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28,500 | What is the largest integer that must divide the product of any $5$ consecutive integers? | 60 | 1.5625 |
28,501 | A point \( M \) is chosen on the diameter \( AB \). Points \( C \) and \( D \), lying on the circumference on one side of \( AB \), are chosen such that \(\angle AMC=\angle BMD=30^{\circ}\). Find the diameter of the circle given that \( CD=12 \). | 8\sqrt{3} | 28.90625 |
28,502 | Given the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$, the focus of the hyperbola is symmetric with respect to the asymptote line and lies on the hyperbola. Calculate the eccentricity of the hyperbola. | \sqrt{5} | 48.4375 |
28,503 | Point $Q$ is located inside triangle $DEF$ such that angles $QDE, QEF,$ and $QFD$ are all congruent. The sides of the triangle have lengths $DE = 15, EF = 16,$ and $FD = 17.$ Find $\tan \angle QDE.$ | \frac{272}{385} | 0 |
28,504 | Given $a$ and $b$ are positive numbers such that $a^b = b^a$ and $b = 4a$, solve for the value of $a$. | \sqrt[3]{4} | 82.8125 |
28,505 | Let \( AEF \) be a triangle with \( EF = 20 \) and \( AE = AF = 21 \). Let \( B \) and \( D \) be points chosen on segments \( AE \) and \( AF \), respectively, such that \( BD \) is parallel to \( EF \). Point \( C \) is chosen in the interior of triangle \( AEF \) such that \( ABCD \) is cyclic. If \( BC = 3 \) and \( CD = 4 \), then the ratio of areas \(\frac{[ABCD]}{[AEF]}\) can be written as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \). | 5300 | 1.5625 |
28,506 | The slope angle of the tangent line to the curve $y=\frac{1}{2}x^2+2$ at the point $\left(-1, \frac{5}{2}\right)$ is what? | \frac{3\pi}{4} | 1.5625 |
28,507 | The distance a dog covers in 3 steps is the same as the distance a fox covers in 4 steps and the distance a rabbit covers in 12 steps. In the time it takes the rabbit to run 10 steps, the dog runs 4 steps and the fox runs 5 steps. Initially, the distances between the dog, fox, and rabbit are as shown in the diagram. When the dog catches up to the fox, the rabbit says: "That was close! If the dog hadn’t caught the fox, I would have been caught by the fox after running $\qquad$ more steps." | 40 | 3.125 |
28,508 | Find the number of six-digit palindromes. | 9000 | 2.34375 |
28,509 | Let the two foci of the conic section $C$ be $F_1$ and $F_2$, respectively. If there exists a point $P$ on curve $C$ such that the ratio $|PF_1| : |F_1F_2| : |PF_2| = 4 : 3 : 2$, determine the eccentricity of curve $C$. | \frac{3}{2} | 20.3125 |
28,510 | There are 2008 red cards and 2008 white cards. 2008 players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.
$ (*)$ If you have more than one red card, then you will pass one red card to the left-neighbouring player.
If you have no red card, then you will pass one white card to the left -neighbouring player.
Find the maximum value of the number of turn required for the state such that all person will have one red card and one white card first. | 1004 | 0 |
28,511 | Two runners start simultaneously and run on a circular track in opposite directions at constant speeds. One runner completes a lap in 5 minutes, and the other runner completes a lap in 8 minutes. Find the number of distinct meeting points of the runners on the track if they run for at least one hour. | 19 | 49.21875 |
28,512 | How many values of $x$, $-17<x<100$, satisfy $\cos^2 x + 3\sin^2 x = \cot^2 x$? (Note: $x$ is measured in radians.) | 37 | 1.5625 |
28,513 | A sample is divided into 5 groups, with a total of 160 data points in the first, second, and third groups, and a total of 260 data points in the third, fourth, and fifth groups, and the frequency of the third group is 0.20. Calculate the frequency of the third group. | 70 | 36.71875 |
28,514 | Express $7.\overline{123}$ as a common fraction in lowest terms. | \frac{593}{111} | 0 |
28,515 | Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}
\]
over all real numbers \( x \). | 2\sqrt{5} | 64.0625 |
28,516 | The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$. | 555 | 0 |
28,517 | Petya's bank account contains $500. The bank allows only two types of transactions: withdrawing $300 or adding $198. What is the maximum amount Petya can withdraw from the account, if he has no other money? | 300 | 14.0625 |
28,518 | How many positive integers less that $200$ are relatively prime to either $15$ or $24$ ? | 120 | 42.1875 |
28,519 | A square with sides 8 inches is shown. If $Q$ is a point such that the segments $\overline{QA}$, $\overline{QB}$, $\overline{QC}$ are equal in length, and segment $\overline{QC}$ is perpendicular to segment $\overline{HD}$, find the area, in square inches, of triangle $AQB$. [asy]
pair A, B, C, D, H, Q;
A = (0,0); B= (2,0); C = (1,2); D = (2,2); H = (0,2); Q = (1,1);
draw(A--B--D--H--cycle);
draw(C--Q); draw(Q--A); draw(Q--B);
label("$A$",A,SW); label("$B$",B,SE);label("$C$",C,N);label("$D$",D,NE);label("$Q$",Q,NW);label("$H$",H,NW);
label("$8''$",(1,0),S);
[/asy] | 12 | 3.125 |
28,520 | Given that $F$ is the right focus of the hyperbola $C$: $x^{2}- \frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \sqrt {6})$ is a point on the $y$-axis. The minimum area of $\triangle APF$ is $\_\_\_\_\_\_$. | 6+9 \sqrt {6} | 0 |
28,521 | A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph. | 22 | 0.78125 |
28,522 | Given that $\textstyle\binom{2k}k$ results in a number that ends in two zeros, find the smallest positive integer $k$. | 13 | 99.21875 |
28,523 | What is the maximum number of cells on an $8 \times 8$ chessboard that can be cut by a single straight line? | 15 | 59.375 |
28,524 | Let points $A(x_1, y_1)$ and $B(x_2, y_2)$ be on the graph of $f(x) = x^2$. The points $C$ and $D$ trisect the segment $\overline{AB}$ with $AC < CB$. A horizontal line drawn through $C$ intersects the curve at another point $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x_2 = 4$. | -2 | 0 |
28,525 | If $\log_3 (x+5)^2 + \log_{1/3} (x - 1) = 4,$ compute $x.$ | \frac{71 + \sqrt{4617}}{2} | 3.90625 |
28,526 | Given the function $f(x) = \sqrt{\log_{3}(4x-1)} + \sqrt{16-2^{x}}$, its domain is A.
(1) Find the set A;
(2) If the function $g(x) = (\log_{2}x)^{2} - 2\log_{2}x - 1$, and $x \in A$, find the maximum and minimum values of the function $g(x)$ and the corresponding values of $x$. | -2 | 20.3125 |
28,527 | Riley has 64 cubes with dimensions \(1 \times 1 \times 1\). Each cube has its six faces labeled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a \(4 \times 4 \times 4\) cube. Riley determines the total of the numbers on the outside of the \(4 \times 4 \times 4\) cube. How many different possibilities are there for this total? | 49 | 4.6875 |
28,528 | When $0.73\overline{864}$ is expressed as a fraction in the form $\frac{y}{999900}$, what is the value of $y$? | 737910 | 41.40625 |
28,529 | A company's capital increases by a factor of two each year compared to the previous year after dividends have been paid, with a fixed dividend of 50 million yuan paid to shareholders at the end of each year. The company's capital after dividends were paid at the end of 2010 was 1 billion yuan.
(i) Find the capital of the company at the end of 2014 after dividends have been paid;
(ii) Find the year from which the company's capital after dividends have been paid exceeds 32.5 billion yuan. | 2017 | 18.75 |
28,530 | Find the minimum value of
\[
\sqrt{x^2 + (2 - x)^2} + \sqrt{(2 - x)^2 + (2 + x)^2}
\]
over all real numbers $x$. | 2\sqrt{5} | 60.15625 |
28,531 | Find the sum of all integral values of \( c \) with \( c \le 30 \) for which the equation \( y=x^2-11x-c \) has two rational roots. | 38 | 6.25 |
28,532 | Given that $\tan\alpha=3$, calculate the following:
(1) $\frac{\sin\alpha+\cos\alpha}{2\sin\alpha-\cos\alpha}$
(2) $\sin^2\alpha+\sin\alpha\cos\alpha+3\cos^2\alpha$ | 15 | 4.6875 |
28,533 | If the line $ax - by + 2 = 0$ $(a > 0, b > 0)$ passes through the center of the circle ${x}^{2} + {y}^{2} + 4x - 4y - 1 = 0$, find the minimum value of $\frac{2}{a} + \frac{3}{b}$. | 5 + 2 \sqrt{6} | 60.9375 |
28,534 | We make colored cubes according to the following specifications:
1. Each face of the cube is divided into two triangles by a diagonal, such that every drawn diagonal connects with two other such diagonals at each end point.
2. Each face's two triangles must be of different colors.
3. Triangles that are adjacent along an edge must be of the same color.
How many different cubes can be made using 6 colors? (Two cubes are not considered different if they can be positioned so that their painted faces look the same from any viewpoint.) | 30 | 67.96875 |
28,535 | Let's consider the number 2023. If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 48 | 30.46875 |
28,536 | Given points $P(\cos \alpha, \sin \alpha)$, $Q(\cos \beta, \sin \beta)$, and $R(\cos \alpha, -\sin \alpha)$ in a two-dimensional space, where $O$ is the origin, if the cosine distance between $P$ and $Q$ is $\frac{1}{3}$, and $\tan \alpha \cdot \tan \beta = \frac{1}{7}$, determine the cosine distance between $Q$ and $R$. | \frac{1}{2} | 59.375 |
28,537 | A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 4 - 2\sqrt{2} | 10.15625 |
28,538 | There are 10,001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of \( k \) societies. Suppose that the following conditions hold:
1. Each pair of students are in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with \( 2m + 1 \) students (where \( m \) is a positive integer) is in exactly \( m \) societies.
Find all possible values of \( k \). | 5000 | 28.125 |
28,539 | Let $A$ , $B$ , $C$ , and $P$ be points in the plane such that no three of them are collinear. Suppose that the areas of triangles $BPC$ , $CPA$ , and $APB$ are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle $ABC$ .
*Proposed by Ankan Bhattacharya* | 84 | 2.34375 |
28,540 | Given a function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$ which is a power function and is increasing on the interval $(0, \infty)$, find the value of the real number $m$. | -1 | 7.8125 |
28,541 | In rectangle $ABCD$, $AB = 10$ cm, $BC = 14$ cm, and $DE = DF$. The area of triangle $DEF$ is one-fifth the area of rectangle $ABCD$. What is the length in centimeters of segment $EF$? Express your answer in simplest radical form. | 4\sqrt{7} | 66.40625 |
28,542 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 8.8 | 0 |
28,543 | Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$ ? | 333 | 85.9375 |
28,544 |
On the image, there are several circles connected by segments. Kostya chooses a natural number \( n \) and places different natural numbers not exceeding \( n \) in the circles so that for all the placed numbers the following property is satisfied: if the numbers \( a \) and \( b \) are connected by a segment, then the difference \( a - b \) must be coprime with \( n \); if they are not connected, then the numbers \( a - b \) and \( n \) must have a common natural divisor greater than 1. For example, in the image (fig. 2) Kostya took \( n = 75 \) and arranged the numbers accordingly, as shown in fig. 3.
a) What is the smallest \( n \) for which the required arrangement of numbers exists in fig. 2?
b) Is it possible to place the numbers in the circles in fig. 4 for \( n = 49 \)?
c) Is it possible to place the numbers in the circles in fig. 4 for \( n = 33 \)?
d) What is the smallest \( n \) for which the arrangement of numbers exists in the circles in fig. 4? | 105 | 0 |
28,545 | A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares. | 27 | 20.3125 |
28,546 | It is known that the optimal amount of a certain material to be added is between 100g and 1100g. If the 0.618 method is used to arrange the experiment and the first and second trials are at points $x_1$ and $x_2$ ($x_1 > x_2$), then when $x_2$ is considered the better point, the third trial point $x_3$ should be __g (answer with a number). | 336 | 4.6875 |
28,547 | There are 4 pieces of part $A$ weighing 5 tons each, 6 pieces of part $B$ weighing 4 tons each, 11 pieces of part $C$ weighing 3 tons each, and 7 pieces of part $D$ weighing 1 ton each. If all the parts are to be transported at once, what is the minimum number of trucks, each with a capacity of 6 tons, required? | 16 | 10.9375 |
28,548 | Given that the rhombus has diagonals of length $8$ and $30$, calculate the radius of the circle inscribed in the rhombus. | \frac{30}{\sqrt{241}} | 0 |
28,549 | There are 99 positive integers, and their sum is 101101. Find the greatest possible value of the greatest common divisor of these 99 positive integers. | 101 | 42.1875 |
28,550 | The hypotenuse of a right triangle whose legs are consecutive even numbers is 34 units. What is the sum of the lengths of the two legs? | 46 | 6.25 |
28,551 | Given $x+m≤{e}^{\frac{2x}{m}+n}$ holds for any $x\in \left(-m,+\infty \right)$, then the minimum value of $m\cdot n$ is ______. | -\frac{2}{e^2} | 30.46875 |
28,552 | What is the sum of the 2023 fractions of the form $\frac{2}{n(n+3)}$ for $n$ as the positive integers from 1 through 2023? Express your answer as a decimal to the nearest thousandth. | 1.222 | 0 |
28,553 | Suppose $a$, $b$, $c$, and $d$ are integers such that:
- $a - b + c = 7$
- $b - c + d = 8$
- $c - d + a = 4$
- $d - a + b = 3$
- $a + b + c - d = 10$
Find the value of $a + b + c + d$. | 16 | 3.125 |
28,554 | How many positive integers $n\leq100$ satisfy $\left\lfloor n\pi\right\rfloor=\left\lfloor\left(n-1\right)\pi\right\rfloor+3$ ? Here $\left\lfloor x\right\rfloor$ is the greatest integer less than or equal to $x$ ; for example, $\left\lfloor\pi\right\rfloor=3$ .
*2018 CCA Math Bonanza Lightning Round #3.2* | 86 | 89.84375 |
28,555 | Given the parabola $C$: $y^{2}=4x$ with focus $F$, a line $l$ passing through $F$ intersects $C$ at points $A$ and $B$. Let $M$ be the midpoint of segment $AB$, and $O$ be the origin. The extensions of $AO$ and $BO$ intersect the line $x=-4$ at points $P$ and $Q$ respectively.
(Ⅰ) Find the equation of the trajectory of the moving point $M$;
(Ⅱ) Connect $OM$, and find the ratio of the areas of $\triangle OPQ$ and $\triangle BOM$. | 32 | 1.5625 |
28,556 | A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | 4 - 2\sqrt{2} | 3.90625 |
28,557 | Consider the largest solution to the equation \[\log_{5x^3} 5 + \log_{25x^4} 5 = -1.\] Find the value of \( x^{10} \). | 0.0000001024 | 0 |
28,558 | Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. | 504 | 0.78125 |
28,559 | An ATM password at Fred's Bank consists of four digits from $0$ to $9$. No password may begin with the sequence "123", and if a password begins with "123", the fourth digit cannot be $4$ or $5$. Calculate the number of valid passwords that are possible. | 9992 | 48.4375 |
28,560 | A circular spinner used in a game has a radius of 15 cm. The probability of winning on one spin of this spinner is $\frac{1}{3}$ for the WIN sector and $\frac{1}{4}$ for the BONUS sector. What is the area, in square centimeters, of both the WIN sector and the BONUS sector? Express your answers in terms of $\pi$. | 56.25\pi | 46.09375 |
28,561 | Find the number of six-digit palindromes. | 9000 | 2.34375 |
28,562 | A finite non-empty set of integers is called $3$ -*good* if the sum of its elements is divisible by $3$ . Find the number of $3$ -good subsets of $\{0,1,2,\ldots,9\}$ . | 351 | 72.65625 |
28,563 | Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 10$ and $\angle O_{1}PO_{2} = 90^{\circ}$, then $AP = \sqrt{c} + \sqrt{d}$, where $c$ and $d$ are positive integers. Find $c + d$. | 100 | 7.03125 |
28,564 | What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month? | 17 | 3.90625 |
28,565 | Given a sector with a radius of 16, and the arc length of the sector is $16\pi$, calculate the central angle and the area of the sector. | 128\pi | 82.03125 |
28,566 | A passenger car traveling at a speed of 66 km/h arrives at its destination at 6:53, while a truck traveling at a speed of 42 km/h arrives at the same destination via the same route at 7:11. How many kilometers before the destination did the passenger car overtake the truck? | 34.65 | 64.84375 |
28,567 | Given the parabola $C: x^{2}=8y$ and its focus $F$, the line $PQ$ and $MN$ intersect the parabola $C$ at points $P$, $Q$, and $M$, $N$, respectively. If the slopes of the lines $PQ$ and $MN$ are $k_{1}$ and $k_{2}$, and satisfy $\frac{1}{{k_1^2}}+\frac{4}{{k_2^2}}=1$, then the minimum value of $|PQ|+|MN|$ is ____. | 88 | 5.46875 |
28,568 | Consider a 4x4 grid with points that are equally spaced horizontally and vertically, where the distance between two neighboring points is 1 unit. Two triangles are formed: Triangle A connects points at (0,0), (3,2), and (2,3), while Triangle B connects points at (0,3), (3,3), and (3,0). What is the area, in square units, of the region where these two triangles overlap? | 0.5 | 10.9375 |
28,569 | The ratio of the areas of two squares is $\frac{50}{98}$. After rationalizing the denominator, express the simplified form of the ratio of their side lengths in the form $\frac{a \sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Find the sum $a+b+c$. | 14 | 1.5625 |
28,570 | In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}-4x=0$. The parameter equation of curve $C_{2}$ is $\left\{\begin{array}{l}x=\cos\beta\\ y=1+\sin\beta\end{array}\right.$ ($\beta$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive $x$-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equations of curves $C_{1}$ and $C_{2}$;<br/>$(2)$ If the ray $\theta =\alpha (\rho \geqslant 0$, $0＜\alpha＜\frac{π}{2})$ intersects curve $C_{1}$ at point $P$, the line $\theta=\alpha+\frac{π}{2}(\rho∈R)$ intersects curves $C_{1}$ and $C_{2}$ at points $M$ and $N$ respectively, and points $P$, $M$, $N$ are all different from point $O$, find the maximum value of the area of $\triangle MPN$. | 2\sqrt{5} + 2 | 0.78125 |
28,571 | Let $\mathbf{a} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 6 \end{pmatrix}.$ Calculate the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}$. Also, determine the vector $\mathbf{a} + \mathbf{b}$. | 15 | 0 |
28,572 | Define mutually externally tangent circles $\omega_1$ , $\omega_2$ , and $\omega_3$ . Let $\omega_1$ and $\omega_2$ be tangent at $P$ . The common external tangents of $\omega_1$ and $\omega_2$ meet at $Q$ . Let $O$ be the center of $\omega_3$ . If $QP = 420$ and $QO = 427$ , find the radius of $\omega_3$ .
*Proposed by Tanishq Pauskar and Mahith Gottipati* | 77 | 32.03125 |
28,573 | Given \( A \subseteq \{1, 2, \ldots, 25\} \) such that \(\forall a, b \in A\), \(a \neq b\), then \(ab\) is not a perfect square. Find the maximum value of \( |A| \) and determine how many such sets \( A \) exist where \( |A| \) achieves this maximum value.
| 16 | 4.6875 |
28,574 | Solve the quadratic equation $x^{2}-2x+3=4x$. | 3-\sqrt{6} | 3.125 |
28,575 | Evaluate the infinite geometric series: $$\frac{5}{3} - \frac{3}{5} + \frac{9}{25} - \frac{27}{125} + \dots$$ | \frac{125}{102} | 44.53125 |
28,576 | Given positive numbers $a$ and $b$ satisfying $a+b=1$, $c\in R$, find the minimum value of $\frac{3a}{b{c}^{2}+b}+\frac{1}{ab{c}^{2}+ab}+3c^{2}$. | 6\sqrt{2} - 3 | 0 |
28,577 | Given that the sequence $\{a_n\}$ is an arithmetic sequence, and $a_2=-1$, the sequence $\{b_n\}$ satisfies $b_n-b_{n-1}=a_n$ ($n\geqslant 2, n\in \mathbb{N}$), and $b_1=b_3=1$
(I) Find the value of $a_1$;
(II) Find the general formula for the sequence $\{b_n\}$. | -3 | 2.34375 |
28,578 | Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$ . Determine the area of $\vartriangle DEF$ . | \frac{9\sqrt{3}}{16} | 0 |
28,579 | The number of four-digit even numbers formed without repeating digits from the numbers $2$, $0$, $1$, $7$ is ______. | 10 | 19.53125 |
28,580 | The longest side of a right triangle is 13 meters and one of the other sides is 5 meters. What is the area and the perimeter of the triangle? Also, determine if the triangle has any acute angles. | 30 | 3.125 |
28,581 | The solutions to the equation $x^2 - 3|x| - 2 = 0$ are. | \frac{-3 - \sqrt{17}}{2} | 0 |
28,582 | In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 15 and 30 units respectively, and the altitude is 18 units. Points $E$ and $F$ divide legs $AD$ and $BC$ into thirds respectively, with $E$ one third from $A$ to $D$ and $F$ one third from $B$ to $C$. Calculate the area of quadrilateral $EFCD$. | 360 | 5.46875 |
28,583 | What is $\frac{2468_{10}}{123_{5}} \times 107_{8} + 4321_{9}$? Express your answer in base 10. | 7789 | 0 |
28,584 | If the integers \( a, b, c \) satisfy \( 0 \leq a \leq 10 \), \( 0 \leq b \leq 10 \), \( 0 \leq c \leq 10 \), and \( 10 \leq a + b + c \leq 20 \), then how many ordered triples \((a, b, c)\) meet the conditions? | 286 | 0 |
28,585 | Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \dots + 12100$, where the signs change after each perfect square. | 1100000 | 0 |
28,586 | (Elective 4-4: Coordinate Systems and Parametric Equations)
In the rectangular coordinate system $xOy$, a pole coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C_{1}$ is $\rho \cos \theta =4$.
(1) Let $M$ be a moving point on the curve $C_{1}$, and let $P$ be a point on the line segment $OM$ such that $|OM|\cdot |OP|=16$. Determine the rectangular coordinate equation of the trajectory $C_{2}$ of point $P$.
(2) Let point $A$ have polar coordinates $(2,\dfrac{\pi }{3})$, and let point $B$ be on the curve $C_{2}$. Determine the maximum area of the triangle $OAB$. | \sqrt{3}+2 | 2.34375 |
28,587 | Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$ . Let $D$ and $E$ be points on the circle such that $DC\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle{DCE}$ to the area of $\triangle{ABD}$ ? | 1/6 | 5.46875 |
28,588 | How many different ways can the five vertices S, A, B, C, and D of a square pyramid S-ABCD be colored using four distinct colors so that each vertex is assigned one color and no two vertices sharing an edge have the same color? | 72 | 57.03125 |
28,589 | Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its right focus $F$, and the line passing through $F$ with a slope of $1$ intersects the ellipse at points $M$ and $N$. The perpendicular bisector of $MN$ intersects the $x$-axis at point $P$. If $\frac{|MN|}{|PF|}=4$, find the eccentricity of the ellipse $C$. | \frac{1}{2} | 8.59375 |
28,590 | How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | 51 | 1.5625 |
28,591 | Given that point $P$ is the intersection point of the lines $l_{1}$: $mx-ny-5m+n=0$ and $l_{2}$: $nx+my-5m-n=0$ ($m$,$n\in R$, $m^{2}+n^{2}\neq 0$), and point $Q$ is a moving point on the circle $C$: $\left(x+1\right)^{2}+y^{2}=1$, calculate the maximum value of $|PQ|$. | 6 + 2\sqrt{2} | 0 |
28,592 | Using the method of base prime representation, where the place of each digit represents an exponent in the prime factorization (starting with the smallest prime on the right), convert the number $196$ into base prime. | 2002 | 34.375 |
28,593 | The opposite number of $-1 \frac{1}{2}$ is ______, its reciprocal is ______, and its absolute value is ______. | 1.5 | 0 |
28,594 | Four tour guides are leading eight tourists. Each tourist must choose one of the guides, however, each guide must take at least two tourists. How many different groupings of guides and tourists are possible? | 105 | 0 |
28,595 | Given that a normal vector of line $l$ is $\overrightarrow{n}=(\sqrt{3}, -1)$, find the size of the slope angle of line $l$. | \frac{\pi}{3} | 75.78125 |
28,596 | Given a solid $\Omega$ which is the larger part obtained by cutting a sphere $O$ with radius $4$ by a plane $\alpha$, and $\triangle ABC$ is an inscribed triangle of the circular section $O'$ with $\angle A=90^{\circ}$. Point $P$ is a moving point on the solid $\Omega$, and the projection of $P$ on the circle $O'$ lies on the circumference of $O'$. Given $OO'=1$, the maximum volume of the tetrahedron $P-ABC$ is \_\_\_\_\_\_ | 10 | 5.46875 |
28,597 | From the center \( O \) of the inscribed circle of a right triangle, the half of the hypotenuse that is closer to \( O \) appears at a right angle. What is the ratio of the sides of the triangle? | 3 : 4 : 5 | 3.90625 |
28,598 | Given $f(x) = 2\sqrt{3}\sin x \cos x + 2\cos^2x - 1$,
(1) Find the maximum value of $f(x)$, as well as the set of values of $x$ for which $f(x)$ attains its maximum value;
(2) In $\triangle ABC$, if $a$, $b$, and $c$ are the lengths of sides opposite the angles $A$, $B$, and $C$ respectively, with $a=1$, $b=\sqrt{3}$, and $f(A) = 2$, determine the angle $C$. | \frac{\pi}{2} | 89.84375 |
28,599 | In the Cartesian coordinate system $xoy$, point $P(0, \sqrt{3})$ is given. The parametric equation of curve $C$ is $\begin{cases} x = \sqrt{2} \cos \varphi \\ y = 2 \sin \varphi \end{cases}$ (where $\varphi$ is the parameter). A polar coordinate system is established with the origin as the pole and the positive half-axis of $x$ as the polar axis. The polar equation of line $l$ is $\rho = \frac{\sqrt{3}}{2\cos(\theta - \frac{\pi}{6})}$.
(Ⅰ) Determine the positional relationship between point $P$ and line $l$, and explain the reason;
(Ⅱ) Suppose line $l$ intersects curve $C$ at two points $A$ and $B$, calculate the value of $\frac{1}{|PA|} + \frac{1}{|PB|}$. | \sqrt{14} | 0 |
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