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40.3k
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stringlengths 10
5.15k
| ground_truth
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float64 0
100
|
|---|---|---|---|
17,400 |
Given that $α∈[\dfrac{π}{2}, \dfrac{3π}{2}]$, $β∈[-\dfrac{π}{2}, 0]$, and the equations $(α-\dfrac{π}{2})^{3}-\sin α-2=0$ and $8β^{3}+2\cos^{2}β+1=0$ hold, find the value of $\sin(\dfrac{α}{2}+β)$.
|
\dfrac{\sqrt{2}}{2}
| 41.40625 |
17,401 |
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$
|
-2006
| 56.25 |
17,402 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction.
|
\frac{335}{2011}
| 53.90625 |
17,403 |
Given two lines $l_1: ax+3y-1=0$ and $l_2: 2x+(a^2-a)y+3=0$, and $l_1$ is perpendicular to $l_2$, find the value of $a$.
|
a = \frac{1}{3}
| 71.875 |
17,404 |
Let $\{a, b, c, d, e, f, g, h\}$ be a permutation of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . What is the probability that $\overline{abc} +\overline{def}$ is even?
|
3/7
| 28.90625 |
17,405 |
Given that \(a\) and \(b\) are real numbers, and the polynomial \(x^{4} + a x^{3} + b x^{2} + a x + 1 = 0\) has at least one real root, determine the minimum value of \(a^{2} + b^{2}\).
|
4/5
| 27.34375 |
17,406 |
Given a positive integer \( A \) whose prime factorization can be written as \( A = 2^{\alpha} \times 3^{\beta} \times 5^{\gamma} \), where \(\alpha, \beta, \gamma\) are natural numbers. If one-half of \( A \) is a perfect square, one-third of \( A \) is a perfect cube, and one-fifth of \( A \) is a perfect fifth power of some natural number, what is the minimum value of \(\alpha + \beta + \gamma\)?
|
31
| 71.09375 |
17,407 |
Solve the equations:<br/>$(1)2\left(x-1\right)^{2}=1-x$;<br/>$(2)4{x}^{2}-2\sqrt{3}x-1=0$.
|
\frac{\sqrt{3} - \sqrt{7}}{4}
| 0 |
17,408 |
Using the 3 vertices of a triangle and 7 points inside it (a total of 10 points), how many smaller triangles can the original triangle be divided into?
(1985 Shanghai Junior High School Math Competition, China;
1988 Jiangsu Province Junior High School Math Competition, China)
|
15
| 72.65625 |
17,409 |
What is the maximum number of L-shaped 3-cell pieces that can be cut out from a rectangular grid of size:
a) $5 \times 10$ cells;
b) $5 \times 9$ cells?
|
15
| 82.8125 |
17,410 |
Fill in the blanks with appropriate numbers.
6.8 + 4.1 + __ = 12 __ + 6.2 + 7.6 = 20 19.9 - __ - 5.6 = 10
|
4.3
| 84.375 |
17,411 |
A function \( f \), defined on the set of integers, satisfies the following conditions:
1) \( f(1) + 1 > 0 \)
2) \( f(x + y) - x f(y) - y f(x) = f(x)f(y) - x - y + xy \) for any \( x, y \in \mathbb{Z} \)
3) \( 2f(x) = f(x + 1) - x + 1 \) for any \( x \in \mathbb{Z} \)
Find \( f(10) \).
|
1014
| 25.78125 |
17,412 |
Let \( x, y \) be nonnegative integers such that \( x + 2y \) is a multiple of 5, \( x + y \) is a multiple of 3, and \( 2x + y \geq 99 \). Find the minimum possible value of \( 7x + 5y \).
|
366
| 67.1875 |
17,413 |
How many positive multiples of 6 that are less than 150 have a units digit of 6?
|
25
| 0 |
17,414 |
Given $\sin\left(\theta - \frac{\pi}{6}\right) = \frac{1}{4}$ with $\theta \in \left( \frac{\pi}{6}, \frac{2\pi}{3}\right)$, calculate the value of $\cos\left(\frac{3\pi}{2} + \theta\right)$.
|
\frac{\sqrt{15} + \sqrt{3}}{8}
| 4.6875 |
17,415 |
Given that the left focus of the ellipse $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 (a＞b＞0) is F, the left vertex is A, and the upper vertex is B. If the distance from point F to line AB is $$\frac {2b}{ \sqrt {17}}$$, then the eccentricity of the ellipse is \_\_\_\_\_\_.
|
\frac {1}{3}
| 39.84375 |
17,416 |
In triangle $XYZ$, side lengths are $XY = 30$, $YZ = 45$, and $XZ = 51$. Points P and Q are on $XY$ and $XZ$ respectively, such that $XP = 18$ and $XQ = 15$. Determine the ratio of the area of triangle $XPQ$ to the area of quadrilateral $PQZY$.
A) $\frac{459}{625}$
B) $\frac{1}{2}$
C) $\frac{3}{5}$
D) $\frac{459}{675}$
|
\frac{459}{625}
| 36.71875 |
17,417 |
What is $(3^{12} \times 9^{-3})^2$? Write your answer as an integer.
|
531441
| 100 |
17,418 |
One end of a bus route is at Station $A$ and the other end is at Station $B$. The bus company has the following rules:
(1) Each bus must complete a one-way trip within 50 minutes (including the stopping time at intermediate stations), and it stops for 10 minutes when reaching either end.
(2) A bus departs from both Station $A$ and Station $B$ every 6 minutes. Determine the minimum number of buses required for this bus route.
|
20
| 65.625 |
17,419 |
The integers from 1 to \( n \), inclusive, are equally spaced in order around a circle. The diameter through the position of the integer 7 also goes through the position of 23. What is the value of \( n \)?
|
32
| 64.0625 |
17,420 |
**Circle $T$ has a circumference of $12\pi$ inches, and segment $XY$ is a diameter. If the measure of angle $TXZ$ is $45^{\circ}$, what is the length, in inches, of segment $XZ$?**
|
6\sqrt{2}
| 42.1875 |
17,421 |
Given that $x$ is a multiple of $2520$, what is the greatest common divisor of $g(x) = (4x+5)(5x+2)(11x+8)(3x+7)$ and $x$?
|
280
| 24.21875 |
17,422 |
Given point $O$ inside $\triangle ABC$, and $\overrightarrow{OA}+\overrightarrow{OC}+2 \overrightarrow{OB}=0$, calculate the ratio of the area of $\triangle AOC$ to the area of $\triangle ABC$.
|
1:2
| 0 |
17,423 |
Alice and Bob play a game around a circle divided into 15 equally spaced points, numbered 1 through 15. Alice moves 7 points clockwise per turn, and Bob moves 4 points counterclockwise per turn. Determine how many turns will be required for Alice and Bob to land on the same point for the first time.
|
15
| 83.59375 |
17,424 |
A pyramid-like stack with a rectangular base containing $6$ apples by $9$ apples is constructed, with each apple above the first level fitting into a pocket formed by four apples below, until no more apples can be fit in a new layer. Determine the total number of apples in the completed stack.
|
154
| 40.625 |
17,425 |
The distance between locations A and B is 135 kilometers. Two cars, a large one and a small one, travel from A to B. The large car departs 4 hours earlier than the small car, but the small car arrives 30 minutes earlier than the large car. The speed ratio of the small car to the large car is 5:2. Find the speeds of both cars.
|
18
| 0.78125 |
17,426 |
Given the function $f(x)=\sin x+\cos x$, where $x\in \mathbb{R}$.
- (I) Find the value of $f\left( \frac{\pi}{2}\right)$;
- (II) Find the smallest positive period of the function $f(x)$;
- (III) Find the minimum value of the function $g(x)=f\left(x+ \frac{\pi}{4}\right)+f\left(x+ \frac{3\pi}{4}\right)$.
|
-2
| 78.90625 |
17,427 |
Among all triangles $ABC$, find the maximum value of $\cos A + \cos B \cos C$.
|
\frac{3}{2}
| 3.90625 |
17,428 |
Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected to serve. Calculate the number of ways to select exactly one person to serve for both days.
|
60
| 57.03125 |
17,429 |
Given the function $f(x)= \begin{cases} 2^{x}, & x < 2 \\ f(x-1), & x\geqslant 2 \end{cases}$, then $f(\log_{2}7)=$ ______.
|
\frac {7}{2}
| 35.15625 |
17,430 |
An element is randomly chosen from among the first $20$ rows of Pascal’s Triangle. What is the probability that the value of the element chosen is $1$?
|
\frac{39}{210}
| 0 |
17,431 |
Given non-negative real numbers $a_{1}, a_{2}, \cdots, a_{2008}$ whose sum equals 1, determine the maximum value of $a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2007} a_{2008} + a_{2008} a_{1}$.
|
\frac{1}{4}
| 66.40625 |
17,432 |
Express the decimal $0.7\overline{56}$ as a common fraction.
|
\frac{749}{990}
| 92.96875 |
17,433 |
Given vectors $\overrightarrow{m}=(\sin A, \frac {1}{2})$ and $\overrightarrow{n}=(3,\sin A+ \sqrt {3}\cos A)$ are collinear, where $A$ is an internal angle of $\triangle ABC$.
$(1)$ Find the size of angle $A$;
$(2)$ If $BC=2$, find the maximum value of the area $S$ of $\triangle ABC$, and determine the shape of $\triangle ABC$ when $S$ reaches its maximum value.
|
\sqrt {3}
| 0 |
17,434 |
Triangle $ABC$ is right angled at $A$ . The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$ , determine $AC^2$ .
|
936
| 2.34375 |
17,435 |
The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$ , where $N$ is a positive integer. Find $N$ .
|
25
| 96.875 |
17,436 |
If the function $G$ has a maximum value of $M$ and a minimum value of $N$ on $m\leqslant x\leqslant n\left(m \lt n\right)$, and satisfies $M-N=2$, then the function is called the "range function" on $m\leqslant x\leqslant n$. <br/>$(1)$ Functions ① $y=2x-1$; ② $y=x^{2}$, of which function ______ is the "range function" on $1\leqslant x\leqslant 2$; (Fill in the number) <br/>$(2)$ Given the function $G:y=ax^{2}-4ax+3a\left(a \gt 0\right)$. <br/>① When $a=1$, the function $G$ is the "range function" on $t\leqslant x\leqslant t+1$, find the value of $t$; <br/>② If the function $G$ is the "range function" on $m+2\leqslant x\leqslant 2m+1(m$ is an integer), and $\frac{M}{N}$ is an integer, find the value of $a$.
|
\frac{1}{8}
| 1.5625 |
17,437 |
A retailer purchases a gadget at $50$ minus $10%$. He aims to sell the gadget at a gain of $25%$ on his cost price after offering a $15%$ discount on the marked price. Determine the gadget's marked price in dollars.
|
66.18
| 86.71875 |
17,438 |
Given triangle $ABC$ with sides $AB = 7$, $AC = 8$, and $BC = 5$, find the value of
\[\frac{\cos \frac{A - B}{2}}{\sin \frac{C}{2}} - \frac{\sin \frac{A - B}{2}}{\cos \frac{C}{2}}.\]
|
\frac{16}{7}
| 30.46875 |
17,439 |
In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the two points $\left(0, -\sqrt{3}\right)$ and $\left(0, \sqrt{3}\right)$ is $4$. Let the trajectory of point $P$ be $C$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) Find the coordinates of the vertices, the lengths of the major and minor axes, and the eccentricity of the ellipse.
|
\dfrac{\sqrt{3}}{2}
| 71.875 |
17,440 |
Given $\sin (x-\frac{5π}{12})=\frac{1}{3}$, find $\cos (\frac{2021π}{6}-2x)$.
|
\frac{7}{9}
| 32.8125 |
17,441 |
On the game show $\text{\emph{Wheel of Fortune II}}$, you observe a spinner with the labels ["Bankrupt", "$\$700$", "$\$900$", "$\$200$", "$\$3000$", "$\$800$"]. Given that each region has equal area, determine the probability of earning exactly $\$2400$ in your first three spins.
|
\frac{1}{36}
| 32.8125 |
17,442 |
A product originally priced at \$120 receives a discount of 8%. Calculate the percentage increase needed to return the reduced price to its original amount.
|
8.7\%
| 17.96875 |
17,443 |
If 18 bahs are equal to 30 rahs, and 6 rahs are equivalent to 10 yahs, how many bahs are equal to 1200 yahs?
|
432
| 77.34375 |
17,444 |
If the equation $x^{2}+(k^{2}-4)x+k-1=0$ has two roots that are opposite numbers, solve for $k$.
|
-2
| 32.03125 |
17,445 |
If $(X-2)^8 = a + a_1(x-1) + \ldots + a_8(x-1)^8$, then the value of $\left(a_2 + a_4 + \ldots + a_8\right)^2 - \left(a_1 + a_3 + \ldots + a_7\right)^2$ is (Answer in digits).
|
-255
| 23.4375 |
17,446 |
Rectangle $EFGH$ has area $4024$. An ellipse with area $4024\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle?
|
8\sqrt{2012}
| 0.78125 |
17,447 |
Given the power function $f(x)=kx^{\alpha}$, its graph passes through the point $(\frac{1}{2}, \frac{\sqrt{2}}{2})$. Find the value of $k+\alpha$.
|
\frac{3}{2}
| 69.53125 |
17,448 |
If $a \lt 0$, the graph of the function $f\left(x\right)=a^{2}\sin 2x+\left(a-2\right)\cos 2x$ is symmetric with respect to the line $x=-\frac{π}{8}$. Find the maximum value of $f\left(x\right)$.
|
4\sqrt{2}
| 54.6875 |
17,449 |
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^2\) and \(13^2\)?
|
165
| 1.5625 |
17,450 |
Evaluate $\frac{7}{3} + \frac{11}{5} + \frac{19}{9} + \frac{37}{17} - 8$.
|
\frac{628}{765}
| 6.25 |
17,451 |
Given the inequality (e-a)e^x + x + b + 1 ≤ 0, where e is the natural constant, find the maximum value of $\frac{b+1}{a}$.
|
\frac{1}{e}
| 31.25 |
17,452 |
Given points $A(\sin\theta, 1)$, $B(\cos\theta, 0)$, $C(-\sin\theta, 2)$, and $\overset{→}{AB}=\overset{→}{BP}$.
(I) Consider the function $f\left(\theta\right)=\overset{→}{BP}\cdot\overset{→}{CA}$, $\theta\in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, discuss the monotonicity of the function and find its range.
(II) If points $O$, $P$, $C$ are collinear, find the value of $\left|\overset{→}{OA}+\overset{→}{OB}\right|$.
|
\frac{\sqrt{74}}{5}
| 51.5625 |
17,453 |
A set \( \mathcal{T} \) of distinct positive integers has the property that for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 3003 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have?
|
30
| 0 |
17,454 |
Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored?
|
402
| 1.5625 |
17,455 |
Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$ , both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$ .
|
865
| 16.40625 |
17,456 |
In a version of SHORT BINGO, a $5\times5$ card has specific ranges for numbers placed in each column. In the first column, 5 distinct numbers must be chosen from the set $1-15$, but they must all be prime numbers. As before, the middle square is labeled as WILD and the other columns have specific number ranges as in the original game. How many distinct possibilities are there for the values in the first column of this SHORT BINGO card?
|
720
| 54.6875 |
17,457 |
Consider one positive even integer and one positive odd integer less than $16$, where the even integer is a multiple of $3$. Compute how many different possible values result when the sum of these two numbers is added to their product.
|
16
| 44.53125 |
17,458 |
How many integers $n$ satisfy $(n+2)(n-8) \le 0$?
|
11
| 100 |
17,459 |
In right triangle \\(ABC\\), where \\(\angle C = 90^{\circ}\\) and \\(\angle A = 30^{\circ}\\), the eccentricity of the ellipse, which has \\(A\\) and \\(B\\) as its foci and passes through point \\(C\\), is \_\_\_\_\_.
|
\sqrt{3} - 1
| 87.5 |
17,460 |
Subtract $555.55$ from $888.88.$ Express the result as a decimal to the nearest hundredth.
|
333.33
| 100 |
17,461 |
Construct a cylindrical iron barrel with a volume of $V$. The lid of the barrel is made of aluminum alloy, and the price of aluminum alloy per unit area is three times that of iron. To minimize the cost of this container, the ratio of the bottom radius $r$ of the iron barrel to its height $h$ should be _______.
|
\frac{1}{4}
| 2.34375 |
17,462 |
Cara is sitting at a circular table with her six friends. How many different sets of two friends can Cara be directly sitting between?
|
15
| 89.84375 |
17,463 |
Blind boxes are a new type of product. Merchants package different styles of products from the same series in boxes with the same appearance, so that consumers do not know which style of product they are buying. A merchant has designed three types of dolls, $A$, $B$, and $C$, in the same series, and sells them in blind boxes. It is known that the production ratio of the three types of dolls $A$, $B$, and $C$ is $6:3:1$. Using frequency to estimate probability, calculate the probability that a consumer randomly buys $4$ blind boxes at once and finds all three types of dolls inside.
|
0.216
| 2.34375 |
17,464 |
Place 6 balls, labeled from 1 to 6, into 3 different boxes. If each box is to contain 2 balls, and the balls labeled 1 and 2 are to be placed in the same box, calculate the total number of different ways to do this.
|
18
| 65.625 |
17,465 |
A parallelogram has a base of 6 cm and a height of 20 cm. Its area is \_\_\_\_\_\_ square centimeters. If both the base and the height are tripled, its area will increase by \_\_\_\_\_\_ times, resulting in \_\_\_\_\_\_ square centimeters.
|
1080
| 54.6875 |
17,466 |
Suppose we need to divide 15 dogs into three groups, one with 4 dogs, one with 7 dogs, and one with 4 dogs. We want to form the groups such that Fluffy is in the 4-dog group, Nipper is in the 7-dog group, and Daisy is in the other 4-dog group. How many ways can we arrange the remaining dogs into these groups?
|
18480
| 77.34375 |
17,467 |
Given the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n=1-5+9-13+17-21+......+{(-1)}^{n-1}(4n-3)$, calculate the value of $S_{15}$.
|
29
| 72.65625 |
17,468 |
A trapezoid has side lengths 4, 6, 8, and 10. The trapezoid can be rearranged to form different configurations with sides 4 and 8 as the parallel bases. Calculate the total possible area of the trapezoid with its different configurations.
A) $24\sqrt{2}$
B) $36\sqrt{2}$
C) $42\sqrt{2}$
D) $48\sqrt{2}$
E) $54\sqrt{2}$
|
48\sqrt{2}
| 18.75 |
17,469 |
The points \((2,3)\) and \((3, 7)\) lie on a circle whose center is on the \(x\)-axis. What is the radius of the circle?
|
\frac{\sqrt{1717}}{2}
| 82.8125 |
17,470 |
Simplify the expression: $$\dfrac{\sqrt{450}}{\sqrt{288}} + \dfrac{\sqrt{245}}{\sqrt{96}}.$$ Express your answer as a common fraction.
|
\frac{30 + 7\sqrt{30}}{24}
| 48.4375 |
17,471 |
Given that the left focus of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ is $F$, the eccentricity is $\frac{\sqrt{2}}{2}$, and the distance between the left intersection point of the ellipse with the $x$-axis and point $F$ is $\sqrt{2} - 1$.
(I) Find the equation of the ellipse;
(II) The line $l$ passing through point $P(0, 2)$ intersects the ellipse at two distinct points $A$ and $B$. When the area of triangle $OAB$ is $\frac{\sqrt{2}}{2}$, find $|AB|$.
|
\frac{3}{2}
| 24.21875 |
17,472 |
In triangle $ABC$, point $A$ is at $(1, 1)$, point $B$ is at $(4, 2)$, and point $C$ is at $(-4, 6)$.
(1) Determine the equation of the line where the median to side $BC$ lies;
(2) Determine the length of the altitude to side $BC$ and the area of triangle $ABC$.
|
10
| 82.8125 |
17,473 |
Li is ready to complete the question over the weekend: Simplify and evaluate $(3-2x^{2}-5x)-(\square x^{2}+3x-4)$, where $x=-2$, but the coefficient $\square$ is unclearly printed.<br/>$(1)$ She guessed $\square$ as $8$. Please simplify $(3-2x^{2}-5x)-(8x^{2}+3x-4)$ and find the value of the expression when $x=-2$;<br/>$(2)$ Her father said she guessed wrong, the standard answer's simplification does not contain quadratic terms. Please calculate and determine the value of $\square$ in the original question.
|
-2
| 91.40625 |
17,474 |
What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $15(3x-4) \equiv 30 \pmod{10}$?
|
240
| 90.625 |
17,475 |
Shift the graph of the function $y=\cos(\frac{π}{2}-2x)$ by an amount corresponding to the difference between the arguments of $y=\sin(2x-\frac{π}{4})$ and $y=\cos(\frac{π}{2}-2x)$.
|
\frac{\pi}{8}
| 87.5 |
17,476 |
At a school fundraiser, $3109 was raised. The money was shared equally among 25 charities. The amount received by each charity from the school was:
|
$124.36
| 0 |
17,477 |
Determine the value of $A + B + C$, where $A$, $B$, and $C$ are the dimensions of a three-dimensional rectangular box with faces having areas $40$, $40$, $90$, $90$, $100$, and $100$ square units.
|
\frac{83}{3}
| 22.65625 |
17,478 |
Given the function $f(x) = \ln x + \ln (ax+1) - \frac {3a}{2}x + 1$ ($a \in \mathbb{R}$).
(1) Discuss the intervals of monotonicity for the function $f(x)$.
(2) When $a = \frac {2}{3}$, if the inequality $xe^{x-\frac {1}{2}} + m \geqslant f(x)$ holds true for all $x$, find the minimum value of $m$, where $e$ is the base of the natural logarithm.
|
\ln \frac {2}{3}
| 0 |
17,479 |
For any positive integer \( n \), define
\( g(n) =\left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right. \)
What is \( \sum_{n = 1}^{1023}{g(n)} \)?
**A** \( \frac{40}{2} \)
**B** \( \frac{42}{2} \)
**C** \( \frac{45}{2} \)
**D** \( \frac{48}{2} \)
**E** \( \frac{50}{2} \)
|
\frac{45}{2}
| 4.6875 |
17,480 |
Given sets $A=\{2,3,4\}$ and $B=\{a+2,a\}$, if $A \cap B = B$, find $A^cB$ ___.
|
\{3\}
| 23.4375 |
17,481 |
A company is planning to increase the annual production of a product by implementing technical reforms in 2013. According to the survey, the product's annual production volume $x$ (in ten thousand units) and the technical reform investment $m$ (in million yuan, where $m \ge 0$) satisfy the equation $x = 3 - \frac{k}{m + 1}$ ($k$ is a constant). Without the technical reform, the annual production volume can only reach 1 ten thousand units. The fixed investment for producing the product in 2013 is 8 million yuan, and an additional investment of 16 million yuan is required for each ten thousand units produced. Due to favorable market conditions, all products produced can be sold. The company sets the selling price of each product at 1.5 times its production cost (including fixed and additional investments).
1. Determine the value of $k$ and express the profit $y$ (in million yuan) of the product in 2013 as a function of the technical reform investment $m$ (profit = sales revenue - production cost - technical reform investment).
2. When does the company's profit reach its maximum with the technical reform investment in 2013? Calculate the maximum profit.
|
21
| 75 |
17,482 |
(1) If $\cos (\frac{\pi}{4}+x) = \frac{3}{5}$, and $\frac{17}{12}\pi < x < \frac{7}{4}\pi$, find the value of $\frac{\sin 2x + 2\sin^2 x}{1 - \tan x}$.
(2) Given the function $f(x) = 2\sqrt{3}\sin x\cos x + 2\cos^2 x - 1 (x \in \mathbb{R})$, if $f(x_0) = \frac{6}{5}$, and $x_0 \in [\frac{\pi}{4}, \frac{\pi}{2}]$, find the value of $\cos 2x_0$.
|
\frac{3 - 4\sqrt{3}}{10}
| 73.4375 |
17,483 |
According to the latest revision of the "Regulations on the Application and Use of Motor Vehicle Driving Licenses" by the Ministry of Public Security: each driving license applicant must pass the "Subject One" (theoretical subject) and "Comprehensive Subject" (combination of driving skills and some theoretical knowledge of "Subject One") exams. It is known that Mr. Li has passed the "Subject One" exam, and the score of "Subject One" is not affected by the "Comprehensive Subject". The "Comprehensive Subject" offers 5 chances to take the exam within three years. Once an exam is passed, the driving license will be issued, and the applicant will no longer participate in subsequent exams; otherwise, the exams will continue until the 5th attempt. The probabilities of Mr. Li passing the "Comprehensive Subject" in each attempt are 0.5, 0.6, 0.7, 0.8, and 0.9, respectively.
(1) Calculate the distribution and mathematical expectation of the number of times ξ that Mr. Li takes the driving license test within three years.
(2) Calculate the probability of Mr. Li obtaining a driving license within three years.
|
0.9988
| 34.375 |
17,484 |
Given $\overrightarrow{m}=(2\sqrt{3},1)$, $\overrightarrow{n}=(\cos^2 \frac{A}{2},\sin A)$, where $A$, $B$, and $C$ are the interior angles of $\triangle ABC$;
$(1)$ When $A= \frac{\pi}{2}$, find the value of $|\overrightarrow{n}|$;
$(2)$ If $C= \frac{2\pi}{3}$ and $|AB|=3$, when $\overrightarrow{m} \cdot \overrightarrow{n}$ takes the maximum value, find the magnitude of $A$ and the length of side $BC$.
|
\sqrt{3}
| 53.90625 |
17,485 |
Given that a match between two people is played with a best-of-five-games format, where the winner is the first to win three games, and that the probability of person A winning a game is $\dfrac{2}{3}$, calculate the probability that person A wins with a score of $3:1$.
|
\dfrac{8}{27}
| 53.125 |
17,486 |
Select 4 out of 6 sprinters to participate in a 4×100 relay race. If neither A nor B runs the first leg, then there are $\boxed{\text{different}}$ possible team compositions.
|
240
| 50 |
17,487 |
Given point O is the circumcenter of triangle ABC, and |BA|=2, |BC|=6, calculate the value of the dot product of the vectors BO and AC.
|
16
| 19.53125 |
17,488 |
Given that $E, U, L, S, R,$ and $T$ represent the digits $1, 2, 3, 4, 5, 6$ (each letter represents a unique digit), and the following conditions are satisfied:
1. $E + U + L = 6$
2. $S + R + U + T = 18$
3. $U \times T = 15$
4. $S \times L = 8$
Determine the six-digit number $\overline{EULSRT}$.
|
132465
| 89.0625 |
17,489 |
Compute: \(93 \times 107\).
|
9951
| 100 |
17,490 |
What is the largest five-digit number whose digits add up to 20?
|
99200
| 85.9375 |
17,491 |
Given two points A (-2, 0), B (0, 2), and point C is any point on the circle $x^2+y^2-2x=0$, determine the minimum area of $\triangle ABC$.
|
3- \sqrt{2}
| 74.21875 |
17,492 |
In triangle $\triangle ABC$, a line passing through the midpoint $E$ of the median $AD$ intersects sides $AB$ and $AC$ at points $M$ and $N$ respectively. Let $\overrightarrow{AM} = x\overrightarrow{AB}$ and $\overrightarrow{AN} = y\overrightarrow{AC}$ ($x, y \neq 0$), then the minimum value of $4x+y$ is \_\_\_\_\_\_.
|
\frac{9}{4}
| 8.59375 |
17,493 |
Given that the terminal side of angle $\alpha$ passes through the fixed point $P$ on the function $y=\log _{a}(x-3)+2$, find the value of $\sin 2\alpha+\cos 2\alpha$.
|
\frac{7}{5}
| 75 |
17,494 |
Find the coefficient of $\frac{1}{x}$ in the expansion of $((1-x^{2})^{4}(\frac{x+1}{x})^{5})$.
|
-29
| 23.4375 |
17,495 |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{5}}{5}$, and its right latus rectum equation is $x=5$.
(1) Find the equation of the ellipse;
(2) A line $l$ with a slope of $1$ passes through the right focus $F$ of the ellipse, intersecting the ellipse $C$ at points $A$ and $B$. $P$ is a moving point on the ellipse. Find the maximum area of $\triangle PAB$.
|
\frac{16\sqrt{10}}{9}
| 2.34375 |
17,496 |
Given $\vec{a}=(\cos \alpha, \sin \alpha), \vec{b}=(\cos \beta, \sin \beta)$, and $|\vec{a}-\vec{b}|=\frac{2 \sqrt{5}}{5}$. If $0 < \alpha < \frac{\pi}{2}$, $-\frac{\pi}{2} < \beta < 0$, and $\sin \beta=-\frac{5}{13}$, then $\sin \alpha=$
|
$\frac{33}{65}$
| 0 |
17,497 |
A student typed out several circles on the computer as follows: ○●○○●○○○●○○○○●○○○○○●… If this pattern continues, forming a series of circles, then the number of ● in the first 120 circles is ______.
|
14
| 34.375 |
17,498 |
Let the numbers x and y satisfy the conditions $\begin{cases} x^2 + y^2 - xy = 2
x^4 + y^4 + x^2y^2 = 8 \end{cases}$ The value of $P = x^8 + y^8 + x^{2014}y^{2014}$ is:
|
48
| 70.3125 |
17,499 |
Given the function $f(x)=\sin ^{2}x+ \frac{ \sqrt{3}}{2}\sin 2x$.
(1) Find the interval(s) where the function $f(x)$ is monotonically decreasing.
(2) In $\triangle ABC$, if $f(\frac{A}{2})=1$ and the area of the triangle is $3\sqrt{3}$, find the minimum value of side $a$.
|
2\sqrt{3}
| 75.78125 |
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