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24,600 | There are 4 college entrance examination candidates entering the school through 2 different intelligent security gates. Each security gate can only allow 1 person to pass at a time. It is required that each security gate must have someone passing through. Then there are ______ different ways for the candidates to enter the school. (Answer in numbers) | 72 | 3.125 |
24,601 | For the curve $ C: y = \frac {1}{1 + x^2}$ , Let $ A(\alpha ,\ f(\alpha)),\ B\left( - \frac {1}{\alpha},\ f\left( - \frac {1}{\alpha} \right)\right)\ (\alpha > 0).$ Find the minimum area bounded by the line segments $ OA,\ OB$ and $ C,$ where $ O$ is the origin.
Note that you are not allowed to use the integral formula of $ \frac {1}{1 + x^2}$ for the problem. | \frac{\pi}{2} - \frac{1}{2} | 0 |
24,602 | Let the set \( A = \{1, 2, \cdots, 2016\} \). For any 1008-element subset \( X \) of \( A \), if there exist \( x \) and \( y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a "good set". Find the largest positive integer \( a \) (where \( a \in A \)) such that any 1008-element subset containing \( a \) is a good set. | 1008 | 33.59375 |
24,603 | Given that the sum of the first n terms of a geometric sequence {a_n} is denoted as S_n, if S_4 = 2 and S_8 = 6, calculate the value of S_{12}. | 14 | 90.625 |
24,604 | The set containing three real numbers can be represented as $\{a, \frac{b}{a}, 1\}$, and also as $\{a^2, a+b, 0\}$. Find the value of $a^{2009} + b^{2009}$. | -1 | 70.3125 |
24,605 | A sphere is inscribed in a right cone with base radius \(15\) cm and height \(30\) cm. Find the radius \(r\) of the sphere, which can be expressed as \(b\sqrt{d} - b\) cm. What is the value of \(b + d\)? | 12.5 | 7.8125 |
24,606 | Suppose that $x^{10} + x + 1 = 0$ and $x^100 = a_0 + a_1x +... + a_9x^9$ . Find $a_5$ . | -252 | 36.71875 |
24,607 | In a basketball match, Natasha attempted only three-point shots, two-point shots, and free-throw shots. She was successful on $25\%$ of her three-point shots and $40\%$ of her two-point shots. Additionally, she had a free-throw shooting percentage of $50\%$. Natasha attempted $40$ shots in total, given that she made $10$ free-throw shot attempts. How many points did Natasha score? | 28 | 6.25 |
24,608 | If the square roots of a number are $2a+3$ and $a-18$, then this number is ____. | 169 | 71.09375 |
24,609 | Determine how many integers are in the list when Eleanor writes down the smallest positive multiple of 15 that is a perfect square, the smallest positive multiple of 15 that is a perfect cube, and all the multiples of 15 between them. | 211 | 63.28125 |
24,610 | Given $A=3x^{2}-x+2y-4xy$, $B=x^{2}-2x-y+xy-5$.
$(1)$ Find $A-3B$.
$(2)$ If $(x+y-\frac{4}{5})^{2}+|xy+1|=0$, find the value of $A-3B$.
$(3)$ If the value of $A-3B$ is independent of $y$, find the value of $x$. | \frac{5}{7} | 82.03125 |
24,611 | A store owner bought 2000 markers at $0.20 each. To make a minimum profit of $200, if he sells the markers for $0.50 each, calculate the number of markers he must sell at least to achieve or exceed this profit. | 1200 | 69.53125 |
24,612 | For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$ . $z$ is the maximum real number such that [ $\frac{5}{z}$ ] + [ $\frac{6}{z}$ ] = 7. Find the value of $ 20z$ . | 30 | 23.4375 |
24,613 | Given that in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\sin A+a\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $b=2$, find the maximum area of $\triangle ABC$. | \sqrt{2}-1 | 67.96875 |
24,614 | Kevin Kangaroo begins his journey on a number line at 0 with a goal of reaching a rock located at 1. However, he hops only $\frac{1}{4}$ of the distance toward the rock with each leap. After each leap, he tires, reducing his subsequent hop to $\frac{1}{4}$ of the remaining distance to the rock. Calculate the total distance Kevin hops after seven leaps. Express your answer as a common fraction. | \frac{14197}{16384} | 35.9375 |
24,615 | In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{{\begin{array}{l}{x=2\cos\alpha,}\\{y=\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $2\rho \cos \theta -\rho \sin \theta +2=0$.
$(1)$ Find the Cartesian equation of curve $C$ and the rectangular coordinate equation of line $l$;
$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and point $P(0,2)$, find the value of $\frac{1}{{|PA|}}+\frac{1}{{|PB|}}$. | \frac{8\sqrt{5}}{15} | 2.34375 |
24,616 | Given that the random variable $ξ$ follows a normal distribution $N(1,σ^{2})$, and $P(ξ\leqslant 4)=0.79$, determine the value of $P(-2\leqslant ξ\leqslant 1)$. | 0.29 | 47.65625 |
24,617 | 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 $m$? | 12 | 7.8125 |
24,618 | Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_. | 12\pi | 92.96875 |
24,619 | The first two terms of a given sequence are 1 and 1 respectively, and each successive term is the sum of the two preceding terms. What is the value of the first term which exceeds 1000? | 1597 | 100 |
24,620 | Given an ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, the distance from a point $M$ on the ellipse to the left focus $F_1$ is 8. Find the distance from $M$ to the right directrix. | \frac{5}{2} | 64.0625 |
24,621 | A circle $C$ is defined by the equation $x^2 + y^2 = 1$. After the transformation $\begin{cases} x' = 2x \\ y' = \sqrt{2}y \end{cases}$, we obtain the curve $C_1$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate of the line $l$ is $\rho \cos \left( \theta + \frac{\pi}{3} \right) = \frac{1}{2}$.
(1) Write the parametric equation of $C_1$ and the normal equation of $l$.
(2) Let point $M(1,0)$. The line $l$ intersects with the curve $C_1$ at two points $A$ and $B$. Compute $|MA| \cdot |MB|$ and $|AB|$. | \frac{12\sqrt{2}}{5} | 40.625 |
24,622 | The number of sets $A$ that satisfy $\{1, 2\} \subset A \subseteq \{1, 2, 3, 4, 5, 6\}$ must be determined. | 15 | 3.90625 |
24,623 | Given a fixed point P (-2, 0) and a line $l: (1+3\lambda)x + (1+2\lambda)y - (2+5\lambda) = 0$, where $\lambda \in \mathbb{R}$, find the maximum distance $d$ from point P to line $l$. | \sqrt{10} | 5.46875 |
24,624 | If the variance of the sample $a_1, a_2, \ldots, a_n$ is 3, then the variance of the sample $3a_1+1, 3a_2+2, \ldots, 3a_n+1$ is ____. | 27 | 87.5 |
24,625 | Calculate: $\sqrt{25}-\left(-1\right)^{2}+|2-\sqrt{5}|$. | 2+\sqrt{5} | 63.28125 |
24,626 | Given that line $l$ intersects circle $C$: $x^2+y^2+2x-4y+a=0$ at points $A$ and $B$, and the midpoint of chord $AB$ is $P(0,1)$.
(I) If the radius of circle $C$ is $\sqrt{3}$, find the value of the real number $a$;
(II) If the length of chord $AB$ is $6$, find the value of the real number $a$;
(III) When $a=1$, circles $O$: $x^2+y^2=4$ and $C$ intersect at points $M$ and $N$, find the length of the common chord $MN$. | \sqrt{11} | 66.40625 |
24,627 | A 9 by 9 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | 91 | 10.9375 |
24,628 | Given a tetrahedron P-ABC, if PA, PB, and PC are mutually perpendicular, and PA=2, PB=PC=1, then the radius of the inscribed sphere of the tetrahedron P-ABC is \_\_\_\_\_\_. | \frac {1}{4} | 38.28125 |
24,629 | What is the probability, expressed as a fraction, of drawing a marble that is either green or white from a bag containing 4 green, 3 white, 5 red, and 6 blue marbles? | \frac{7}{18} | 85.9375 |
24,630 | A square and four circles, each with a radius of 8 inches, are arranged as in the previous problem. What is the area, in square inches, of the square? | 1024 | 56.25 |
24,631 | 24×12, my approach is to first calculate \_\_\_\_\_\_, then calculate \_\_\_\_\_\_, and finally calculate \_\_\_\_\_\_. | 288 | 21.09375 |
24,632 | Calculate:<br/>$(1)-3+\left(-9\right)+10-\left(-18\right)$;<br/>$(2)12÷2×(-\frac{1}{2})-75÷(-5)$;<br/>$(3)-4^3-2×(-5)^2+6÷(-\frac{1}{3})^2$;<br/>$(4)(-1\frac{1}{4}-1\frac{5}{6}+2\frac{8}{9})÷(-\frac{1}{6})^2$. | -7 | 63.28125 |
24,633 | Given that $a > 2b$ ($a, b \in \mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\infty)$. Determine the minimum value of $$\frac{a^2 + 4b^2}{a - 2b}$$. | \sqrt{2} | 2.34375 |
24,634 | In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C_{1}$ are given by $\begin{cases} x=2\cos \alpha \\ y=2+2\sin \alpha \end{cases}$ ($\alpha$ is the parameter). Point $M$ moves on curve $C_{1}$, and point $P$ satisfies $\overrightarrow{OP}=2\overrightarrow{OM}$. The trajectory of point $P$ forms the curve $C_{2}$.
(I) Find the equation of $C_{2}$;
(II) In the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the ray $\theta = \dfrac{\pi}{3}$ intersects $C_{1}$ at point $A$ and $C_{2}$ at point $B$. Find the length of the segment $|AB|$. | 2 \sqrt{3} | 60.9375 |
24,635 | Arrange 8 people, including A and B, to work for 4 days. If 2 people are arranged each day, the probability that A and B are arranged on the same day is ______. (Express the result as a fraction) | \frac{1}{7} | 59.375 |
24,636 | Six friends earn $25, $30, $35, $45, $50, and $60. Calculate the amount the friend who earned $60 needs to distribute to the others when the total earnings are equally shared among them. | 19.17 | 35.15625 |
24,637 | In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the interior angles $A$, $B$, and $C$, respectively. If $a\cos \left(B-C\right)+a\cos A=2\sqrt{3}c\sin B\cos A$ and $b^{2}+c^{2}-a^{2}=2$, then the area of $\triangle ABC$ is ____. | \frac{\sqrt{3}}{2} | 24.21875 |
24,638 | Teams A and B are playing a volleyball match. Currently, Team A needs to win one more game to become the champion, while Team B needs to win two more games to become the champion. If the probability of each team winning each game is 0.5, then the probability of Team A becoming the champion is $\boxed{\text{answer}}$. | 0.75 | 50 |
24,639 | Given that the function f(x) (x ∈ R) satisfies f(x + π) = f(x) + sin(x), and f(x) = 0 when 0 ≤ x ≤ π. Find f(23π/6). | \frac{1}{2} | 37.5 |
24,640 | Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$ . | 13 | 94.53125 |
24,641 | In a new arcade game, the "monster" is the shaded region of a semicircle with radius 2 cm as shown in the diagram. The mouth, which is an unshaded piece within the semicircle, subtends a central angle of 90°. Compute the perimeter of the shaded region.
A) $\pi + 3$ cm
B) $2\pi + 2$ cm
C) $\pi + 4$ cm
D) $2\pi + 4$ cm
E) $\frac{5}{2}\pi + 2$ cm | \pi + 4 | 70.3125 |
24,642 | Given $a \gt 0$, $b \gt 0$, and $a+2b=1$, find the minimum value of $\frac{{b}^{2}+a+1}{ab}$. | 2\sqrt{10} + 6 | 0 |
24,643 | In each part of this problem, cups are arranged in a circle and numbered \(1, 2, 3, \ldots\). A ball is placed in cup 1. Then, moving clockwise around the circle, a ball is placed in every \(n\)th cup. The process ends when cup 1 contains two balls.
(a) There are 12 cups in the circle and a ball is placed in every 5th cup, beginning and ending with cup 1. List, in order, the cups in which the balls are placed.
(b) There are 9 cups in the circle and a ball is placed in every 6th cup, beginning and ending with cup 1. List the numbers of the cups that do not receive a ball.
(c) There are 120 cups in the circle and a ball is placed in every 3rd cup, beginning and ending with cup 1. How many cups do not contain at least one ball when the process is complete? Explain how you obtained your answer.
(d) There are 1000 cups in the circle and a ball is placed in every 7th cup, beginning and ending with cup 1. Determine the number of the cup into which the 338th ball is placed. | 360 | 13.28125 |
24,644 | Given that the coefficient of the second term in the expansion of $((x+2y)^{n})$ is $8$, find the sum of the coefficients of all terms in the expansion of $((1+x)+(1+x)^{2}+…+(1+x)^{n})$. | 30 | 99.21875 |
24,645 | The integer sequence \(a_1, a_2, a_3, \dots\) is defined as follows: \(a_1 = 1\). For \(n \geq 1\), \(a_{n+1}\) is the smallest integer greater than \(a_n\) such that for all \(i, j, k \in \{1, 2, \dots, n+1\}\), the condition \(a_i + a_j \neq 3a_k\) is satisfied. Find the value of \(a_{22006}\). | 66016 | 5.46875 |
24,646 | Complex numbers $d$, $e$, and $f$ are zeros of a polynomial $Q(z) = z^3 + sz^2 + tz + u$, and $|d|^2 + |e|^2 + |f|^2 = 300$. The points corresponding to $d$, $e$, and $f$ in the complex plane are the vertices of an equilateral triangle. Find the square of the length of each side of the triangle. | 300 | 60.15625 |
24,647 | Two arithmetic sequences $C$ and $D$ both begin with 50. Sequence $C$ has a common difference of 12 and is increasing, while sequence $D$ has a common difference of 8 and is decreasing. What is the absolute value of the difference between the 31st term of sequence $C$ and the 31st term of sequence $D$? | 600 | 96.875 |
24,648 | Consider a dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square, which is at a distance of 1 unit from the center, contains 8 unit squares. The second ring, at a distance of 2 units from the center, contains 16 unit squares. If the pattern of increasing distance continues, what is the number of unit squares in the $50^{th}$ ring, if each ring’s distance from the center increases linearly (i.e., $n^{th}$ ring is at $n$ units from the center), and every second ring starting from the first is colored red? | 400 | 100 |
24,649 | Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
[asy]
draw(circle((0,0),13));
draw(circle((5,-6.2),5));
draw(circle((-5,-6.2),5));
label(" $B$ ", (9.5,-9.5), S);
label(" $A$ ", (-9.5,-9.5), S);
[/asy] | 69 | 1.5625 |
24,650 | There are 2 boys and 3 girls, a total of 5 students standing in a row. If boy A does not stand at either end, and among the 3 girls, exactly 2 girls stand next to each other, calculate the number of different arrangements. | 72 | 24.21875 |
24,651 | Given vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ satisfying $\overrightarrow{a} + \overrightarrow{b} + 2\overrightarrow{c} = \overrightarrow{0}$, and the magnitudes $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=3$, $|\overrightarrow{c}|=2$, find the value of $\overrightarrow{a} \cdot \overrightarrow{b} + 2\overrightarrow{a} \cdot \overrightarrow{c} + 2\overrightarrow{b} \cdot \overrightarrow{c}$. | -13 | 91.40625 |
24,652 | In the expansion of $(1+x)^3 + (1+x)^4 + \ldots + (1+x)^{12}$, the coefficient of the term containing $x^2$ is _____. (Provide your answer as a number) | 285 | 82.03125 |
24,653 | Let $x, y, z$ be positive real numbers such that $x + 2y + 3z = 1$. Find the maximum value of $x^2 y^2 z$. | \frac{4}{16807} | 1.5625 |
24,654 | Given circles $P, Q,$ and $R$ where $P$ has a radius of 1 unit, $Q$ a radius of 2 units, and $R$ a radius of 1 unit. Circles $Q$ and $R$ are tangent to each other externally, and circle $R$ is tangent to circle $P$ externally. Compute the area inside circle $Q$ but outside circle $P$ and circle $R$. | 2\pi | 71.875 |
24,655 | Which of the following numbers is equal to 33 million? | 33000000 | 13.28125 |
24,656 | Ria writes down the numbers $1,2,\cdots, 101$ in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen? | 68 | 78.90625 |
24,657 | Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$, where $S_n=n^2+n$, and the general term of the sequence $\{b_n\}$ is given by $b_n=x^{n-1}$.
(1) Find the general term formula for the sequence $\{a_n\}$;
(2) Let $c_n=a_nb_n$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$.
(i) Find $T_n$;
(ii) If $x=2$, find the minimum value of the sequence $\left\{\dfrac{nT_{n+1}-2n}{T_{n+2}-2}\right\}$. | \dfrac{1}{4} | 46.875 |
24,658 | Start with a three-digit positive integer $A$ . Obtain $B$ by interchanging the two leftmost digits of $A$ . Obtain $C$ by doubling $B$ . Obtain $D$ by subtracting $500$ from $C$ . Given that $A + B + C + D = 2014$ , fi nd $A$ . | 344 | 71.09375 |
24,659 | Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$. Let $P$ be a point on the ellipse, and $\triangle F\_1 P F\_2$ have centroid $G$ and incenter $I$. If $\overrightarrow{IG} = λ(1,0) (λ ≠ 0)$, find the eccentricity $e$ of the ellipse.
A) $\frac{1}{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{1}{4}$
D) $\frac{\sqrt{5}-1}{2}$ | \frac{1}{2} | 27.34375 |
24,660 | In a particular sequence, the first term is $a_1 = 1009$ and the second term is $a_2 = 1010$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \ge 1$. Determine $a_{1000}$. | 1675 | 14.84375 |
24,661 | Determine the number of different arrangements for assigning 6 repatriated international students to work in Jinan and Qingdao, given that at least 2 students must be assigned to Jinan and at least 3 students must be assigned to Qingdao. | 35 | 4.6875 |
24,662 | In a triangle with integer side lengths, one side is four times as long as a second side, and the length of the third side is 20. What is the greatest possible perimeter of the triangle? | 50 | 75 |
24,663 | Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$ such that it is tangent to circle $C$ at point $R$. Meanwhile, a secant from $Q$ intersects $C$ at points $D$ and $E$, such that $QD < QE$. If $QD = 4$ and $QR = ED - QD$, then what is $QE$? | 16 | 86.71875 |
24,664 | In a weekend volleyball tournament, Team E plays against Team F, and Team G plays against Team H on Saturday. On Sunday, the winners of Saturday's matches face off in a final, while the losers compete for the consolation prize. Furthermore, there is a mini tiebreaker challenge between the losing teams on Saturday to decide the starting server for Sunday's consolation match. One possible ranking of the team from first to fourth at the end of the tournament is EGHF. Determine the total number of possible four-team ranking sequences at the end of the tournament. | 16 | 73.4375 |
24,665 | Using the digits 1, 2, 3, 4, how many distinct four-digit even numbers can be formed? | 12 | 99.21875 |
24,666 | Given that $\sin \alpha = 3 \sin \left(\alpha + \frac{\pi}{6}\right)$, find the value of $\tan \left(\alpha + \frac{\pi}{12}\right)$. | 2 \sqrt{3} - 4 | 24.21875 |
24,667 | In a household, when someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.3, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability of the phone call being answered within the first four rings. | 0.9 | 95.3125 |
24,668 | Given Jones traveled 100 miles on his first trip and 500 miles on a subsequent trip at a speed four times as fast, compare his new time to the old time. | 1.25 | 34.375 |
24,669 | The cost price of a certain product is 1360 yuan. When it is sold at 80% of the marked price, the profit margin is 15%. What should be the marked price of the product in yuan? | 1955 | 79.6875 |
24,670 | Rectangle $ABCD$ has $AB = 8$ and $BC = 13$ . Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \parallel BC$ . Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \parallel AB$ . Find the area of quadrilateral $P_1Q_1P_2Q_2$ . | 52 | 1.5625 |
24,671 | To meet market demand, a supermarket purchased a brand of zongzi before the arrival of the Dragon Boat Festival on May 5th. The cost of each box is $40. The supermarket stipulates that the selling price of each box must not be less than $45. Based on past sales experience, it was found that when the selling price is set at $45 per box, 700 boxes can be sold per day. For every $1 increase in the selling price per box, 20 fewer boxes are sold per day.
$(1)$ Find the functional relationship between the daily sales volume $y$ (boxes) and the selling price per box $x$ (in dollars).
$(2)$ At what price per box should the selling price be set to maximize the daily profit $P$ (in dollars)? What is the maximum profit?
$(3)$ To stabilize prices, the relevant management department has set a maximum selling price of $58 per box for this type of zongzi. If the supermarket wants to make a profit of at least $6000 per day, how many boxes of zongzi must be sold per day at least? | 440 | 40.625 |
24,672 | Given that the slant height of a certain cone is $4$ and the height is $2\sqrt{3}$, calculate the total surface area of the cone. | 12\pi | 99.21875 |
24,673 | Given the function $y=\sin (2x+\frac{π}{3})$, determine the horizontal shift required to obtain this graph from the graph of the function $y=\sin 2x$. | \frac{\pi}{6} | 20.3125 |
24,674 | The graph of the function $y=\sin (2x+\varphi)$ is translated to the left by $\dfrac {\pi}{8}$ units along the $x$-axis and results in a graph of an even function, then determine one possible value of $\varphi$. | \dfrac {\pi}{4} | 91.40625 |
24,675 | A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 7, and 8. What is the area of the triangle? | \frac{119.84}{\pi^2} | 0 |
24,676 | Right $\triangle PQR$ has sides $PQ = 5$, $QR = 12$, and $PR = 13$. Rectangle $ABCD$ is inscribed in $\triangle PQR$ such that $A$ and $B$ are on $\overline{PR}$, $D$ on $\overline{PQ}$, and $C$ on $\overline{QR}$. If the height of the rectangle (parallel to side $\overline{PQ}$) is half its length (parallel to side $\overline{PR}$), find the length of the rectangle. | 7.5 | 0 |
24,677 | A box contains $12$ ping-pong balls, of which $9$ are new and $3$ are old. Three balls are randomly drawn from the box for use, and then returned to the box. Let $X$ denote the number of old balls in the box after this process. What is the value of $P(X = 4)$? | \frac{27}{220} | 3.125 |
24,678 | Calculate: ${2}^{0}-|-3|+(-\frac{1}{2})=\_\_\_\_\_\_$. | -2\frac{1}{2} | 0 |
24,679 | The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no numbers are left. What is the last number to be erased? | 1598 | 55.46875 |
24,680 | Given a point $Q$ on a rectangular piece of paper $DEF$, where $D, E, F$ are folded onto $Q$. Let $Q$ be a fold point of $\triangle DEF$ if the creases, which number three unless $Q$ is one of the vertices, do not intersect within the triangle. Suppose $DE=24, DF=48,$ and $\angle E=90^\circ$. Determine the area of the set of all possible fold points $Q$ of $\triangle DEF$. | 147 | 0 |
24,681 | In a WeChat group, five members simultaneously grab for four red envelopes, each person can grab at most one, and all red envelopes are claimed. Among the four red envelopes, there are two containing 2 yuan and two containing 3 yuan. Determine the number of scenarios in which both members A and B have grabbed a red envelope. | 18 | 19.53125 |
24,682 | Determine the value of $a + b$ if the points $(2,a,b),$ $(a,3,b),$ and $(a,b,4)$ are collinear. | -2 | 0 |
24,683 | Twelve chairs are evenly spaced around a round table and numbered clockwise from $1$ through $12$. Six married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse or next to someone of the same profession. Determine the number of seating arrangements possible. | 2880 | 4.6875 |
24,684 | Points are drawn on the sides of a square, dividing each side into \( n \) equal parts. The points are joined to form several small squares and some triangles. How many small squares are formed when \( n=7 \)? | 84 | 0.78125 |
24,685 | For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ a point $P$ on the ellipse makes lines connecting it to the two foci $F1$ and $F2$ perpendicular to each other. Then, the area of $\triangle PF1F2$ is ________. | 12 | 0 |
24,686 | In quadrilateral $EFGH$, $\angle F$ is a right angle, diagonal $\overline{EG}$ is perpendicular to $\overline{GH}$, $EF=20$, $FG=24$, and $GH=16$. Find the perimeter of $EFGH$. | 60 + 8\sqrt{19} | 0 |
24,687 | Consider a cube with side length 4 units. Determine the volume of the set of points that are inside or within 2 units outside of the cube. | 1059 | 0 |
24,688 | The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$. | \frac{1}{2} | 15.625 |
24,689 | How many ways are there to arrange numbers from 1 to 8 in circle in such way the adjacent numbers are coprime?
Note that we consider the case of rotation and turn over as distinct way. | 72 | 0.78125 |
24,690 | I'm going to dinner at a large restaurant which my friend recommended, unaware that I am vegan and have both gluten and dairy allergies. Initially, there are 6 dishes that are vegan, which constitutes one-sixth of the entire menu. Unfortunately, 4 of those vegan dishes contain either gluten or dairy. How many dishes on the menu can I actually eat? | \frac{1}{18} | 0 |
24,691 | A huge number $y$ is given by $2^33^24^65^57^88^39^{10}11^{11}$. What is the smallest positive integer that, when multiplied with $y$, results in a product that is a perfect square? | 110 | 1.5625 |
24,692 | In the country Betia, there are 125 cities, some of which are connected by express trains that do not stop at intermediate stations. It is known that any four cities can be visited in a circular order. What is the minimum number of city pairs connected by express trains? | 7688 | 7.8125 |
24,693 | Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed, leaving the remaining six cards in either ascending or descending order. | 74 | 0 |
24,694 | In the triangle \( \triangle ABC \), it is given that the angles are in the ratio \(\angle A : \angle B : \angle C = 3 : 5 : 10\). Also, it is known that \(\triangle A'B'C \cong \triangle ABC\). What is the ratio \(\angle BCA' : \angle BCB'\)? | 1:4 | 0 |
24,695 | Two machine tools, A and B, produce the same product. The products are divided into first-class and second-class according to quality. In order to compare the quality of the products produced by the two machine tools, each machine tool produced 200 products. The quality of the products is as follows:<br/>
| | First-class | Second-class | Total |
|----------|-------------|--------------|-------|
| Machine A | 150 | 50 | 200 |
| Machine B | 120 | 80 | 200 |
| Total | 270 | 130 | 400 |
$(1)$ What are the frequencies of first-class products produced by Machine A and Machine B, respectively?<br/>
$(2)$ Can we be $99\%$ confident that there is a difference in the quality of the products produced by Machine A and Machine B?<br/>
Given: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.<br/>
| $P(K^{2}\geqslant k)$ | 0.050 | 0.010 | 0.001 |
|-----------------------|-------|-------|-------|
| $k$ | 3.841 | 6.635 | 10.828 | | 99\% | 7.03125 |
24,696 | Each square of an $n \times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form $S_B$ . He then performs the same calculation on the red cells to compute $S_R$ .
If $S_B- S_R = 50$ , determine (with proof) all possible values of $k$ . | 313 | 0 |
24,697 | Technology changes the world. Express sorting robots have become popular from Weibo to Moments. According to the introduction, these robots can not only automatically plan the optimal route, accurately place packages in the corresponding compartments, sense and avoid obstacles, automatically return to the team to pick up packages, but also find charging piles to charge themselves when they run out of battery. A certain sorting warehouse plans to sort an average of 200,000 packages per day. However, the actual daily sorting volume may deviate from the plan. The table below shows the situation of sorting packages in the third week of October in this warehouse (the part exceeding the planned amount is recorded as positive, and the part that does not reach the planned amount is recorded as negative):
| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|-----------------|--------|---------|-----------|----------|--------|----------|--------|
| Sorting Situation (in 10,000s) | +6 | +4 | -6 | +8 | -1 | +7 | -4 |
$(1)$ The day with the most sorted packages in the warehouse this week is ______; the day with the least sorted packages is ______; the day with the most sorted packages has ______ more packages than the day with the least sorted packages;<br/>$(2)$ How many packages, on average, did the warehouse actually sort per day this week? | 22 | 0 |
24,698 | Given that the Riemann function defined on the interval $\left[0,1\right]$ is: $R\left(x\right)=\left\{\begin{array}{l}{\frac{1}{q}, \text{when } x=\frac{p}{q} \text{(p, q are positive integers, } \frac{p}{q} \text{ is a reduced proper fraction)}}\\{0, \text{when } x=0,1, \text{or irrational numbers in the interval } (0,1)}\end{array}\right.$, and the function $f\left(x\right)$ is an odd function defined on $R$ with the property that for any $x$ we have $f\left(2-x\right)+f\left(x\right)=0$, and $f\left(x\right)=R\left(x\right)$ when $x\in \left[0,1\right]$, find the value of $f\left(-\frac{7}{5}\right)-f\left(\frac{\sqrt{2}}{3}\right)$. | \frac{5}{3} | 0 |
24,699 | Given the data in the table, where the number of No. 5 batteries is $x$ and the number of No. 7 batteries is $y$, the masses of one No. 5 battery and one No. 7 battery are $x$ grams and $y$ grams, respectively. By setting up and solving a system of equations, express the value of $x$ obtained by elimination. | 24 | 3.125 |
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