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21,900 | Suppose you have an unlimited number of pennies, nickels, dimes, and quarters. Determine the number of ways to make 30 cents using these coins. | 17 | 0 |
21,901 | Given the function f(x) = 2sin(x - π/6)sin(x + π/3), x ∈ R.
(1) Find the smallest positive period of the function f(x) and the center of symmetry of its graph.
(2) In △ABC, if A = π/4, and acute angle C satisfies f(C/2 + π/6) = 1/2, find the value of BC/AB. | \sqrt{2} | 74.21875 |
21,902 | Let $M \Theta N$ represent the remainder when the larger of $M$ and $N$ is divided by the smaller one. For example, $3 \Theta 10 = 1$. For non-zero natural numbers $A$ less than 40, given that $20 \Theta(A \bigodot 20) = 7$, find $A$. | 13 | 33.59375 |
21,903 | Given that the monogram of a person's first, middle, and last initials is in alphabetical order with no letter repeated, and the last initial is 'M', calculate the number of possible monograms. | 66 | 35.15625 |
21,904 | There are two equilateral triangles with a vertex at $(0, 1)$ , with another vertex on the line $y = x + 1$ and with the final vertex on the parabola $y = x^2 + 1$ . Find the area of the larger of the two triangles. | 26\sqrt{3} + 45 | 0 |
21,905 | In $\triangle ABC$, the sides corresponding to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $a=2$, $b=3$, $\cos B= \frac{1}{3}$.
(1) Find the value of side $c$;
(2) Find the value of $\cos (A-C)$. | \frac{23}{27} | 86.71875 |
21,906 | There are $8$ balls of the same size, including $4$ different black balls, $2$ different red balls, and $2$ different yellow balls.<br/>$(1)$ Arrange these $8$ balls in a line, with the black balls together, the 2 red balls adjacent, and the 2 yellow balls not adjacent. Find the number of ways to arrange them;<br/>$(2)$ Take out $4$ balls from these $8$ balls, ensuring that balls of each color are taken. Find the number of ways to do so;<br/>$(3)$ Divide these $8$ balls into three groups, each group having at least $2$ balls. Find the number of ways to divide them. | 490 | 14.84375 |
21,907 | In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 10 and 18 units, respectively, and the altitude is 15 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively, and $G$ is the midpoint of $CD$. Determine the area of triangle $EFG$. | 52.5 | 83.59375 |
21,908 | How many positive integers between 10 and 2016 are divisible by 3 and have all of their digits the same? | 12 | 34.375 |
21,909 | How many positive 3-digit numbers are multiples of 25, but not of 60? | 33 | 50 |
21,910 | Define the operation "" such that $ab = a^2 + 2ab - b^2$. Let the function $f(x) = x2$, and the equation $f(x) = \lg|x + 2|$ (where $x \neq -2$) has exactly four distinct real roots $x_1, x_2, x_3, x_4$. Find the value of $x_1 + x_2 + x_3 + x_4$. | -8 | 24.21875 |
21,911 | Let $u,$ $v,$ and $w$ be the roots of the equation $x^3 - 18x^2 + 20x - 8 = 0.$ Find the value of $(2+u)(2+v)(2+w).$ | 128 | 45.3125 |
21,912 | Given the function $$f(x)= \begin{cases} \overset{2^{x},x\geq 3,}{f(x+1),x<3}\end{cases}$$, find the value of $f(\log_2 6)$. | 12 | 78.125 |
21,913 | Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\cdots+(m+10)^2$ is the square of a positive integer $n$ . Find $m+n$ | 95 | 75 |
21,914 | Calculate the sum of $5.46$, $2.793$, and $3.1$ as a decimal. | 11.353 | 100 |
21,915 | The probability that the random variable $X$ follows a normal distribution $N\left( 3,{{\sigma }^{2}} \right)$ and $P\left( X\leqslant 4 \right)=0.84$ can be expressed in terms of the standard normal distribution $Z$ as $P(Z\leqslant z)=0.84$, where $z$ is the z-score corresponding to the upper tail probability $1-0.84=0.16$. Calculate the value of $P(2<X<4)$. | 0.68 | 40.625 |
21,916 | Given the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{b^{2}} = 1$ with $a > b > 0$ and eccentricity $\frac{\sqrt{3}}{2}$, find the slope of the line that intersects the ellipse at points $A$ and $B$, where the midpoint of segment $AB$ is $M(-2, 1)$. | \frac{1}{2} | 61.71875 |
21,917 | Two distinct positive integers $a$ and $b$ are factors of 48. If $a\cdot b$ is not a factor of 48, what is the smallest possible value of $a\cdot b$? | 32 | 10.9375 |
21,918 | Given that $ab= \frac{1}{4}$, $a$, $b \in (0,1)$, find the minimum value of $\frac{1}{1-a}+ \frac{2}{1-b}$. | 4+ \frac{4 \sqrt{2}}{3} | 20.3125 |
21,919 | There are integers $x$ that satisfy the inequality $|x-2000|+|x| \leq 9999$. Find the number of such integers $x$. | 9999 | 55.46875 |
21,920 | Given real numbers \( a, b, c \) satisfy
\[
a^{2}+b^{2}-4a \leqslant 1, \quad b^{2}+c^{2}-8b \leqslant -3, \quad c^{2}+a^{2}-12c \leqslant -26,
\]
what is the value of \( (a+b)^{c} \)? | 27 | 20.3125 |
21,921 | How many distinct four-digit positive integers are there such that the product of their digits equals 18? | 48 | 0 |
21,922 | The perimeter of a square with side length $y$ inches is equal to the circumference of a circle with radius 5 centimeters. If 1 inch equals 2.54 centimeters, what is the value of $y$ in inches? Express your answer as a decimal to the nearest hundredth. | 3.09 | 5.46875 |
21,923 | Count the total number of possible scenarios in a table tennis match between two players, where the winner is the first one to win three games and they play until a winner is determined. | 20 | 78.125 |
21,924 | Given a geometric sequence $\{a_n\}$ with the first term $\frac{3}{2}$ and common ratio $-\frac{1}{2}$, and the sum of the first $n$ terms is $S_n$, then when $n\in N^*$, the sum of the maximum and minimum values of $S_n - \frac{1}{S_n}$ is ______. | \frac{1}{4} | 39.84375 |
21,925 | An influenza outbreak occurred in three areas, $A$, $B$, and $C$, where $6\%$, $5\%$, and $4\%$ of the population in each area have the flu, respectively. Assuming the population ratios in these three areas are $6:5:4$, if a person is randomly selected from these three areas, the probability that this person has the flu is ______. | \frac{77}{1500} | 34.375 |
21,926 | If the integers $m,n,k$ hold the equation $221m+247n+323k=2001$, find the smallest possible value of $k$ greater than 100. | 111 | 0 |
21,927 | (1) Compute: $\sin 6^\circ \sin 42^\circ \sin 66^\circ \sin 78^\circ$
(2) Given that $\alpha$ is an angle in the second quadrant, and $\sin \alpha = \frac{\sqrt{15}}{4}$, find the value of $\frac{\sin(\alpha + \frac{\pi}{4})}{\sin 2\alpha + \cos 2\alpha + 1}$. | -\sqrt{2} | 35.15625 |
21,928 | The largest prime factor of $999999999999$ is greater than $2006$ . Determine the remainder obtained when this prime factor is divided by $2006$ . | 1877 | 94.53125 |
21,929 | The sequence $b_1, b_2, b_3, \dots$ satisfies $b_1 = 25$, $b_9 = 125$, and for $n \ge 3$, $b_n$ is the geometric mean of the first $n - 1$ terms. Find $b_2$. | 625 | 35.9375 |
21,930 | Simplify $$\frac{13!}{11! + 3 \cdot 9!}$$ | \frac{17160}{113} | 9.375 |
21,931 | What is the sum of the real roots of the equation $4x^4-3x^2+7x-3=0$? | -1 | 31.25 |
21,932 | Five people, named A, B, C, D, and E, stand in a row. If A and B must be adjacent, and B must be to the left of A, what is the total number of different arrangements? | 24 | 74.21875 |
21,933 | Evaluate the ratio $\frac{10^{3000} + 10^{3004}}{10^{3001} + 10^{3003}}$ and determine which whole number it is closest to. | 10 | 89.84375 |
21,934 | A hall is organizing seats in rows for a lecture. Each complete row must contain $13$ chairs. Initially, the hall has $169$ chairs arranged. To maintain fully occupied rows with minimal empty seats, if $100$ students are expected to attend, how many chairs should be removed or added? | 65 | 26.5625 |
21,935 | Given that Nayla has an index card measuring $5 \times 7$ inches, and she shortens the length of one side by $2$ inches, resulting in a card with an area of $21$ square inches, determine the area of the card if instead, she shortens the length of the other side by the same amount. | 25 | 89.0625 |
21,936 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\sin(C+\frac{π}{3})-c\sin B=0$.
$(1)$ Find the value of angle $C$.
$(2)$ If the area of $\triangle ABC$ is $10\sqrt{3}$ and $D$ is the midpoint of $AC$, find the minimum value of $BD$. | 2\sqrt{5} | 31.25 |
21,937 | The robot vacuum cleaner is programmed to move on the floor according to the law:
$$\left\{\begin{array}{l} x = t(t-6)^2 \\ y = 0, \quad 0 \leq t \leq 7 \\ y = (t-7)^2, \quad t \geq 7 \end{array}\right.$$
where the axes are chosen parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and coordinates are measured in meters. Find the distance traveled by the robot in the first 7 minutes and the absolute change in the velocity vector during the eighth minute. | \sqrt{445} | 32.03125 |
21,938 | If \( e^{i \theta} = \frac{3 + i \sqrt{2}}{4}, \) then find \( \cos 3\theta. \) | \frac{9}{64} | 0 |
21,939 | Given lines $l_{1}$: $\rho\sin(\theta-\frac{\pi}{3})=\sqrt{3}$ and $l_{2}$: $\begin{cases} x=-t \\ y=\sqrt{3}t \end{cases}$ (where $t$ is a parameter), find the polar coordinates of the intersection point $P$ of $l_{1}$ and $l_{2}$. Additionally, three points $A$, $B$, and $C$ lie on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, with $O$ being the coordinate origin. If $\angle{AOB}=\angle{BOC}=\angle{COA}=120^{\circ}$, find the value of $\frac{1}{|OA|^{2}}+\frac{1}{|OB|^{2}}+\frac{1}{|OC|^{2}}$. | \frac{15}{8} | 7.8125 |
21,940 | Find the last three digits of $7^{103}.$ | 343 | 85.9375 |
21,941 | Two people are playing a game. One person thinks of a ten-digit number, and the other can ask questions about which digits are in specific sets of positions in the number's sequence. The first person answers the questions without indicating which digits are in which exact positions. What is the minimum number of questions needed to reliably guess the number? | 10 | 55.46875 |
21,942 | Given the function $f(x)=2\sin ωx\cos ωx-2\sqrt{3} \cos ^{2}ωx+\sqrt{3} (ω > 0)$, and the distance between two adjacent symmetry axes of the graph of $y=f(x)$ is $\frac{π}{2}$.
(I) Find the period of the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $C$ is acute, and $f(C)=\sqrt{3}$, $c=3\sqrt{2}$, $\sin B=2\sin A$. Find the area of $\triangle ABC$. | 3\sqrt{3} | 87.5 |
21,943 | Jamie King invested some money in real estate and mutual funds. The total amount he invested was $\$200,\!000$. If he invested 5.5 times as much in real estate as he did in mutual funds, what was his total investment in real estate? | 169,230.77 | 0 |
21,944 | A number in the set $\{50, 51, 52, 53, ... , 500\}$ is randomly selected. What is the probability that it is a two-digit number divisible by 3? Express your answer as a common fraction. | \frac{17}{451} | 54.6875 |
21,945 | Given a triangle $ABC$ with internal angles $A$, $B$, and $C$ opposite to the sides $a$, $b$, and $c$ respectively. If $(2a-c)\cos B=b\cos C$, and the dot product $\vec{AB}\cdot \vec{BC} = -3$,
1. Find the area of $\triangle ABC$;
2. Find the minimum value of side $AC$. | \sqrt{6} | 73.4375 |
21,946 | Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. How many frogs are in the bucket? | 15 | 63.28125 |
21,947 | Given a function $f(x)$ defined on $R$ such that $f(x) + f(x+4) = 23$. When $x \in (0,4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $(-4,2023]$. | 506 | 17.96875 |
21,948 | Given $f(\alpha)= \dfrac {\sin (\pi-\alpha)\cos (2\pi-\alpha)\cos (-\alpha+ \dfrac {3}{2}\pi)}{\cos ( \dfrac {\pi}{2}-\alpha)\sin (-\pi-\alpha)}$
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos (\alpha- \dfrac {3}{2}\pi)= \dfrac {1}{5}$, find the value of $f(\alpha)$;
$(3)$ If $\alpha=- \dfrac {31}{3}\pi$, find the value of $f(\alpha)$. | -\dfrac {1}{2} | 94.53125 |
21,949 | Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with eccentricity $\frac{1}{2}$, a circle $\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\sqrt{6}=0$. <br/>$(1)$ Find the equation of the ellipse $C$; <br/>$(2)$ A line passing through the fixed point $Q(1,0)$ with slope $k$ intersects the ellipse $C$ at points $M$ and $N$. If $\overrightarrow{OM}•\overrightarrow{ON}=-2$, find the value of the real number $k$ and the area of $\triangle MON$. | \frac{6\sqrt{6}}{11} | 53.125 |
21,950 | Given the function $f(x)=\sin ^{2}x+a\sin x\cos x-\cos ^{2}x$, and $f(\frac{\pi }{4})=1$.
(1) Find the value of the constant $a$;
(2) Find the smallest positive period and minimum value of $f(x)$. | -\sqrt{2} | 91.40625 |
21,951 | The variables \(a, b, c, d, e\), and \(f\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that
\[
\begin{aligned}
& a + b = c, \\
& b + c = d, \\
& c + e = f,
\end{aligned}
\]
Determine the value of \(a + c + f\). | 73 | 2.34375 |
21,952 | What is the area of the smallest square that can contain a circle of radius 5? | 100 | 100 |
21,953 | Simplify the expression: $(a-\frac{3a}{a+1}$) $÷\frac{{a}^{2}-4a+4}{a+1}$, then choose a number you like from $-2$, $-1$, and $2$ to substitute for $a$ and calculate the value. | \frac{1}{2} | 10.9375 |
21,954 | A system of inequalities defines a region on a coordinate plane as follows:
$$
\begin{cases}
x+y \leq 5 \\
3x+2y \geq 3 \\
x \geq 1 \\
y \geq 1
\end{cases}
$$
Determine the number of units in the length of the longest side of the quadrilateral formed by the region satisfying all these conditions. Express your answer in simplest radical form. | 3\sqrt{2} | 79.6875 |
21,955 | If the line $2x+my=2m-4$ is parallel to the line $mx+2y=m-2$, find the value of $m$. | -2 | 38.28125 |
21,956 | The graph of $y = ax^2 + bx + c$ is described, where $a$, $b$, and $c$ are integers. Given that the vertex of this parabola is at $(-2, 3)$ and one point on the graph is $(1, 6)$, determine the value of $a$. | \frac{1}{3} | 47.65625 |
21,957 | A digital watch displays time in a 24-hour format showing only hours and minutes. Find the largest possible sum of the digits in the display. | 24 | 0 |
21,958 | Given equations of the form $x^2 + bx + c = 0$, determine the number of such equations that have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1, 2, 3, 4, 5, 7\}$. | 18 | 0 |
21,959 | What is the value of $a^3 - b^3$ given that $a+b=12$ and $ab=20$? | 992 | 76.5625 |
21,960 | Given \\(\cos \left(\alpha+ \frac{\pi}{6}\right)-\sin \alpha= \frac{2 \sqrt{3}}{3} \\), determine the value of \\(\sin \left(\alpha- \frac{7\pi}{6}\right) \\). | \frac{2}{3} | 67.96875 |
21,961 | Given the function $f(x)=4\cos (ωx- \frac {π}{6})\sin (π-ωx)-\sin (2ωx- \frac {π}{2})$, where $ω > 0$.
(1) Find the range of the function $f(x)$.
(2) If $y=f(x)$ is an increasing function in the interval $[- \frac {3π}{2}, \frac {π}{2}]$, find the maximum value of $ω$. | \frac{1}{6} | 22.65625 |
21,962 | Given two lines $$l_{1}: \sqrt {3}x+y-1=0$$ and $$l_{2}: ax+y=1$$, and $l_{1}$ is perpendicular to $l_{2}$, then the slope angle of $l_{1}$ is \_\_\_\_\_\_, and the distance from the origin to $l_{2}$ is \_\_\_\_\_\_. | \frac { \sqrt {3}}{2} | 0 |
21,963 | How many integer quadruples $a,b,c,d$ are there such that $7$ divides $ab-cd$ where $0\leq a,b,c,d < 7$ ? | 385 | 0 |
21,964 | Let $P = (\sqrt{2007}+\sqrt{2008}),$ $Q = (-\sqrt{2007}-\sqrt{2008}),$ $R = (\sqrt{2007}-\sqrt{2008}),$ and $S = (-\sqrt{2008}+\sqrt{2007}).$ Calculate $PQRS.$ | -1 | 39.84375 |
21,965 | Let $ABC$ be a triangle with $AB = 5$ , $AC = 8$ , and $BC = 7$ . Let $D$ be on side $AC$ such that $AD = 5$ and $CD = 3$ . Let $I$ be the incenter of triangle $ABC$ and $E$ be the intersection of the perpendicular bisectors of $\overline{ID}$ and $\overline{BC}$ . Suppose $DE = \frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$ .
*Proposed by Ray Li* | 13 | 3.125 |
21,966 | An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the selected element is $1$? | \frac{13}{70} | 65.625 |
21,967 | Suppose $ABCD$ is a rectangle whose diagonals meet at $E$ . The perimeter of triangle $ABE$ is $10\pi$ and the perimeter of triangle $ADE$ is $n$ . Compute the number of possible integer values of $n$ . | 47 | 1.5625 |
21,968 | The graph of the function $f(x)=\sin({ωx-\frac{π}{6}})$, where $0<ω<6$, is shifted to the right by $\frac{π}{6}$ units to obtain the graph of the function $g(x)$. If $\left(0,\frac{π}{ω}\right)$ is a monotone interval of $g(x)$, and $F(x)=f(x)+g(x)$, determine the maximum value of $F(x)$. | \sqrt{3} | 17.1875 |
21,969 | Let \( p \) and \( q \) be positive integers such that
\[
\frac{6}{11} < \frac{p}{q} < \frac{5}{9}
\]
and \( q \) is as small as possible. What is \( p+q \)? | 31 | 4.6875 |
21,970 | The line $y=2b$ intersects the left and right branches of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ at points $B$ and $C$ respectively, with $A$ being the right vertex and $O$ the origin. If $\angle AOC = \angle BOC$, then calculate the eccentricity of the hyperbola. | \frac{\sqrt{19}}{2} | 0.78125 |
21,971 | Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two axes of symmetry of the graph of the function $y=f(x)$, evaluate the value of $f(-\frac{{5π}}{{12}})$. | \frac{\sqrt{3}}{2} | 20.3125 |
21,972 | A sphere with center $O$ has radius $10$. A right triangle with sides $8, 15,$ and $17$ is situated in 3D space such that each side is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?
- **A)** $\sqrt{84}$
- **B)** $\sqrt{85}$
- **C)** $\sqrt{89}$
- **D)** $\sqrt{91}$
- **E)** $\sqrt{95}$ | \sqrt{91} | 43.75 |
21,973 | The coefficient of the $x^2$ term in the expansion of $(1+x)(1+ \sqrt{x})^5$ is \_\_\_\_\_\_. | 15 | 84.375 |
21,974 | Let \( f \) be a function taking positive integers to positive integers, such that:
(i) \( f \) is increasing (\( f(n + 1) > f(n) \) for all positive integers \( n \))
(ii) \( f(mn) = f(m) f(n) \) for all positive integers \( m \) and \( n \)
(iii) if \( m \neq n \) and \( m^n = n^m \), then \( f(m) = n \) or \( f(n) = m \)
Find all possible values of \( f(60) \). | 3600 | 29.6875 |
21,975 | A certain store sells a batch of helmets for $80 each. It can sell 200 helmets per month. During the "Creating a Civilized City" period, the store plans to reduce the price of the helmets for sale. After investigation, it was found that for every $1 decrease in price, an additional 20 helmets are sold per month. It is known that the cost price of the helmets is $50 each.
$(1)$ If the price of each helmet is reduced by $10, the store can sell ______ helmets per month, and the monthly profit from sales is ______ dollars.
$(2)$ If the store plans to reduce the price of these helmets to reduce inventory while ensuring a monthly profit of $7500, find the selling price of the helmets. | 65 | 23.4375 |
21,976 | The sequence $3, 8, 13, a, b, 33$ is arithmetic. What is the sum of values $a$ and $b$? | 41 | 77.34375 |
21,977 | The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. Find the smallest possible difference between the two 2-digit numbers. | 14 | 5.46875 |
21,978 | The largest and smallest possible three-digit numbers that can be formed using the digits 5, 1, and 9 are found by maximizing and minimizing the order of the digits, respectively. Calculate the difference between these two numbers. | 792 | 100 |
21,979 | There are 5 people standing in a row, where A and B must stand next to each other, and C and D cannot stand next to each other. How many different arrangements are there? | 24 | 54.6875 |
21,980 | Given that the central angle of a sector is $\frac{3}{2}$ radians, and its radius is 6 cm, then the arc length of the sector is \_\_\_\_\_\_ cm, and the area of the sector is \_\_\_\_\_\_ cm<sup>2</sup>. | 27 | 50.78125 |
21,981 | As shown in the diagram, rectangle \(ABCD\) is inscribed in a semicircle, with \(EF\) as the diameter of the semicircle. Given that \(DA = 16\), and \(FD = AE = 9\), find the area of rectangle \(ABCD\). | 240 | 45.3125 |
21,982 | A wholesaler gives different discounts based on the size of the customer's order. The steps to calculate the amount payable by the customer are as follows:
S1 Input the order amount x (unit: pieces); input the unit price A (unit: yuan);
S2 If x < 250, then the discount rate d=0;
If 250 ≤ x < 500, then the discount rate d=0.05;
If 500 ≤ x < 1000, then the discount rate d=0.10;
If x ≥ 1000, then the discount rate d=0.15;
S3 Calculate the payable amount T=Ax(1-d) (unit: yuan);
S4 Output the payable amount T.
It is known that a customer pays 38000 yuan for 400 pieces, then the order amount is pieces when the payable amount is 88200 yuan. | 980 | 31.25 |
21,983 | A flower shop buys a number of roses from a farm at a price of 5 yuan per rose each day and sells them at a price of 10 yuan per rose. If the roses are not sold by the end of the day, they are discarded.
(1) If the shop buys 16 roses in one day, find the profit function \( y \) (in yuan) with respect to the demand \( n \) for that day (in roses, \( n \in \mathbf{N} \)).
(2) The shop recorded the daily demand for roses (in roses) for 100 days and summarized the data in Table 1.
Using the frequencies of the demands recorded over the 100 days as probabilities for each demand:
(i) If the shop buys 16 roses in one day, let \( X \) represent the profit (in yuan) for that day. Find the distribution, expected value, and variance of \( X \).
(ii) If the shop plans to buy either 16 or 17 roses in one day, which would you recommend they buy? Please explain your reasoning. | 16 | 25 |
21,984 | Given that $\dfrac{\pi}{2} < \alpha < \beta < \dfrac{3\pi}{4}, \cos(\alpha - \beta) = \dfrac{12}{13}, \sin(\alpha + \beta) = -\dfrac{3}{5}$, find the value of $\sin 2\alpha$. | -\dfrac{56}{65} | 5.46875 |
21,985 | Eight people are sitting around a circular table for a meeting, and the recorder is sitting between the leader and the deputy leader. Calculate the total number of different seating arrangements possible, considering arrangements that can be made identical through rotation as the same. | 240 | 76.5625 |
21,986 | A digital watch displays hours and minutes in a 24-hour format. Find the largest possible sum of the digits in the display. | 24 | 0 |
21,987 | Positive real numbers $a$ , $b$ , $c$ satisfy $a+b+c=1$ . Find the smallest possible value of $$ E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}. $$ | \frac{1}{8} | 97.65625 |
21,988 | Given an angle measuring $54^{\circ}$, use only a compass to divide it into three equal parts (that is, find such points that rays passing through the vertex of the given angle and these points divide the angle into three equal parts). | 18 | 71.09375 |
21,989 | The school's boys basketball team has 16 players, including a set of twins, Bob and Bill, and a set of triplets, Chris, Craig, and Carl. In how many ways can we choose 7 starters if all three triplets must be in the starting lineup and both twins must either both be in the lineup or both not be in the lineup? | 385 | 24.21875 |
21,990 | As shown in the diagram, circles \( \odot O_{1} \) and \( \odot O_{2} \) are externally tangent. The line segment \( O_{1}O_{2} \) intersects \( \odot O_{1} \) at points \( A \) and \( B \), and intersects \( \odot O_{2} \) at points \( C \) and \( D \). Circle \( \odot O_{3} \) is internally tangent to \( \odot O_{1} \) at point \( B \), and circle \( \odot O_{4} \) is internally tangent to \( \odot O_{2} \) at point \( C \). The common external tangent of \( \odot O_{2} \) and \( \odot O_{3} \) passes through point \( A \), tangent to \( \odot O_{3} \) at point \( E \) and tangent to \( \odot O_{2} \) at point \( F \). The common external tangent of \( \odot O_{1} \) and \( \odot O_{4} \) passes through point \( D \). If the radius of circle \( \odot O_{3} \) is 1.2, what is the radius of circle \( \odot O_{4} \)? | 1.2 | 70.3125 |
21,991 | Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. If the polar equation of curve $C$ is $\rho\cos^2\theta-4\sin\theta=0$, and the polar coordinates of point $P$ are $(3, \frac{\pi}{2})$, in the Cartesian coordinate system, line $l$ passes through point $P$ with a slope of $\sqrt{3}$.
(Ⅰ) Write the Cartesian coordinate equation of curve $C$ and the parametric equation of line $l$;
(Ⅱ) Suppose line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $\frac{1}{|PA|}+ \frac{1}{|PB|}$. | \frac{\sqrt{6}}{6} | 21.09375 |
21,992 | Compute the sum of $302^2 - 298^2$ and $152^2 - 148^2$. | 3600 | 100 |
21,993 | Find the value of $x$ in the following expressions:
(1) $8x^3 = 27$; (2) $(x-2)^2 = 3$. | -\sqrt{3} + 2 | 0 |
21,994 | A factory produces a type of instrument. Due to limitations in production capacity and technical level, some defective products are produced. According to experience, the defect rate $p$ of the factory producing this instrument is generally related to the daily output $x$ (pieces) as follows:
$$
P= \begin{cases}
\frac {1}{96-x} & (1\leq x\leq 94, x\in \mathbb{N}) \\
\frac {2}{3} & (x>94, x\in \mathbb{N})
\end{cases}
$$
It is known that for every qualified instrument produced, a profit of $A$ yuan can be made, but for every defective product produced, a loss of $\frac {A}{2}$ yuan will be incurred. The factory wishes to determine an appropriate daily output.
(1) Determine whether producing this instrument can be profitable when the daily output (pieces) exceeds 94 pieces, and explain the reason;
(2) When the daily output $x$ pieces does not exceed 94 pieces, try to express the daily profit $T$ (yuan) of producing this instrument as a function of the daily output $x$ (pieces);
(3) To obtain the maximum profit, how many pieces should the daily output $x$ be? | 84 | 9.375 |
21,995 | Two sectors of a circle of radius $15$ overlap in the same manner as the original problem, with $P$ and $R$ as the centers of the respective circles. The angle at the centers for both sectors is now $45^\circ$. Determine the area of the shaded region. | \frac{225\pi - 450\sqrt{2}}{4} | 8.59375 |
21,996 | Let $\triangle XYZ$ be a right triangle with $\angle Y$ as the right angle. A circle with a diameter of $YZ$ intersects side $XZ$ at point $W$. Given that $XW = 2$ and $YW = 3$, find the length of $ZW$. | 4.5 | 0.78125 |
21,997 | If the inequality system about $x$ is $\left\{\begin{array}{l}{\frac{x+3}{2}≥x-1}\\{3x+6＞a+4}\end{array}\right.$ has exactly $3$ odd solutions, and the solution to the equation about $y$ is $3y+6a=22-y$ is a non-negative integer, then the product of all integers $a$ that satisfy the conditions is ____. | -3 | 30.46875 |
21,998 | Sunshine High School is planning to order a batch of basketballs and jump ropes from an online store. After checking on Tmall, they found that each basketball is priced at $120, and each jump rope is priced at $25. There are two online stores, Store A and Store B, both offering free shipping and their own discount schemes:<br/>Store A: Buy one basketball and get one jump rope for free;<br/>Store B: Pay 90% of the original price for both the basketball and jump rope.<br/>It is known that they want to buy 40 basketballs and $x$ jump ropes $\left(x \gt 40\right)$.<br/>$(1)$ If they purchase from Store A, the payment will be ______ yuan; if they purchase from Store B, the payment will be ______ yuan; (express in algebraic expressions with $x$)<br/>$(2)$ If $x=80$, through calculation, determine which store is more cost-effective to purchase from at this point.<br/>$(3)$ If $x=80$, can you provide a more cost-effective purchasing plan? Write down your purchasing method and calculate the amount to be paid. | 5700 | 61.71875 |
21,999 | In a bus station in the city, there are 10 waiting seats arranged in a row. Now, if 4 passengers randomly choose some seats to wait, the number of ways to arrange them so that there are exactly 5 consecutive empty seats is $\boxed{480}$. | 480 | 62.5 |
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