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40.3k
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100
|
|---|---|---|---|
19,900 |
How many arithmetic sequences, where the common difference is a natural number greater than 2, satisfy the conditions that the first term is 1783, the last term is 1993, and the number of terms is at least 3?
|
13
| 34.375 |
19,901 |
Given points $P(\sqrt{3}, 1)$, $Q(\cos x, \sin x)$, and $O$ as the origin of coordinates, the function $f(x) = \overrightarrow{OP} \cdot \overrightarrow{QP}$
(Ⅰ) Find the smallest positive period of the function $f(x)$;
(Ⅱ) If $A$ is an internal angle of $\triangle ABC$, $f(A) = 4$, $BC = 3$, and the area of $\triangle ABC$ is $\frac{3\sqrt{3}}{4}$, find the perimeter of $\triangle ABC$.
|
3 + 2\sqrt{3}
| 54.6875 |
19,902 |
Given a function $f(x)$ such that for any $x$, $f(x+2)=f(x+1)-f(x)$, and $f(1)=\log_3-\log_2$, $f(2)=\log_3+\log_5$, calculate the value of $f(2010)$.
|
-1
| 31.25 |
19,903 |
A company has a total of 60 employees. In order to carry out club activities, a questionnaire survey was conducted among all employees. There are 28 people who like sports, 26 people who like literary and artistic activities, and 12 people who do not like either sports or literary and artistic activities. How many people like sports but do not like literary and artistic activities?
|
22
| 78.90625 |
19,904 |
In the rectangular coordinate system $(xOy)$, the polar coordinate system is established with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of circle $C$ is $ρ=2 \sqrt{2}\cos \left(θ+\frac{π}{4} \right)$, and the parametric equation of line $l$ is $\begin{cases} x=t \\ y=-1+2 \sqrt{2}t \end{cases}(t\text{ is the parameter})$. Line $l$ intersects circle $C$ at points $A$ and $B$, and $P$ is any point on circle $C$ different from $A$ and $B$.
(1) Find the rectangular coordinates of the circle center.
(2) Find the maximum area of $\triangle PAB$.
|
\frac{10 \sqrt{5}}{9}
| 14.0625 |
19,905 |
Add $254_{9} + 627_{9} + 503_{9}$. Express your answer in base 9.
|
1485_{9}
| 5.46875 |
19,906 |
If the square roots of a number are $2a+3$ and $a-18$, then this number is ____.
|
169
| 62.5 |
19,907 |
Calculate:<br/>$(1)(\sqrt{50}-\sqrt{8})÷\sqrt{2}$;<br/>$(2)\sqrt{\frac{3}{4}}×\sqrt{2\frac{2}{3}}$.
|
\sqrt{2}
| 78.90625 |
19,908 |
Given a right triangle $PQR$ with $\angle PQR = 90^\circ$, suppose $\cos Q = 0.6$ and $PQ = 15$. What is the length of $QR$?
|
25
| 8.59375 |
19,909 |
Given that $abc$ represents a three-digit number, if it satisfies $a \lt b$ and $b \gt c$, then we call this three-digit number a "convex number". The number of three-digit "convex" numbers without repeated digits is ______.
|
204
| 95.3125 |
19,910 |
There are two arithmetic sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, with respective sums of the first $n$ terms denoted by $S_{n}$ and $T_{n}$. Given that $\dfrac{S_{n}}{T_{n}} = \dfrac{3n}{2n+1}$, find the value of $\dfrac{a_{1}+a_{2}+a_{14}+a_{19}}{b_{1}+b_{3}+b_{17}+b_{19}}$.
A) $\dfrac{27}{19}$
B) $\dfrac{18}{13}$
C) $\dfrac{10}{7}$
D) $\dfrac{17}{13}$
|
\dfrac{17}{13}
| 25 |
19,911 |
Calculate:<br/>$(1)(1\frac{3}{4}-\frac{3}{8}+\frac{5}{6})÷(-\frac{1}{24})$;<br/>$(2)-2^2+(-4)÷2×\frac{1}{2}+|-3|$.
|
-2
| 79.6875 |
19,912 |
In triangle $PQR$, let $PQ = 15$, $PR = 20$, and $QR = 25$. The line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y$. Determine the perimeter of $\triangle PXY$.
|
35
| 14.0625 |
19,913 |
Chen, Ruan, Lu, Tao, and Yang did push-ups. It is known that Chen, Lu, and Yang together averaged 40 push-ups per person, Ruan, Tao, and Chen together averaged 28 push-ups per person, and Ruan, Lu, Tao, and Yang together averaged 33 push-ups per person. How many push-ups did Chen do?
|
36
| 46.875 |
19,914 |
Insert two numbers between 1 and 2 to form an arithmetic sequence. What is the common difference?
|
\frac{1}{3}
| 14.0625 |
19,915 |
Let $f(x) = x^2 + px + q$ and $g(x) = x^2 + rx + s$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g$, and the $x$-coordinate of the vertex of $g$ is a root of $f$. If both $f$ and $g$ have the same minimum value and the graphs of the two polynomials intersect at the point $(50, -50)$, what is the value of $p + r$?
|
-200
| 53.125 |
19,916 |
How many 4-digit positive integers, where each digit is odd, are divisible by 3?
|
208
| 3.125 |
19,917 |
If we divide $2020$ by a prime $p$ , the remainder is $6$ . Determine the largest
possible value of $p$ .
|
53
| 92.96875 |
19,918 |
The polynomial $Q(x) = 3x^3 + dx^2 + ex + f$ has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. What is $e$?
|
-42
| 66.40625 |
19,919 |
Given the functions $f(x)=2(x+1)$ and $g(x)=x+ \ln x$, points $A$ and $B$ are located on the graphs of $f(x)$ and $g(x)$ respectively, and their y-coordinates are always equal. Calculate the minimum distance between points $A$ and $B$.
|
\frac{3}{2}
| 32.8125 |
19,920 |
Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \dots , 4025 $ . Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $ ?
|
1006/4025
| 38.28125 |
19,921 |
One corner of a cube is cut off, creating a new triangular face. How many edges does this new solid have?
|
15
| 45.3125 |
19,922 |
How many positive integers which divide $5n^{11}-2n^5-3n$ for all positive integers $n$ are there?
|
12
| 46.09375 |
19,923 |
Observation: Given $\sqrt{5}≈2.236$, $\sqrt{50}≈7.071$, $\sqrt[3]{6.137}≈1.8308$, $\sqrt[3]{6137}≈18.308$; fill in the blanks:<br/>① If $\sqrt{0.5}\approx \_\_\_\_\_\_.$<br/>② If $\sqrt[3]{x}≈-0.18308$, then $x\approx \_\_\_\_\_\_$.
|
-0.006137
| 51.5625 |
19,924 |
Find the product of the roots of the equation $24x^2 + 60x - 750 = 0$.
|
-\frac{125}{4}
| 28.90625 |
19,925 |
Given $x \gt 0$, $y \gt 0$, when $x=$______, the maximum value of $\sqrt{xy}(1-x-2y)$ is _______.
|
\frac{\sqrt{2}}{16}
| 14.84375 |
19,926 |
For positive real numbers \(a\), \(b\), and \(c\), compute the maximum value of
\[
\frac{abc(a + b + c + ab)}{(a + b)^3 (b + c)^3}.
\]
|
\frac{1}{16}
| 82.8125 |
19,927 |
In the arithmetic sequence $\{a_{n}\}$, if $\frac{{a}_{9}}{{a}_{8}}＜-1$, and its sum of the first $n$ terms $S_{n}$ has a minimum value, determine the minimum value of $n$ for which $S_{n} \gt 0$.
|
16
| 20.3125 |
19,928 |
Calculate $7 \cdot 9\frac{2}{5}$.
|
65\frac{4}{5}
| 82.03125 |
19,929 |
Given the function $y=2\sin \left(3x+ \dfrac{\pi}{4}\right)$, determine the shift required to obtain its graph from the graph of the function $y=2\sin 3x$.
|
\dfrac{\pi}{12}
| 64.0625 |
19,930 |
Given that M is a point on the parabola $y^2 = 2px$ ($p > 0$), F is the focus of the parabola $C$, and $|MF| = p$. K is the intersection point of the directrix of the parabola $C$ and the x-axis. Calculate the measure of angle $\angle MKF$.
|
45
| 33.59375 |
19,931 |
Given the function $f(x) = \log_{m}(m - x)$, if the maximum value in the interval $[3, 5]$ is 1 greater than the minimum value, determine the real number $m$.
|
3 + \sqrt{6}
| 96.875 |
19,932 |
Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits.
|
397
| 63.28125 |
19,933 |
Given that -1, a, b, -4 form an arithmetic sequence, and -1, c, d, e, -4 form a geometric sequence, calculate the value of $$\frac{b-a}{d}$$.
|
\frac{1}{2}
| 96.875 |
19,934 |
Determine the sum of coefficients $A$, $B$, $C$, and $D$ for the simplified polynomial form of the function
\[ y = \frac{x^3 - 4x^2 - 9x + 36}{x - 3} \]
which is defined everywhere except at $x = D$.
|
-9
| 51.5625 |
19,935 |
Calculate $46_8 - 27_8$ and express your answer in base 8.
|
17_8
| 96.875 |
19,936 |
Two distinct numbers are selected from the set $\{1,2,3,4,\dots,38\}$ so that the sum of the remaining $36$ numbers equals the product of these two selected numbers plus one. Find the difference of these two numbers.
|
20
| 82.8125 |
19,937 |
Find the smallest integer \( k \) such that when \( 5 \) is multiplied by a number consisting of \( k \) digits of \( 7 \) (i.e., \( 777\ldots7 \) with \( k \) sevens), the resulting product has digits summing to \( 800 \).
A) 86
B) 87
C) 88
D) 89
E) 90
|
88
| 36.71875 |
19,938 |
Any seven points are taken inside or on a square with side length $2$. Determine $b$, the smallest possible number with the property that it is always possible to select one pair of points from these seven such that the distance between them is equal to or less than $b$.
|
\sqrt{2}
| 96.09375 |
19,939 |
The reciprocal of $-2024$ is ______; $7.8963\approx$ ______ (rounded to the nearest hundredth).
|
7.90
| 97.65625 |
19,940 |
There are two rows of seats, with 4 seats in the front row and 5 seats in the back row. Now, we need to arrange seating for 2 people, and these 2 people cannot sit next to each other (sitting one in front and one behind is also considered as not adjacent). How many different seating arrangements are there?
|
58
| 7.8125 |
19,941 |
In the next 3 days, a meteorological station forecasts the weather with an accuracy rate of 0.8. The probability that the forecast is accurate for at least two consecutive days is ___.
|
0.768
| 24.21875 |
19,942 |
If the function $f(x) = C_8^0x + C_8^1x^1 + C_8^2x^2 + \ldots + C_8^8x^8$ ($x \in \mathbb{R}$), then $\log_2f(3) = \ $.
|
16
| 100 |
19,943 |
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|+1?$
A) $\frac{\pi}{2} + 2$
B) $\frac{3\pi}{2}$
C) $\frac{3\pi}{2} + 2$
D) $2\pi + 2$
|
\frac{3\pi}{2} + 2
| 59.375 |
19,944 |
Given the sequence $\{a_n\}$ that satisfies $a_1=1$, $a_2=2$, and $2na_n=(n-1)a_{n-1}+(n+1)a_{n+1}$ for $n \geq 2$ and $n \in \mathbb{N}^*$, find the value of $a_{18}$.
|
\frac{26}{9}
| 94.53125 |
19,945 |
There are three environmental knowledge quiz questions, $A$, $B$, and $C$. The table below shows the statistics of the quiz results. The number of people who answered exactly two questions correctly is $\qquad$, and the number of people who answered only one question correctly is $\qquad$.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline & Total number of people & Correctly answered $A$ & Correctly answered $B$ & Correctly answered $C$ & All incorrect & All correct \\
\hline Number of people & 40 & 10 & 13 & 15 & 15 & 1 \\
\hline
\end{tabular}
|
13
| 37.5 |
19,946 |
Given that in a class test, $15\%$ of the students scored $60$ points, $50\%$ scored $75$ points, $20\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test.
|
2.75
| 29.6875 |
19,947 |
Given any two positive integers, a certain operation (denoted by the operator $\oplus$) is defined as follows: when $m$ and $n$ are both positive even numbers or both positive odd numbers, $m \oplus n = m + n$; when one of $m$ and $n$ is a positive even number and the other is a positive odd number, $m \oplus n = m \cdot n$. The number of elements in the set $M = {(a, b) \mid a \oplus b = 12, a, b \in \mathbb{N}^*}$ is $\_\_\_\_\_\_$.
|
15
| 25 |
19,948 |
Translate the graph of the function $f(x)=\sin(2x+\varphi)$ ($|\varphi| < \frac{\pi}{2}$) to the left by $\frac{\pi}{6}$ units. If the resulting graph is symmetric about the origin, determine the minimum value of the function $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$.
|
-\frac{\sqrt{3}}{2}
| 74.21875 |
19,949 |
An acronym XYZ is drawn within a 2x4 rectangular grid with grid lines spaced 1 unit apart. The letter X is formed by two diagonals crossing in a $1 \times 1$ square. Y consists of a vertical line segment and two slanted segments each forming 45° with the vertical line, making up a symmetric letter. Z is formed by a horizontal segment at the top and bottom of a $1 \times 2$ rectangle, with a diagonal connecting these segments. In units, what is the total length of the line segments forming the acronym XYZ?
A) $5 + 4\sqrt{2} + \sqrt{5}$
B) $5 + 2\sqrt{2} + 3\sqrt{5}$
C) $6 + 3\sqrt{2} + 2\sqrt{5}$
D) $7 + 4\sqrt{2} + \sqrt{3}$
E) $4 + 5\sqrt{2} + \sqrt{5}$
|
5 + 4\sqrt{2} + \sqrt{5}
| 28.90625 |
19,950 |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2a^{2}\sin B\sin C=\sqrt{3}(a^{2}+b^{2}-c^{2})\sin A$. Find:
$(1)$ Angle $C$;
$(2)$ If $a=1$, $b=2$, and the midpoint of side $AB$ is $D$, find the length of $CD$.
|
\frac{\sqrt{7}}{2}
| 79.6875 |
19,951 |
There are $2020\times 2020$ squares, and at most one piece is placed in each square. Find the minimum possible number of pieces to be used when placing a piece in a way that satisfies the following conditions.
・For any square, there are at least two pieces that are on the diagonals containing that square.
Note : We say the square $(a,b)$ is on the diagonals containing the square $(c,d)$ when $|a-c|=|b-d|$ .
|
2020
| 7.03125 |
19,952 |
A linear function \( f(x) \) is given. It is known that the distance between the points of intersection of the graphs \( y = x^{2} \) and \( y = f(x) \) is \( 2 \sqrt{3} \), and the distance between the points of intersection of the graphs \( y = x^{2}-2 \) and \( y = f(x)+1 \) is \( \sqrt{60} \). Find the distance between the points of intersection of the graphs \( y = x^{2}-1 \) and \( y = f(x)+1 \).
|
2 \sqrt{11}
| 22.65625 |
19,953 |
Given $\log_9 \Big(\log_4 (\log_3 x) \Big) = 1$, calculate the value of $x^{-2/3}$.
|
3^{-174762.6667}
| 3.90625 |
19,954 |
Let $a$ and $b$ be positive integers such that $(2a+b)(2b+a)=4752$ . Find the value of $ab$ .
*Proposed by James Lin*
|
520
| 0.78125 |
19,955 |
Given a number \\(x\\) randomly selected from the interval \\(\left[-\frac{\pi}{4}, \frac{2\pi}{3}\right]\\), find the probability that the function \\(f(x)=3\sin\left(2x- \frac{\pi}{6}\right)\\) is not less than \\(0\\).
|
\frac{6}{11}
| 71.875 |
19,956 |
Given a quadratic function $f(x)$ with a second-degree coefficient $a$, and the inequality $f(x) > -2x$ has the solution set $(1,3)$:
(1) If the function $y = f(x) + 6a$ has exactly one zero, find the explicit form of $f(x)$.
(2) Let $h(a)$ be the maximum value of $f(x)$, find the minimum value of $h(a)$.
|
-2
| 20.3125 |
19,957 |
What is the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$ ?
|
21
| 67.96875 |
19,958 |
For some constants \( c \) and \( d \), let
\[ g(x) = \left\{
\begin{array}{cl}
cx + d & \text{if } x < 3, \\
10 - 2x & \text{if } x \ge 3.
\end{array}
\right.\]
The function \( g \) has the property that \( g(g(x)) = x \) for all \( x \). What is \( c + d \)?
|
4.5
| 0 |
19,959 |
In a triangle $XYZ$, $\angle XYZ = \angle YXZ$. If $XZ=8$ and $YZ=11$, what is the perimeter of $\triangle XYZ$?
|
30
| 12.5 |
19,960 |
Let $x_1$ , $x_2$ , …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$ ?
|
171
| 46.09375 |
19,961 |
Given a geometric sequence with positive terms $\{a_n\}$, let $S_n$ be the sum of the first $n$ terms. If $S_3 + a_2 = 9a_3$, determine the common ratio.
|
\frac{1}{2}
| 97.65625 |
19,962 |
Zhang Hua has to go through four traffic posts A, B, C, and D on his way to school. The probability of encountering a red light at posts A and B is $\frac{1}{2}$ each, and at posts C and D, it is $\frac{1}{3}$ each. Assuming that the events of encountering red lights at the four traffic posts are independent, let X represent the number of times he encounters red lights.
(1) If x≥3, he will be late. Find the probability that Zhang Hua is not late.
(2) Find $E(X)$, the expected number of times he encounters red lights.
|
\frac{5}{3}
| 55.46875 |
19,963 |
Given that $x = \frac{3}{4}$ is a solution to the equation $108x^2 - 35x - 77 = 0$, what is the other value of $x$ that will solve the equation? Express your answer as a common fraction.
|
-\frac{23}{54}
| 38.28125 |
19,964 |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 24$ and $X$ is an integer, what is the smallest possible value of $X$?
|
4625
| 32.03125 |
19,965 |
Given $\sin a= \frac{ \sqrt{5}}{5}$, $a\in\left( \frac{\pi}{2},\pi\right)$, find:
$(1)$ The value of $\sin 2a$;
$(2)$ The value of $\tan \left( \frac{\pi}{3}+a\right)$.
|
5 \sqrt{3}-8
| 71.875 |
19,966 |
The number of natural numbers from 1 to 1992 that are multiples of 3, but not multiples of 2 or 5, is
(Ninth "Jinyun Cup" Middle School Mathematics Invitational Competition, 1992)
|
266
| 17.1875 |
19,967 |
Rationalize the denominator of \(\frac{5}{4\sqrt{7} - 3\sqrt{2}}\) and write your answer in the form \(\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}\), where \(B < D\), the fraction is in lowest terms, and all radicals are in simplest radical form. What is \(A+B+C+D+E\)?
|
138
| 85.9375 |
19,968 |
Consider a rectangle with dimensions 6 units by 8 units. Points $A$, $B$, and $C$ are located on the sides of this rectangle such that the coordinates of $A$, $B$, and $C$ are $(0,2)$, $(6,0)$, and $(3,8)$ respectively. What is the area of triangle $ABC$ in square units?
|
21
| 97.65625 |
19,969 |
In the polar coordinate system, the equation of curve C is $\rho^2\cos2\theta=9$. Point P is $(2\sqrt{3}, \frac{\pi}{6})$. Establish a Cartesian coordinate system with the pole O as the origin and the positive half-axis of the x-axis as the polar axis.
(1) Find the parametric equation of line OP and the Cartesian equation of curve C;
(2) If line OP intersects curve C at points A and B, find the value of $\frac{1}{|PA|} + \frac{1}{|PB|}$.
|
\sqrt{2}
| 11.71875 |
19,970 |
In trapezoid $ABCD$, the sides $AB$ and $CD$ are equal. The length of the base $BC$ is 10 units, and the height from $AB$ to $CD$ is 5 units. The base $AD$ is 22 units. Calculate the perimeter of $ABCD$.
|
2\sqrt{61} + 32
| 39.84375 |
19,971 |
In three independent repeated trials, the probability of event $A$ occurring in each trial is the same. If the probability of event $A$ occurring at least once is $\frac{63}{64}$, then the probability of event $A$ occurring exactly once is $\_\_\_\_\_\_$.
|
\frac{9}{64}
| 53.125 |
19,972 |
The sum of three different numbers is 100. The two larger numbers differ by 8 and the two smaller numbers differ by 5. What is the value of the largest number?
|
\frac{121}{3}
| 7.8125 |
19,973 |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder?
|
\sqrt{55}
| 24.21875 |
19,974 |
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, and point $M$ is on $E$ with $MF_1$ perpendicular to the x-axis and $\sin \angle MF_2F_1 = \frac{1}{3}$, find the eccentricity of $E$.
|
\sqrt{2}
| 91.40625 |
19,975 |
In the Metropolitan County, there are $25$ cities. From a given bar chart, the average population per city is indicated midway between $5,200$ and $5,800$. If two of these cities, due to a recent demographic survey, were found to exceed the average by double, calculate the closest total population of all these cities.
|
148,500
| 0 |
19,976 |
On an $8 \times 8$ chessboard, some squares are marked with asterisks such that:
(1) No two squares with asterisks share a common edge or vertex;
(2) Every unmarked square shares a common edge or vertex with at least one marked square.
What is the minimum number of squares that need to be marked with asterisks? Explain the reasoning.
|
16
| 9.375 |
19,977 |
Given that \\(\theta\\) is an angle in the fourth quadrant, and \\(\sin (\theta+ \frac {\pi}{4})= \frac {3}{5}\\), then \\(\tan (\theta- \frac {\pi}{4})=\\) \_\_\_\_\_\_ .
|
- \frac {4}{3}
| 84.375 |
19,978 |
A high school has three math teachers. To facilitate students, math teachers are scheduled for duty from Monday to Friday, with two teachers on duty on Monday. If each teacher is on duty for two days a week, then there are ________ possible duty schedules for a week.
|
36
| 14.84375 |
19,979 |
Four fair eight-sided dice (with faces showing 1 to 8) are rolled. What is the probability that the sum of the numbers on the top faces equals 32?
|
\frac{1}{4096}
| 92.96875 |
19,980 |
Let $\eta(m)$ be the product of all positive integers that divide $m$ , including $1$ and $m$ . If $\eta(\eta(\eta(10))) = 10^n$ , compute $n$ .
*Proposed by Kevin Sun*
|
450
| 60.15625 |
19,981 |
Suppose that $\{b_n\}$ is an arithmetic sequence with $$
b_1+b_2+ \cdots +b_{150}=150 \quad \text{and} \quad
b_{151}+b_{152}+ \cdots + b_{300}=450.
$$What is the value of $b_2 - b_1$? Express your answer as a common fraction.
|
\frac{1}{75}
| 83.59375 |
19,982 |
A quadrilateral is inscribed in a circle with a radius of 13. The diagonals of the quadrilateral are perpendicular to each other. One of the diagonals is 18, and the distance from the center of the circle to the point where the diagonals intersect is \( 4 \sqrt{6} \).
Find the area of the quadrilateral.
|
18 \sqrt{161}
| 0.78125 |
19,983 |
Sets $A, B$ , and $C$ satisfy $|A| = 92$ , $|B| = 35$ , $|C| = 63$ , $|A\cap B| = 16$ , $|A\cap C| = 51$ , $|B\cap C| = 19$ . Compute the number of possible values of $ |A \cap B \cap C|$ .
|
10
| 3.90625 |
19,984 |
Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 60^\circ$, and $DF=12$. Find the radius of the incircle of $\triangle DEF$.
|
6(\sqrt{3}-1)
| 17.1875 |
19,985 |
Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator?
|
36
| 20.3125 |
19,986 |
$\frac{x^{2}}{9} + \frac{y^{2}}{7} = 1$, where $F_{1}$ and $F_{2}$ are the foci of the ellipse. Given that point $A$ lies on the ellipse and $\angle AF_{1}F_{2} = 45^{\circ}$, find the area of triangle $AF_{1}F_{2}$.
|
\frac{7}{2}
| 17.96875 |
19,987 |
Five students from a certain class participated in a speech competition and the order of appearance was determined by drawing lots, under the premise that student A must appear before student B. Calculate the probability of students A and B appearing adjacent to each other.
|
\frac{2}{5}
| 85.9375 |
19,988 |
Given the parabola $y^2=2x$, find the equation of its directrix.
|
-\frac{1}{2}
| 100 |
19,989 |
Given that $\alpha$ is an angle in the second quadrant, simplify the function $f(\alpha) = \frac{\tan(\alpha - \pi) \cos(2\pi - \alpha) \sin(-\alpha + \frac{3\pi}{2})}{\cos(-\alpha - \pi) \tan(\pi + \alpha)}$. Then, if $\cos(\alpha + \frac{\pi}{2}) = -\frac{1}{5}$, find $f(\alpha)$.
|
-\frac{2\sqrt{6}}{5}
| 78.90625 |
19,990 |
Given that the line $x=\dfrac{\pi }{6}$ is the axis of symmetry of the graph of the function $f\left(x\right)=\sin \left(2x+\varphi \right)\left(|\varphi | \lt \dfrac{\pi }{2}\right)$, determine the horizontal shift required to transform the graph of the function $y=\sin 2x$ into the graph of $y=f\left(x\right)$.
|
\dfrac{\pi}{12}
| 42.96875 |
19,991 |
A bag contains 15 red marbles, 10 blue marbles, and 5 green marbles. Four marbles are selected at random and without replacement. What is the probability that two marbles are red, one is blue, and one is green? Express your answer as a common fraction.
|
\frac{350}{1827}
| 9.375 |
19,992 |
Find the smallest positive real number $d,$ such that for all nonnegative real numbers $x, y,$ and $z,$
\[
\sqrt{xyz} + d |x^2 - y^2 + z^2| \ge \frac{x + y + z}{3}.
\]
|
\frac{1}{3}
| 35.15625 |
19,993 |
Given that the polar coordinate equation of circle C is ρ² + 2$\sqrt {2}$ρsin(θ + $\frac {π}{4}$) + 1 = 0, and the origin O of the rectangular coordinate system xOy coincides with the pole, and the positive semi-axis of the x-axis coincides with the polar axis. (1) Find the standard equation and a parametric equation of circle C; (2) Let P(x, y) be any point on circle C, find the maximum value of xy.
|
\frac {3}{2} + \sqrt {2}
| 0 |
19,994 |
A wire of length $80$cm is randomly cut into three segments. The probability that each segment is no less than $20$cm is $\_\_\_\_\_\_\_.$
|
\frac{1}{16}
| 28.125 |
19,995 |
In $\vartriangle ABC, AB=AC=14 \sqrt2 , D$ is the midpoint of $CA$ and $E$ is the midpoint of $BD$ . Suppose $\vartriangle CDE$ is similar to $\vartriangle ABC$ . Find the length of $BD$ .
|
14
| 17.1875 |
19,996 |
When 2007 bars of soap are packed into \( N \) boxes, where \( N \) is a positive integer, there is a remainder of 5. How many possible values of \( N \) are there?
|
14
| 47.65625 |
19,997 |
Each face of a regular tetrahedron is labeled with one of the numbers $1, 2, 3, 4$. Four identical regular tetrahedrons are simultaneously rolled onto a table. What is the probability that the product of the four numbers on the faces touching the table is divisible by 4?
|
$\frac{13}{16}$
| 0 |
19,998 |
Given the hyperbola $\frac{x^{2}}{4}-y^{2}=1$ with its right focus $F$, and points $P_{1}$, $P_{2}$, …, $P_{n}$ on its right upper part where $2\leqslant x\leqslant 2 \sqrt {5}, y\geqslant 0$. The length of the line segment $|P_{k}F|$ is $a_{k}$, $(k=1,2,3,…,n)$. If the sequence $\{a_{n}\}$ is an arithmetic sequence with the common difference $d\in( \frac{1}{5}, \frac{ {\sqrt {5}}}{5})$, find the maximum value of $n$.
|
14
| 19.53125 |
19,999 |
If an integer \( n \) that is greater than 8 is a solution to the equation \( x^2 - ax + b = 0 \), and the representation of \( a \) in base \( n \) is 18, then the representation of \( b \) in base \( n \) is:
|
80
| 82.8125 |
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