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20,900 | Given an ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{1}{2}$, and a circle centered at the origin with the radius equal to the semi-minor axis of the ellipse is tangent to the line $\sqrt{7}x - \sqrt{5}y + 12 = 0$.
(1) Find the equation of ellipse C;
(2) Let A(-4, 0), and a line l, which does not coincide with the x-axis, passes through point R(3, 0) and intersects ellipse C at points P and Q. Lines AP and AQ intersect the line $x = \frac{16}{3}$ at points M and N, respectively. If the slopes of lines MR and NR are $k_1$ and $k_2$ respectively, is $k_1k_2$ a constant? If so, find the constant. If not, explain why. | -\frac{12}{7} | 42.96875 |
20,901 | First, find the derivative of the following functions and calculate the derivative at \\(x=\pi\\).
\\((1) f(x)=(1+\sin x)(1-4x)\\) \\((2) f(x)=\ln (x+1)-\dfrac{x}{x+1}\\). | \dfrac{\pi}{(\pi+1)^{2}} | 0 |
20,902 | Given the fractional equation about $x$: $\frac{x+m}{x+2}-\frac{m}{x-2}=1$ has a solution not exceeding $6$, and the inequality system about $y$: $\left\{\begin{array}{l}{m-6y>2}\\{y-4\leq 3y+4}\end{array}\right.$ has exactly four integer solutions, then the sum of the integers $m$ that satisfy the conditions is ____. | -2 | 42.1875 |
20,903 | Square A has side lengths each measuring $x$ inches. Square B has side lengths each measuring $5x$ inches. Square C has side lengths each measuring $2x$ inches. What is the ratio of the area of Square A to the area of Square B, and what is the ratio of the area of Square C to the area of Square B? Express each answer as a common fraction. | \frac{4}{25} | 96.875 |
20,904 | Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. | 1 + 3\sqrt{2} | 87.5 |
20,905 | Given: The lengths of the three sides of a triangle, $a$, $b$, and $c$, are integers, and $a \leq b < c$, where $b = 5$. Calculate the number of such triangles. | 10 | 55.46875 |
20,906 | How many groups of integer solutions are there for the equation $xyz = 2009$? | 72 | 18.75 |
20,907 | If exactly three of the balls match the numbers of their boxes, calculate the number of different ways to place the balls. | 10 | 28.125 |
20,908 | Given vectors $\overrightarrow{\alpha}$, $\overrightarrow{\beta}$, $\overrightarrow{\gamma}$ satisfy $|\overrightarrow{\alpha}|=1$, $|\overrightarrow{\alpha}-\overrightarrow{\beta}|=|\overrightarrow{\beta}|$, $(\overrightarrow{\alpha}-\overrightarrow{\gamma}) \cdot (\overrightarrow{\beta}-\overrightarrow{\gamma})=0$. If for every determined $\overrightarrow{\beta}$, the maximum and minimum values of $|\overrightarrow{\gamma}|$ are $m$ and $n$ respectively, then for any $\overrightarrow{\beta}$, the minimum value of $m-n$ is \_\_\_\_\_\_\_\_. | \frac{1}{2} | 3.125 |
20,909 | In convex quadrilateral \(WXYZ\), \(\angle W = \angle Y\), \(WZ = YX = 150\), and \(WX \ne ZY\). The perimeter of \(WXYZ\) is 520. Find \(\cos W\). | \frac{11}{15} | 49.21875 |
20,910 | The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to each teacher in her school. Betsy had 15 pieces of candy left over. Find the least number of pieces of candy the Tasty Candy Company could have placed in each one-pound bag. | 302 | 5.46875 |
20,911 | The constant term in the expansion of the binomial $\left(\frac{1}{\sqrt{x}} - x^2\right)^{10}$ is ______. | 45 | 98.4375 |
20,912 | Consider four functions, labelled from (6) to (9). The domain of function (7) is $\{-6, -5, -4, -3, -2, -1, 0, 1\}$. Determine the product of the labels of the functions that are invertible. The graphs are as follows:
- Function (6): $y = x^3 - 3x$
- Function (7): A set of discrete points: $\{(-6, 3), (-5, -5), (-4, 1), (-3, 0), (-2, -2), (-1, 4), (0, -4), (1, 2)\}$
- Function (8): $y = \tan(x)$, defined over $(-\frac{\pi}{2}, \frac{\pi}{2})$
- Function (9): $y = \frac{5}{x}$, defined over all real numbers except $x = 0$ | 504 | 96.09375 |
20,913 | Given the function $y=\cos (2x-\frac{\pi }{4})$, determine the horizontal translation of the graph of the function $y=\sin 2x$. | \frac{\pi }{8} | 45.3125 |
20,914 | The Great Eighteen Hockey League is divided into two divisions, with nine teams in each division. Each team plays each of the other teams in its own division three times and every team in the other division twice. How many league games are scheduled? | 378 | 85.9375 |
20,915 | Guangcai Kindergarten has a total of 180 books, of which 40% are given to the senior class. The remaining books are divided between the junior and middle classes in a ratio of 4:5. How many books does each of the junior and middle classes get? | 60 | 14.84375 |
20,916 | There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members.
| \binom{2003}{11} | 0 |
20,917 | Point P is any point on the surface of the circumscribed sphere of a cube ABCD-A1B1C1D1 with edge length 2. What is the maximum volume of the tetrahedron P-ABCD? | \frac{4(1+\sqrt{3})}{3} | 2.34375 |
20,918 | Using the side lengths 2, 3, 5, 7, and 11, how many different triangles with exactly two equal sides can be formed? | 14 | 12.5 |
20,919 | An Englishman owns a plot of land in Russia. He knows that, in the units familiar to him, the size of his plot is three acres. The cost of the land is 250,000 rubles per hectare. It is known that 1 acre = 4840 square yards, 1 yard = 0.9144 meters, and 1 hectare = 10,000 square meters. Calculate how much the Englishman will earn from the sale. | 303514 | 3.90625 |
20,920 | We plotted the graph of the function \( f(x) = \frac{1}{x} \) in the coordinate system. How should we choose the new, still equal units on the axes, if we want the curve to become the graph of the function \( g(x) = \frac{2}{x} \)? | \frac{\sqrt{2}}{2} | 0 |
20,921 | Daniel worked for 50 hours per week for 10 weeks during the summer, earning \$6000. If he wishes to earn an additional \$8000 during the school year which lasts for 40 weeks, how many fewer hours per week must he work compared to the summer if he receives the same hourly wage? | 33.33 | 57.03125 |
20,922 | Given triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. It is known that $a=5$ and $\sin A= \frac{\sqrt{5}}{5}$.
(1) If the area of triangle $ABC$ is $\sqrt{5}$, find the minimum value of the perimeter $l$.
(2) If $\cos B= \frac{3}{5}$, find the value of side $c$. | 11 | 74.21875 |
20,923 | In the expansion of $(x^{4}+y^{2}+\frac{1}{2xy})^{7}$, the constant term is ______. | \frac{105}{16} | 76.5625 |
20,924 | Consider the set $\{0.34,0.304,0.034,0.43\}$. The sum of the smallest and largest numbers in the set is | 0.464 | 86.71875 |
20,925 | $2016$ bugs are sitting in different places of $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies away. What is the greatest number of contacts, which can be reached by bugs? | 1008^2 | 0 |
20,926 | A teacher received a number of letters from Monday to Friday, which were 10, 6, 8, 5, 6, respectively. The variance $s^2$ of this set of data is \_\_\_\_\_\_. | 3.2 | 82.03125 |
20,927 | What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers? | 61 | 32.8125 |
20,928 | Given the expressions $(2401^{\log_7 3456})^{\frac{1}{2}}$, calculate its value. | 3456^2 | 0 |
20,929 | For every positive integer \( n \), define \( a_{n} \) as the last digit of the sum of the digits of the number formed by writing "2005" \( n \) times consecutively. For example, \(\mathrm{a}_{1}=7\) and \(\mathrm{a}_{2}=4\).
a) What are the positive integers \( n \) such that \( a_{n}=0 \)?
b) Calculate \( a_{1}+a_{2}+\cdots+a_{2005} \). | 9025 | 74.21875 |
20,930 | Among the four students A, B, C, and D participating in competitions in mathematics, writing, and English, each subject must have at least one participant (and each participant can only choose one subject). If students A and B cannot participate in the same competition, the total number of different participation schemes is _____. (Answer with a number) | 30 | 93.75 |
20,931 | The area of two parallel plane sections of a sphere are $9 \pi$ and $16 \pi$. The distance between the planes is given. What is the surface area of the sphere? | 100\pi | 32.03125 |
20,932 | Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$. | 3\sqrt{2} | 87.5 |
20,933 | Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions. | 36 | 50 |
20,934 | In an opaque bag, there are a total of 50 glass balls in red, black, and white colors. Except for the color, everything else is the same. After several trials of drawing balls, Xiaogang found that the probability of drawing a red or black ball is stable at 15% and 45%, respectively. What could be the possible number of white balls in the bag? | 20 | 46.09375 |
20,935 | Ilya has a one-liter bottle filled with freshly squeezed orange juice and a 19-liter empty jug. Ilya pours half of the bottle's contents into the jug, then refills the bottle with half a liter of water and mixes everything thoroughly. He repeats this operation a total of 10 times. Afterward, he pours all that is left in the bottle into the jug. What is the proportion of orange juice in the resulting drink in the jug? If necessary, round the answer to the nearest 0.01. | 0.05 | 33.59375 |
20,936 | In the polar coordinate system, find the length of the segment cut by the curve $\rho=1$ from the line $\rho\sin\theta-\rho\cos\theta=1$. | \sqrt{2} | 72.65625 |
20,937 | For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{42}$ a repeating decimal? | 20 | 14.0625 |
20,938 | Given 5 balls with 2 identical black balls and one each of red, white, and blue, calculate the number of different arrangements of 4 balls in a row. | 60 | 50.78125 |
20,939 | Let \\(α\\) and \\(β\\) be in \\((0,π)\\), and \\(\sin(α+β) = \frac{5}{13}\\), \\(\tan \frac{α}{2} = \frac{1}{2}\\). Find the value of \\(\cos β\\). | -\frac{16}{65} | 8.59375 |
20,940 | The maximum distance from a point on the ellipse $$\frac {x^{2}}{16}+ \frac {y^{2}}{4}=1$$ to the line $$x+2y- \sqrt {2}=0$$ is \_\_\_\_\_\_. | \sqrt{10} | 51.5625 |
20,941 | Determine how many regions of space are divided by:
a) The six planes of a cube's faces.
b) The four planes of a tetrahedron's faces. | 15 | 92.96875 |
20,942 | How many subsets of the set $\{1, 2, 3, 4, 5, 6\}$ must contain the number 6 and at least one of the numbers 1 or 2? | 24 | 97.65625 |
20,943 | What is the sum of all two-digit positive integers whose squares end with the digits 25? | 495 | 17.96875 |
20,944 | If $P = 3012 \div 4$, $Q = P \div 2$, and $Y = P - Q$, then what is the value of $Y$? | 376.5 | 100 |
20,945 | If the numbers $1, 2, 3, 4, 5, 6$ are randomly arranged in a row, represented as $a, b, c, d, e, f$, what is the probability that the number $a b c + d e f$ is odd? | 1/10 | 21.09375 |
20,946 | Suppose that the angles of triangle $ABC$ satisfy
\[\cos 3A + \cos 3B + \cos 3C = 1.\]
Two sides of the triangle have lengths 8 and 15. Find the maximum length of the third side assuming one of the angles is $150^\circ$. | \sqrt{289 + 120\sqrt{3}} | 31.25 |
20,947 | Among all natural numbers not greater than 200, how many numbers are coprime to both 2 and 3 and are not prime numbers? | 23 | 11.71875 |
20,948 | Nine positive integers $a_1,a_2,...,a_9$ have their last $2$ -digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$ -digit part of the sum of their squares. | 85 | 98.4375 |
20,949 | Given the function $f\left( x \right)=\frac{{{x}^{2}}}{1+{{x}^{2}}}$.
(1) Find the values of $f\left( 2 \right)+f\left( \frac{1}{2} \right),f\left( 3 \right)+f\left( \frac{1}{3} \right),f\left( 4 \right)+f\left( \frac{1}{4} \right)$ and conjecture a general conclusion (proof is not required).
(2) Evaluate: $2f\left(2\right)+2f\left(3\right)+⋯+2f\left(2017\right)+f\left( \frac{1}{2}\right)+f\left( \frac{1}{3}\right)+⋯+f\left( \frac{1}{2017}\right)+ \frac{1}{{2}^{2}}f\left(2\right)+ \frac{1}{{3}^{2}}f\left(3\right)+⋯ \frac{1}{{2017}^{2}}f\left(2017\right)$ | 4032 | 53.125 |
20,950 | A certain company has two research and development teams, Team A and Team B. The probability of success for developing a new product by Team A is $\frac{4}{5}$, and for Team B is $\frac{3}{4}$. Team A is assigned to develop a new product $A$, and Team B is assigned to develop a new product $B$. It is assumed that the research and development of Teams A and B are independent of each other.
$(1)$ Find the probability that exactly one new product is successfully developed.
$(2)$ If the development of new product $A$ is successful, the company will make a profit of $150$ thousand dollars, otherwise it will incur a loss of $60$ thousand dollars. If the development of new product $B$ is successful, the company will make a profit of $120$ thousand dollars, otherwise it will incur a loss of $40$ thousand dollars. Find the probability distribution and the mathematical expectation $E(\xi)$ of the company's profit (in thousand dollars). | 188 | 55.46875 |
20,951 | Given \( \alpha, \beta \in (0, \pi) \), and \( \tan \alpha, \tan \beta \) are the roots of the equation \( x^{2} + 3x + 1 = 0 \), find the value of \( \cos(\alpha - \beta) \). | \frac{2}{3} | 68.75 |
20,952 | Given $\begin{vmatrix} p & q \\ r & s \end{vmatrix} = 6,$ find
\[\begin{vmatrix} p & 9p + 4q \\ r & 9r + 4s \end{vmatrix}.\] | 24 | 96.875 |
20,953 | Compute $63 \times 57$ in your head. | 3591 | 76.5625 |
20,954 | A function $f(x) = a \cos ωx + b \sin ωx (ω > 0)$ has a minimum positive period of $\frac{π}{2}$. The function reaches its maximum value of $4$ at $x = \frac{π}{6}$.
1. Find the values of $a$, $b$, and $ω$.
2. If $\frac{π}{4} < x < \frac{3π}{4}$ and $f(x + \frac{π}{6}) = \frac{4}{3}$, find the value of $f(\frac{x}{2} + \frac{π}{6})$. | -\frac{4\sqrt{6}}{3} | 11.71875 |
20,955 | Draw five lines \( l_1, l_2, \cdots, l_5 \) on a plane such that no two lines are parallel and no three lines pass through the same point.
(1) How many intersection points are there in total among these five lines? How many intersection points are there on each line? How many line segments are there among these five lines?
(2) Considering these line segments as sides, what is the maximum number of isosceles triangles that can be formed? Please briefly explain the reasoning and draw the corresponding diagram. | 10 | 53.125 |
20,956 | Given that $C_{n}^{4}$, $C_{n}^{5}$, and $C_{n}^{6}$ form an arithmetic sequence, find the value of $C_{n}^{12}$. | 91 | 62.5 |
20,957 | Find the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^4}{9^4 - 1} + \frac{3^8}{9^8 - 1} + \cdots.$$ | \frac{1}{2} | 54.6875 |
20,958 | The real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_{k} - y_{k+1} \right| \). | 2000 | 40.625 |
20,959 | The lateral surface of a cylinder unfolds into a square. What is the ratio of its lateral surface area to the base area. | 4\pi | 75.78125 |
20,960 | The inclination angle of the line $x- \sqrt {3}y+3=0$ is \_\_\_\_\_\_. | \frac {\pi}{6} | 79.6875 |
20,961 | A and B are running on a circular track at their respective constant speeds. If both start running from point A in opposite directions, and after their first meeting, B takes another 8 minutes to return to the starting point. Given that A takes 6 minutes to complete a lap, how many minutes does it take for B to complete a lap? | 12 | 56.25 |
20,962 | If $\sqrt[3]{0.3}\approx 0.6694$ and $\sqrt[3]{3}\approx 1.442$, then $\sqrt[3]{300}\approx$____. | 6.694 | 40.625 |
20,963 | Determine how many divisors of \(9!\) are multiples of 3. | 128 | 70.3125 |
20,964 | Given triangle $ABC$ where $AB=6$, $\angle A=30^\circ$, and $\angle B=120^\circ$, find the area of $\triangle ABC$. | 9\sqrt{3} | 78.125 |
20,965 | How many minutes are needed at least to finish these tasks: washing rice for 2 minutes, cooking porridge for 10 minutes, washing vegetables for 3 minutes, and chopping vegetables for 5 minutes. | 12 | 43.75 |
20,966 | Consider a two-digit number in base 12 represented by $AB_{12}$, where $A$ and $B$ are duodecimal digits (0 to 11), and $A \neq B$. When this number is reversed to $BA_{12}$, under what condition is a particular prime number a necessary factor of the difference $AB_{12} - BA_{12}$? | 11 | 100 |
20,967 | Let $(2x+1)^6 = a_0x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6$, which is an identity in $x$ (i.e., it holds for any $x$). Try to find the values of the following three expressions:
(1) $a_0 + a_1 + a_2 + a_3 + a_4 + a_5 + a_6$;
(2) $a_1 + a_3 + a_5$;
(3) $a_2 + a_4$. | 300 | 11.71875 |
20,968 | Find the area of quadrilateral ABCD given that $\angle A = \angle D = 120^{\circ}$, $AB = 5$, $BC = 7$, $CD = 3$, and $DA = 4$. | \frac{47\sqrt{3}}{4} | 39.84375 |
20,969 | Given \\(\sin \theta + \cos \theta = \frac{3}{4}\\), where \\(\theta\\) is an angle of a triangle, the value of \\(\sin \theta - \cos \theta\\) is \_\_\_\_\_. | \frac{\sqrt{23}}{4} | 53.125 |
20,970 | $ABCDEFGH$ is a rectangular prism with $AB=CD=EF=GH=1$, $AD=BC=EH=FG=2$, and $AE=BF=CG=DH=3$. Find $\sin \angle GAC$. | \frac{3}{\sqrt{14}} | 0.78125 |
20,971 | Evaluate the expression given by $$2+\cfrac{3}{4+\cfrac{5}{6+\cfrac{7}{8}}}.$$ | \frac{137}{52} | 3.90625 |
20,972 | A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("fipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it. Student 5 then goes through and "flips" every 5th locker. This process continues with all students with odd numbers $n < 100$ going through and "flipping" every $n$ th locker. How many lockers are open after this process? | 10 | 85.9375 |
20,973 | Eighty bricks, each measuring $3''\times9''\times18''$, are to be stacked one on top of another to form a tower 80 bricks tall. Each brick can be oriented so it contributes $3''$, $9''$, or $18''$ to the total height of the tower. How many different tower heights can be achieved using all eighty of the bricks? | 401 | 21.09375 |
20,974 | $(1)$ Calculate: $\sqrt{9}+2\sin30{}°-(π-3){}°$;<br/>$(2)$ Solve the equation: $\left(2x-3\right)^{2}=2\left(2x-3\right)$. | \frac{5}{2} | 50 |
20,975 | What is the base five product of the numbers $203_{5}$ and $14_{5}$? | 3402_5 | 91.40625 |
20,976 | Given that $D$ is a point on the side $AB$ of $\triangle ABC$, and $\overrightarrow{CD} = \frac{1}{3}\overrightarrow{AC} + \lambda \cdot \overrightarrow{BC}$, determine the value of the real number $\lambda$. | -\frac{4}{3} | 64.0625 |
20,977 | The minimum value of the function $y=|x-1|$ is 0, the minimum value of the function $y=|x-1|+|x-2|$ is 1, and the minimum value of the function $y=|x-1|+|x-2|+|x-3|$ is 2. What is the minimum value of the function $y=|x-1|+|x-2|+\ldots+|x-10|$? | 25 | 90.625 |
20,978 | A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $60\%$ longer than the minor axis, find the length of the major axis. | 6.4 | 78.90625 |
20,979 | Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from the vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(1, 0, 2) \)
\( A_{2}(1, 2, -1) \)
\( A_{3}(2, -2, 1) \)
\( A_{4}(2, 1, 0) \) | \sqrt{\frac{7}{11}} | 0 |
20,980 | Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \log_b 9^6$. Find the value of $b$. | \frac{5}{9} | 3.125 |
20,981 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{2\sqrt{3}\sin A}{3\sin C}$.
(1) Find the value of $b$;
(2) If $B = \frac{\pi}{3}$, find the maximum area of triangle $ABC$. | \frac{3\sqrt{3}}{16} | 6.25 |
20,982 | Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used? | 430 | 38.28125 |
20,983 | In the sequence $\{a_n\}$, $a_n$ is the closest positive integer to $\sqrt{n}$ ($n \in \mathbb{N}^*$). Compute the sum $\sum_{i=1}^{100}\frac{1}{a_i} = \_\_\_\_\_\_\_\_$. | 19 | 16.40625 |
20,984 | Given $\tan (\alpha +\beta )=7$ and $\tan (\alpha -\beta )=1$, find the value of $\tan 2\alpha$. | -\dfrac{4}{3} | 79.6875 |
20,985 | Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane that contains this given edge. | \frac{\sqrt{3}}{4} | 4.6875 |
20,986 | How many ways are there to put 7 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable? | 11 | 24.21875 |
20,987 | Consider a $3 \times 3$ block of squares as the center area in an array of unit squares. The first ring around this center block contains unit squares that directly touch the block. If the pattern continues as before, how many unit squares are in the $10^{th}$ ring? | 88 | 21.875 |
20,988 | In a pocket, there are eight cards of the same size, among which three are marked with the number $1$, three are marked with the number $2$, and two are marked with the number $3$. The first time, a card is randomly drawn from the pocket and then put back. After that, a second card is drawn randomly. Let the sum of the numbers on the cards drawn the first and second times be $\xi$.
$(1)$ When is the probability of $\xi$ the greatest? Please explain your reasoning.
$(2)$ Calculate the expected value $E(\xi)$ of the random variable $\xi$. | \dfrac {15}{4} | 38.28125 |
20,989 | Calculate $7 \cdot 9\frac{2}{5}$. | 65\frac{4}{5} | 78.90625 |
20,990 | Calculate using a simple method:<br/>$(1)100.2\times 99.8$;<br/>$(2)103^{2}$. | 10609 | 99.21875 |
20,991 | Calculate $3(72+76+80+84+88+92+96+100+104+108)$. | 2700 | 92.1875 |
20,992 | The noon temperatures for ten consecutive days were $78^{\circ}$, $80^{\circ}$, $82^{\circ}$, $85^{\circ}$, $88^{\circ}$, $90^{\circ}$, $92^{\circ}$, $95^{\circ}$, $97^{\circ}$, and $95^{\circ}$ Fahrenheit. The increase in temperature over the weekend days (days 6 to 10) is attributed to a local summer festival. What is the mean noon temperature, in degrees Fahrenheit, for these ten days? | 88.2 | 64.84375 |
20,993 | Simplify and find the value: $4(a^{2}b+ab^{2})-3(a^{2}b-1)+2ab^{2}-6$, where $a=1$, $b=-4$. | 89 | 56.25 |
20,994 | Example: The ancients used Heavenly Stems and Earthly Branches to keep track of order. There are 10 Heavenly Stems: Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, Gui; and 12 Earthly Branches: Zi, Chou, Yin, Mao, Chen, Si, Wu, Wei, Shen, You, Xu, Hai. The 10 characters of the Heavenly Stems and the 12 characters of the Earthly Branches are arranged in two rows in a cyclic manner as follows:
Jia Yi Bing Ding Wu Ji Geng Xin Ren Gui Jia Yi Bing Ding Wu Ji Geng Xin Ren Gui…
Zi Chou Yin Mao Chen Si Wu Wei Shen You Xu Hai Zi Chou Yin Mao Chen Si Wu Wei Shen You Xu Hai…
Counting from left to right, the first column is Jia Zi, the third column is Bing Yin…, the question is, when Jia and Zi are in the same column for the second time,
what is the column number? | 61 | 28.90625 |
20,995 | In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho^{2}-2\rho\cos \theta-4\rho\sin \theta+4=0$, and the equation of line $l$ is $x-y-1=0$.
$(1)$ Write the parametric equation of curve $C$;
$(2)$ Find a point $P$ on curve $C$ such that the distance from point $P$ to line $l$ is maximized, and find this maximum value. | 1+ \sqrt{2} | 4.6875 |
20,996 | Among the four-digit numbers composed of the digits $0$, $1$, $2$, $3$, $4$, $5$ without repetition, there are a total of \_\_\_\_\_ numbers that are not divisible by $5$. | 192 | 88.28125 |
20,997 | Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are face cards (Jacks, Queens, or Kings) totaling to a numeric value of 20? | \frac{11}{221} | 10.15625 |
20,998 | Given \( f(x) = a \sin x + b \sqrt[3]{x} + c \ln \left(x + \sqrt{x^{2} + 1}\right) + 1003 \) (where \( a \), \( b \), and \( c \) are real numbers), and \( f\left(\lg^{2} 10\right) = 1 \), what is \( f(\lg \lg 3) \)? | 2005 | 56.25 |
20,999 | A shopping mall sells a batch of branded shirts, with an average daily sales volume of $20$ shirts, and a profit of $40$ yuan per shirt. In order to expand sales and increase profits, the mall decides to implement an appropriate price reduction strategy. After investigation, it was found that for every $1$ yuan reduction in price per shirt, the mall can sell an additional $2$ shirts on average.
$(1)$ If the price reduction per shirt is set at $x$ yuan, and the average daily profit is $y$ yuan, find the functional relationship between $y$ and $x$.
$(2)$ At what price reduction per shirt will the mall have the maximum average daily profit?
$(3)$ If the mall needs an average daily profit of $1200$ yuan, how much should the price per shirt be reduced? | 20 | 59.375 |
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