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23,800 | Given that $f(x)$ has a derivative and satisfies $\lim_{\Delta x \to 0} \, \frac{f(1)-f(1-2\Delta x)}{2\Delta x}=-1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$. | -1 | 55.46875 |
23,801 | Given a sequence $\{a_n\}$, the first $m(m\geqslant 4)$ terms form an arithmetic sequence with a common difference of $2$. Starting from the $(m-1)$-th term, $a_{m-1}$, $a_{m}$, $a_{m+1}$, ... form a geometric sequence with a common ratio of $2$. If $a_{1}=-2$, then $m=$ ______, and the sum of the first $6$ terms of $\{a_n\}$, $S_{6}=$ ______. | 28 | 68.75 |
23,802 | Given that a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ has only one common point with the parabola $y=x^{2}+1$, calculate the eccentricity of the hyperbola. | \sqrt{5} | 29.6875 |
23,803 | Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}+1, n \text{ is odd}}\\{{a}_{n}+2, n \text{ is even}}\end{array}\right.$
$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$.
$(2)$ Find the sum of the first $20$ terms of the sequence $\{a_{n}\}$. | 300 | 73.4375 |
23,804 | Given a sequence ${a_n}$ that satisfies $a_1=1$ and $a_n=a_1+ \frac {1}{2}a_2+ \frac {1}{3}a_3+…+ \frac {1}{n-1}a_{n-1}$ for $n\geqslant 2, n\in\mathbb{N}^*$, if $a_k=2017$, then $k=$ \_\_\_\_\_\_. | 4034 | 56.25 |
23,805 | Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \cap E| = |E \cap F| = |F \cap D| = 1$ and $D \cap E \cap F = \emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\{1,2,3,4,5,6,7,8\}$. Find $M$ modulo $1000$. | 064 | 0.78125 |
23,806 | Given $$(5x- \frac {1}{ \sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion. | 375 | 50 |
23,807 | There is an opaque bag containing 4 identical balls labeled with the numbers 1, 2, 3, and 4.
(Ⅰ) If balls are drawn one by one without replacement twice, calculate the probability that the first ball drawn has an even number and the sum of the two balls’ numbers is divisible by 3.
(Ⅱ) If a ball is randomly taken from the bag and labeled as a, then put back into the bag, followed by randomly taking another ball, labeled as b, calculate the probability that the line $ax + by + 1 = 0$ has no common points with the circle $x^2 + y^2 = \frac{1}{16}$. | \frac{1}{2} | 32.03125 |
23,808 | In $\triangle XYZ$, the medians $\overline{XU}$ and $\overline{YV}$ intersect at right angles. If $XU = 18$ and $YV = 24$, find the area of $\triangle XYZ$. | 288 | 60.9375 |
23,809 | Given the function $f\left(x\right)=\cos \left(\omega x+\varphi \right)\left(\omega\ \ \gt 0,0 \lt \varphi\ \ \lt \pi \right)$, if $f\left(x\right)$ is an odd function and monotonically decreasing on $(-\frac{π}{3},\frac{π}{6})$, then the maximum value of $\omega$ is ______. | \frac{3}{2} | 59.375 |
23,810 | Let $T$ be a subset of $\{1,2,3,...,60\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | 25 | 82.8125 |
23,811 | Given the function $f(x)= \sqrt {3}\sin ^{2}x+\sin x\cos x- \frac { \sqrt {3}}{2}$ $(x\in\mathbb{R})$.
$(1)$ If $x\in(0, \frac {\pi}{2})$, find the maximum value of $f(x)$;
$(2)$ In $\triangle ABC$, if $A < B$ and $f(A)=f(B)= \frac {1}{2}$, find the value of $\frac {BC}{AB}$. | \sqrt {2} | 0 |
23,812 | The product of two consecutive page numbers is $2,156.$ What are the sum of these two page numbers? | 93 | 68.75 |
23,813 | What is the largest number of positive, consecutive integers whose sum is 105? | 14 | 22.65625 |
23,814 | In the diagram, three identical circles touch each other, and each circle has a circumference of 24. Calculate the perimeter of the shaded region within the triangle formed by the centers of the circles. | 12 | 5.46875 |
23,815 | Three different numbers are chosen from the set $\{-4, -3, -1, 3, 5, 8\}$ and multiplied. Find the largest possible positive product. | 120 | 95.3125 |
23,816 | Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 10$, $BC = 5$, and the height of the pyramid, $PA$, while not perpendicular to the plane of $ABCD$, ends at the center of rectangle $ABCD$ and is twice the length of $BC$. What is the volume of $PABCD$? | \frac{500}{3} | 100 |
23,817 | (For science students) In the expansion of $(x^2 - 3x + 2)^4$, the coefficient of the $x^2$ term is __________ (Answer with a number). | 248 | 75 |
23,818 | Given that $α$ is an angle in the third quadrant and $\cos 2α=-\frac{3}{5}$, find $\tan (\frac{π}{4}+2α)$. | -\frac{1}{7} | 10.15625 |
23,819 | Find the value of $m + n$ where $m$ and $n$ are integers such that the positive difference between the roots of the equation $4x^2 - 12x - 9 = 0$ can be expressed as $\frac{\sqrt{m}}{n}$, with $m$ not divisible by the square of any prime number. | 19 | 57.8125 |
23,820 | Consider a dodecahedron, which is made up of 12 pentagonal faces. An ant starts at one of the top vertices and walks to one of the three adjacent vertices (vertex A). From vertex A, the ant walks again to one of its adjacent vertices (vertex B). What is the probability that vertex B is one of the bottom vertices? There are three bottom vertices in total in a dodecahedron. | \frac{1}{3} | 27.34375 |
23,821 | Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 2,$ and $x \ge -\frac{1}{2},$ $y \ge -2,$ and $z \ge -3.$ Find the maximum value of:
\[
\sqrt{4x + 2} + \sqrt{4y + 8} + \sqrt{4z + 12}.
\] | 3\sqrt{10} | 1.5625 |
23,822 | Given that $\sin\alpha + \cos\alpha = \frac{1}{5}$, and $0 \leq \alpha < \pi$, find the value of $\tan\alpha$. | - \frac {4}{3} | 89.84375 |
23,823 | 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 | 44.53125 |
23,824 | Let $P$ be a point on the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{20} = 1$, and let $F_{1}$ and $F_{2}$ be the left and right foci, respectively. If $|PF_{1}| = 9$, then find $|PF_{2}|$. | 17 | 96.09375 |
23,825 | The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$ .
\[ \begin{tabular}{|c|c|c|}\hline A & 1 & B \hline 5 & C & 13 \hline D & E & 3 \hline\end{tabular} \] | 32 | 82.03125 |
23,826 | The mean of the set of numbers $\{106, 102, 95, 103, 100, y, x\}$ is 104. What is the median of this set of seven numbers? | 103 | 56.25 |
23,827 | Let \[f(x) =
\begin{cases}
2x + 4 &\text{if }x<0, \\
6-3x&\text{if }x\ge 0.
\end{cases}
\]Find $f(-2)$ and $f(4)$. | -6 | 67.1875 |
23,828 | Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. If $\frac{S_6}{S_3} = 3$, then calculate $\frac{S_{12}}{S_{9}}$. | \frac{5}{3} | 100 |
23,829 | Given quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at $O$, $BO=4$, $OD=5$, $AO=9$, $OC=2$, and $AB=7$, find the length of $AD$. | \sqrt{166} | 35.9375 |
23,830 | Read the material first, then answer the question.
$(1)$ Xiao Zhang encountered a problem when simplifying a quadratic radical: simplify $\sqrt{5-2\sqrt{6}}$.
After thinking about it, Xiao Zhang's process of solving this problem is as follows:
$\sqrt{5-2\sqrt{6}}=\sqrt{2-2\sqrt{2\times3}+3}$①
$=\sqrt{{(\sqrt{2})}^2}-2\sqrt{2}\times\sqrt{3}+{(\sqrt{3})}^2$②
$=\sqrt{{(\sqrt{2}-\sqrt{3})}^2}$③
$=\sqrt{2}-\sqrt{3}$④
In the above simplification process, an error occurred in step ____, and the correct result of the simplification is ____;
$(2)$ Please simplify $\sqrt{8+4\sqrt{3}}$ based on the inspiration you obtained from the above material. | \sqrt{6}+\sqrt{2} | 44.53125 |
23,831 | Solve the following equations using appropriate methods:
(1) $x^2=49$;
(2) $(2x+3)^2=4(2x+3)$;
(3) $2x^2+4x-3=0$ (using the formula method);
(4) $(x+8)(x+1)=-12$. | -5 | 0 |
23,832 | Given the function $f(x)=\frac{1}{2}x^{2}-a\ln x+b$ where $a\in R$.
(I) If the equation of the tangent line to the curve $y=f(x)$ at $x=1$ is $3x-y-3=0$, find the values of the real numbers $a$ and $b$.
(II) If $x=1$ is the extreme point of the function $f(x)$, find the value of the real number $a$.
(III) If $-2\leqslant a < 0$, for any $x_{1}$, $x_{2}\in(0,2]$, the inequality $|f(x_{1})-f(x_{2})|\leqslant m| \frac{1}{x_{1}}- \frac{1}{x_{2}}|$ always holds. Find the minimum value of $m$. | 12 | 7.03125 |
23,833 | Given $\tan (\theta-\pi)=2$, then $\sin ^{2}\theta+\sin \theta\cos \theta-2\cos ^{2}\theta=$ \_\_\_\_\_\_ . | \frac {4}{5} | 91.40625 |
23,834 | Anton colors a cell in a \(4 \times 50\) rectangle. He then repeatedly chooses an uncolored cell that is adjacent to at most one already colored cell. What is the maximum number of cells that can be colored? | 150 | 0 |
23,835 | How many positive integer multiples of $77$ (product of $7$ and $11$) can be expressed in the form $10^{j}-10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 99$? | 784 | 90.625 |
23,836 | Find the value of the arithmetic series $1-3+5-7+9-11+\cdots +2021-2023+2025$. | 1013 | 56.25 |
23,837 | Given a hyperbola $C$ that shares the same foci with the ellipse $\frac{x^{2}}{27}+ \frac{y^{2}}{36}=1$ and passes through the point $(\sqrt{15},4)$.
(I) Find the equation of the hyperbola $C$.
(II) If $F\_1$ and $F\_2$ are the two foci of the hyperbola $C$, and point $P$ is on the hyperbola $C$ such that $\angle F\_1 P F\_2 = 120^{\circ}$, find the area of $\triangle F\_1 P F\_2$. | \frac{5\sqrt{3}}{3} | 21.875 |
23,838 | In triangle ABC, BR = RC, CS = 3SA, and (AT)/(TB) = p/q. If the area of △RST is twice the area of △TBR, determine the value of p/q. | \frac{7}{3} | 1.5625 |
23,839 | Calculate the value of \[\cot(\cot^{-1}5 + \cot^{-1}11 + \cot^{-1}17 + \cot^{-1}23).\] | \frac{97}{40} | 0 |
23,840 | Given that the random variable X follows a normal distribution N(2, σ²) and P(X≤4)=0.88, find P(0<X<4). | 0.76 | 71.09375 |
23,841 | Given a four-digit positive integer $\overline{abcd}$, if $a+c=b+d=11$, then this number is called a "Shangmei number". Let $f(\overline{abcd})=\frac{{b-d}}{{a-c}}$ and $G(\overline{abcd})=\overline{ab}-\overline{cd}$. For example, for the four-digit positive integer $3586$, since $3+8=11$ and $5+6=11$, $3586$ is a "Shangmei number". Also, $f(3586)=\frac{{5-6}}{{3-8}}=\frac{1}{5}$ and $G(M)=35-86=-51$. If a "Shangmei number" $M$ has its thousands digit less than its hundreds digit, and $G(M)$ is a multiple of $7$, then the minimum value of $f(M)$ is ______. | -3 | 2.34375 |
23,842 | Let $p$, $q$, $r$, $s$, and $t$ be positive integers with $p+q+r+s+t=2025$ and let $N$ be the largest of the sums $p+q$, $q+r$, $r+s$, and $s+t$. Determine the smallest possible value of $N$. | 676 | 19.53125 |
23,843 | Let the function $f(x)=\ln x- \frac{1}{2}ax^{2}-bx$.
$(1)$ When $a=b= \frac{1}{2}$, find the maximum value of the function $f(x)$;
$(2)$ Let $F(x)=f(x)+ \frac{1}{2}ax^{2}+bx+ \frac{a}{x}$, $(0 < x\leqslant 3)$, the slope of the tangent line at any point $P(x_{0},y_{0})$ on its graph is $k\leqslant \frac{1}{2}$ always holds, find the range of the real number $a$;
$(3)$ When $a=0$, $b=-1$, the equation $2mf(x)=x^{2}$ has a unique real solution, find the value of the positive number $m$. | \frac{1}{2} | 15.625 |
23,844 | Use the bisection method to find an approximate solution for $f(x)=0$. Given that $f(1)=-2$, $f(3)=0.625$, and $f(2)=-0.984$, calculate the next $f(m)$, where $m =$ __________. | 2.5 | 35.9375 |
23,845 | (1) If 7 students stand in a row, and students A and B must stand next to each other, how many different arrangements are there?
(2) If 7 students stand in a row, and students A, B, and C must not stand next to each other, how many different arrangements are there?
(3) If 7 students stand in a row, with student A not standing at the head and student B not standing at the tail, how many different arrangements are there? | 3720 | 89.84375 |
23,846 | Jason wishes to purchase some comic books. He has $15 and each comic book costs $1.20, tax included. Additionally, there is a discount of $0.10 on each comic book if he buys more than 10 comic books. What is the maximum number of comic books he can buy? | 12 | 3.125 |
23,847 | The repeating decimal for $\frac{5}{13}$ is $0.cdc\ldots$ What is the value of the sum $c+d$? | 11 | 87.5 |
23,848 | To solve the problem, we need to find the value of $\log_{4}{\frac{1}{8}}$.
A) $-\frac{1}{2}$
B) $-\frac{3}{2}$
C) $\frac{1}{2}$
D) $\frac{3}{2}$ | -\frac{3}{2} | 92.96875 |
23,849 | Find the sum of $642_8$ and $157_8$ in base $8$. | 1021_8 | 97.65625 |
23,850 | Given that $f(x)$ is a periodic function defined on $\mathbb{R}$ with a period of $2$, when $x \in (-1, 1]$, $f(x)=\begin{cases} -4x^{2}+ \frac{9}{8},-1 < x\leqslant 0, \\ \log _{2}x,0 < x\leqslant 1, \end{cases}$, find the value of $f(f( \frac{7}{2}))=\_\_\_\_\_\_\_\_$. | -3 | 71.875 |
23,851 | Charlie is planning to construct a boundary around a rectangular playground using exactly 380 feet of fencing. Regulations specify that the length of the boundary must be at least 100 feet and the width must be at least 60 feet. Charlie wants to maximize the area enclosed by the fence for more play equipment and sitting area. What are the optimal dimensions and the maximum possible area of the playground? | 9000 | 71.09375 |
23,852 | Given vectors $\overrightarrow {a}$=（sinx，cosx）, $\overrightarrow {b}$=（1，$\sqrt {3}$）.
(1) If $\overrightarrow {a}$$∥ \overrightarrow {b}$, find the value of tanx;
(2) Let f(x) = $\overrightarrow {a}$$$\cdot \overrightarrow {b}$, stretch the horizontal coordinates of each point on the graph of f(x) to twice their original length (vertical coordinates remain unchanged), then shift all points to the left by φ units (0 < φ < π), obtaining the graph of function g(x). If the graph of g(x) is symmetric about the y-axis, find the value of φ. | \frac{\pi}{3} | 56.25 |
23,853 | The distance from the point $(3,0)$ to one of the asymptotes of the hyperbola $\frac{{x}^{2}}{16}-\frac{{y}^{2}}{9}=1$ is $\frac{9}{5}$. | \frac{9}{5} | 88.28125 |
23,854 | Given the function $f(x)=ax + a^{-x}$ ($a>0$ and $a\neq1$), and $f(1)=3$, find the value of $f(0)+f(1)+f(2)$. | 12 | 57.03125 |
23,855 | If the scores for innovation capability, innovation value, and innovation impact are $8$ points, $9$ points, and $7$ points, respectively, and the total score is calculated based on the ratio of $5:3:2$ for the three scores, calculate the total score of the company. | 8.1 | 76.5625 |
23,856 | Find $x$ such that $\log_x 49 = \log_2 32$. | 7^{2/5} | 9.375 |
23,857 | Given the hyperbola $C$: $mx^{2}+ny^{2}=1(mn < 0)$, one of its asymptotes is tangent to the circle $x^{2}+y^{2}-6x-2y+9=0$. Determine the eccentricity of $C$. | \dfrac {5}{4} | 71.875 |
23,858 | John wants to find all the five-letter words that begin and end with the same letter. How many combinations of letters satisfy this property? | 456976 | 7.03125 |
23,859 | On June 14, 2018, the 21st FIFA World Cup will kick off in Russia. A local sports channel organized fans to guess the outcomes of the matches for the top four popular teams: Germany, Spain, Argentina, and Brazil. Each fan can choose one team from the four, and currently, three people are participating in the guessing game.
$(1)$ If each of the three people can choose any team and the selection of each team is equally likely, find the probability that exactly two teams are chosen by people.
$(2)$ If one of the three people is a female fan, assuming the probability of the female fan choosing the German team is $\frac{1}{3}$ and the probability of a male fan choosing the German team is $\frac{2}{5}$, let $\xi$ be the number of people choosing the German team among the three. Find the probability distribution and the expected value of $\xi$. | \frac{17}{15} | 56.25 |
23,860 | What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 | 23.4375 |
23,861 | Let \( x, y, z \) be complex numbers such that:
\[
xy + 3y = -9,
yz + 3z = -9,
zx + 3x = -9.
\]
Determine all possible values of \( xyz \). | 27 | 48.4375 |
23,862 | Find the smallest positive integer that is both an integer power of 7 and is not a palindrome. | 2401 | 7.8125 |
23,863 | Find the sum of the squares of the solutions to the equation
\[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\] | \frac{2008}{1005} | 39.84375 |
23,864 | Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. If $a_5=5S_5=15$, find the sum of the first $100$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$. | \frac{100}{101} | 5.46875 |
23,865 | If the shortest distance from a point on the ellipse $\frac{y^2}{16} + \frac{x^2}{9} = 1$ to the line $y = x + m$ is $\sqrt{2}$, find the minimum value of $m$. | -7 | 14.84375 |
23,866 | $ABC$ is triangle. $l_1$ - line passes through $A$ and parallel to $BC$ , $l_2$ - line passes through $C$ and parallel to $AB$ . Bisector of $\angle B$ intersect $l_1$ and $l_2$ at $X,Y$ . $XY=AC$ . What value can take $\angle A- \angle C$ ? | 60 | 20.3125 |
23,867 | If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$ ? | 10 | 77.34375 |
23,868 | A student named Zhang has a set of 6 questions to choose from, with 4 categorized as type A and 2 as type B. Zhang randomly selects 2 questions to solve.
(1) What is the probability that both selected questions are type A?
(2) What is the probability that the selected questions are not of the same type? | \frac{8}{15} | 66.40625 |
23,869 | We flip a fair coin 12 times. What is the probability that we get exactly 9 heads and all heads occur consecutively? | \dfrac{1}{1024} | 70.3125 |
23,870 | Given that Chelsea is ahead by 60 points halfway through a 120-shot archery contest, with each shot scoring 10, 8, 5, 3, or 0 points and Chelsea scoring at least 5 points on every shot, determine the smallest number of bullseyes (10 points) Chelsea needs to shoot in her next n attempts to ensure victory, assuming her opponent can score a maximum of 10 points on each remaining shot. | 49 | 38.28125 |
23,871 | Cagney can frost a cupcake every 15 seconds, while Lacey can frost every 40 seconds. They take a 10-second break after every 10 cupcakes. Calculate the number of cupcakes that they can frost together in 10 minutes. | 50 | 7.03125 |
23,872 | We need to arrange the performance order for 4 singing programs and 2 skit programs. The requirement is that there must be exactly 3 singing programs between the 2 skit programs. The number of possible arrangements is \_\_\_\_\_\_ . (Answer with a number) | 96 | 43.75 |
23,873 | Triangle $PQR$ has side lengths $PQ=6$, $QR=8$, and $PR=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$? | 5.5 | 66.40625 |
23,874 | Given $f(x) = \sin x + a\cos x$,
(1) If $a= \sqrt{3}$, find the maximum value of $f(x)$ and the corresponding value of $x$.
(2) If $f\left(\frac{\pi}{4}\right) = 0$ and $f(x) = \frac{1}{5}$ $(0 < x < \pi)$, find the value of $\tan x$. | \frac{4}{3} | 74.21875 |
23,875 | In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is given by
$$\begin{cases}
x= \dfrac { \sqrt {2}}{2}t \\
y= \dfrac { \sqrt {2}}{2}t+4 \sqrt {2}
\end{cases}
(t \text{ is the parameter}),$$
establishing a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation of circle $C$ is $ρ=2\cos \left(θ+ \dfrac {π}{4}\right)$.
(Ⅰ) Find the Cartesian coordinates of the center $C$ of the circle;
(Ⅱ) From any point on line $l$, draw a tangent to the circle $C$, and find the minimum length of the tangent line. | 2 \sqrt {6} | 0 |
23,876 | Four different balls are to be placed into five boxes numbered 1, 2, 3, 4, and 5. (Write the result as a number)
(1) How many ways can there be with at least one ball in box number 1?
(2) How many ways are there to have exactly two empty boxes?
(3) How many ways are there to have exactly three empty boxes?
(4) How many ways are there such that the box number containing ball A is not less than the box number containing ball B? | 375 | 29.6875 |
23,877 | Given that the merchant purchased $1200$ keychains at $0.15$ each and desired to reach a target profit of $180$, determine the minimum number of keychains the merchant must sell if each is sold for $0.45$. | 800 | 31.25 |
23,878 | Given a sequence $\{a\_n\}$ that satisfies $a\_1=1$ and $a\_n= \frac{2S\_n^2}{2S\_n-1}$ for $n\geqslant 2$, where $S\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\_{2016}$. | \frac{1}{4031} | 82.8125 |
23,879 | In response to the national medical and health system reform and the "Three Visits to the Countryside" cultural and scientific activities nationwide in 2023, to truly implement the concept of "putting the people first" and promote the transfer and decentralization of high-quality medical resources, continuously enhance the "depth" and "warmth" of medical services. The People's Hospital of our city plans to select 3 doctors from the 6 doctors recommended by each department to participate in the activity of "Healthy Countryside Visit, Free Clinic Warming Hearts." Among these 6 doctors, there are 2 surgeons, 2 internists, and 2 ophthalmologists.
- $(1)$ Find the probability that the number of selected surgeons is greater than the number of selected internists.
- $(2)$ Let $X$ represent the number of surgeons selected out of the 3 people. Find the mean and variance of $X$. | \frac{2}{5} | 79.6875 |
23,880 | Given the word 'ARROW', find the probability that a random arrangement of its letters will have both R's next to each other. | \frac{2}{5} | 84.375 |
23,881 | Given Harry has 4 sisters and 6 brothers, and his sister Harriet has S sisters and B brothers, calculate the product of S and B. | 24 | 17.1875 |
23,882 | Given a periodic sequence $\left\{x_{n}\right\}$ that satisfies $x_{n}=\left|x_{n-1}-x_{n-2}\right|(n \geqslant 3)$, if $x_{1}=1$ and $x_{2}=a \geqslant 0$, calculate the sum of the first 2002 terms when the period of the sequence is minimized. | 1335 | 85.9375 |
23,883 | Using systematic sampling to select 20 students from 1000, the students are randomly numbered from 000 to 999 and grouped: the first group ranges from 000 to 049, the second group from 050 to 099, ..., and the twentieth group from 950 to 999. If the number 122 from the third group is selected, then the number of the student selected in the eighteenth group would be: ______. | 872 | 88.28125 |
23,884 | Given the function $f(x)=2\sin x\cos x+2 \sqrt {3}\cos ^{2}x- \sqrt {3}$, where $x\in\mathbb{R}$.
(Ⅰ) Find the smallest positive period and the intervals of monotonic decrease for the function $y=f(-3x)+1$;
(Ⅱ) Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If the acute angle $A$ satisfies $f\left( \frac {A}{2}- \frac {\pi}{6}\right)= \sqrt {3}$, and $a=7$, $\sin B+\sin C= \frac {13 \sqrt {3}}{14}$, find the area of $\triangle ABC$. | 10 \sqrt {3} | 0 |
23,885 | On a bustling street, a middle-aged man is shouting "giving away money" while holding a small black cloth bag in his hand. Inside the bag, there are 3 yellow and 3 white ping-pong balls (which are identical in volume and texture). Next to him, there's a small blackboard stating:
Method of drawing balls: Randomly draw 3 balls from the bag. If the 3 balls drawn are of the same color, the stall owner will give the drawer $10; if the 3 balls drawn are not of the same color, the drawer will pay the stall owner $2.
(1) What is the probability of drawing 3 yellow balls?
(2) What is the probability of drawing 2 yellow balls and 1 white ball?
(3) Assuming there are 80 draws per day, estimate how much money the stall owner can make in a month (30 days) from a probabilistic perspective? | 1920 | 24.21875 |
23,886 | There are 3 small balls of each of the red, yellow, and blue colors, with the same size. Each of the 3 balls of each color is marked with numbers 1, 2, and 3. If 3 balls are randomly drawn, find the probability that neither their colors nor their numbers are the same. | \frac{1}{14} | 12.5 |
23,887 | If the function $f\left(x\right)=\frac{1}{2}\left(m-2\right){x}^{2}+\left(n-8\right)x+1\left(m\geqslant 0,n\geqslant 0\right)$ is monotonically decreasing in the interval $\left[\frac{1}{2},2\right]$, find the maximum value of $mn$. | 18 | 29.6875 |
23,888 | In a geometric sequence $\{a_n\}$, it is known that $a_1 = -2$ and $S_3 = -\frac{7}{2}$. Then, the common ratio $q$ equals \_\_\_\_\_\_ . | -\frac{3}{2} | 27.34375 |
23,889 | 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 $m$? | 60 | 76.5625 |
23,890 | Given that the scores X of 10000 people approximately follow a normal distribution N(100,13^2), it is given that P(61 < X < 139)=0.997, find the number of people scoring no less than 139 points in this exam. | 15 | 60.15625 |
23,891 | Chloe chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Max chooses a real number uniformly at random from the interval $[0, 4500]$. Determine the probability that Max's number is at least twice as large as Chloe's number. | \frac{3}{8} | 7.8125 |
23,892 | Humanity finds 12 habitable planets, of which 6 are Earth-like and 6 are Mars-like. Earth-like planets require 3 units of colonization resources, while Mars-like need 1 unit. If 18 units of colonization resources are available, how many different combinations of planets can be colonized, assuming each planet is unique? | 136 | 7.8125 |
23,893 | If $x = 151$ and $x^3y - 3x^2y + 3xy = 3423000$, what is the value of $y$? | \frac{3423000}{3375001} | 0.78125 |
23,894 | In a batch of 100 products, there are 98 qualified products and 2 defective ones. During product inspection, 3 products are randomly selected from the 100 products.
(1) How many different ways are there to select the 3 products?
(2) How many ways are there to select exactly 1 defective product out of the 3?
(3) How many ways are there to select at least 1 defective product out of the 3? | 9604 | 96.09375 |
23,895 | Given the cyclist encounters red lights at each of 4 intersections with probability $\frac{1}{3}$ and the events of encountering red lights are independent, calculate the probability that the cyclist does not encounter red lights at the first two intersections and encounters the first red light at the third intersection. | \frac{4}{27} | 78.125 |
23,896 | Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$ . Find $3m + n$ .
| 46 | 83.59375 |
23,897 | Given the sequence $\{a_n\}$ where $a_n > 0$, $a_1=1$, $a_{n+2}= \frac {1}{a_n+1}$, and $a_{100}=a_{96}$, find the value of $a_{2014}+a_3$. | \frac{\sqrt{5}}{2} | 37.5 |
23,898 | Given $0 < \beta < \alpha < \frac{\pi}{2}$, point $P(1,4 \sqrt{3})$ is a point on the terminal side of angle $\alpha$, and $\sin \alpha \sin \left(\frac{\pi}{2}-\beta \right)+\cos \alpha \cos \left(\frac{\pi}{2}+\beta \right)= \frac{3 \sqrt{3}}{14}$, calculate the value of angle $\beta$. | \frac{\pi}{3} | 64.84375 |
23,899 | Calculate the area, in square units, of the triangle formed by the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2) (x-1)$. | 45 | 68.75 |
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