of six-letter words where the first and last two letters are the same (e.g., "aabbaa"). | 456976 | 26.5625 |
23,592 | In the diagram, the circle is inscribed in the square. This means that the circle and the square share points \(S, T, U,\) and \(V\), and the width of the square is exactly equal to the diameter of the circle. Rounded to the nearest tenth, what percentage of line segment \(XY\) is outside the circle? | 29.3 | 71.875 |
23,593 | Find the smallest solution to the equation \[\lfloor x^2 \rfloor - \lfloor x \rfloor^2 = 19.\] | \sqrt{119} | 75.78125 |
23,594 | In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written? | 20 | 64.0625 |
23,595 | If $f\left(x\right)=\ln |a+\frac{1}{{1-x}}|+b$ is an odd function, then $a=$____, $b=$____. | \ln 2 | 1.5625 |
23,596 | The Nanjing Youth Olympic Games are about to open, and a clothing store owner, Mr. Chen, spent 3600 yuan to purchase two types of sportswear, A and B, and sold them out quickly. When Mr. Chen went to purchase the same types and quantities of clothing again, he found that the purchase prices of types A and B had increased by 20 yuan/piece and 5 yuan/piece, respectively, resulting in an additional expenditure of 400 yuan. Let the number of type A clothing purchased by Mr. Chen each time be $x$ pieces, and the number of type B clothing be $y$ pieces.
(1) Please write down the function relationship between $y$ and $x$ directly: .
(2) After calculating, Mr. Chen found that the average unit price of types A and B clothing had increased by 8 yuan during the second purchase compared to the first.
① Find the values of $x$ and $y$.
② After selling all the clothing purchased for the second time at a 35% profit, Mr. Chen took all the sales proceeds to purchase more goods. At this time, the prices of both types of clothing had returned to their original prices, so Mr. Chen spent 3000 yuan to purchase type B clothing, and the rest of the money was used to purchase type A clothing. As a result, the quantities of types A and B clothing purchased were exactly equal. How many pieces of clothing did Mr. Chen purchase in total this time? | 80 | 11.71875 |
23,597 | Let $p$, $q$, and $r$ be solutions of the equation $x^3 - 6x^2 + 11x = 14$.
Compute $\frac{pq}{r} + \frac{qr}{p} + \frac{rp}{q}$. | -\frac{47}{14} | 46.875 |
23,598 | How many positive multiples of 7 that are less than 2000 end with the digit 5? | 29 | 58.59375 |
23,599 | A venture capital firm is considering investing in the development of a new energy product, with estimated investment returns ranging from 100,000 yuan to 10,000,000 yuan. The firm is planning to establish a reward scheme for the research team: the reward amount $y$ (in units of 10,000 yuan) increases with the investment return $x$ (in units of 10,000 yuan), with the maximum reward not exceeding 90,000 yuan, and the reward not exceeding 20% of the investment return.
1. Analyze whether the function $y=\frac{x}{150}+1$ meets the company's requirements for the reward function model and explain the reasons.
2. If the company adopts the function model $y=\frac{10x-3a}{x+2}$ as the reward function model, determine the minimum positive integer value of $a$. | 328 | 57.03125 |
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