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40.3k
| problem
stringlengths 10
5.15k
| ground_truth
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float64 0
100
|
|---|---|---|---|
20,100 |
Given that $a$ and $b$ are positive numbers, and $a+b=1$, find the value of $a$ when $a=$____, such that the minimum value of the algebraic expression $\frac{{2{a^2}+1}}{{ab}}-2$ is ____.
|
2\sqrt{3}
| 32.8125 |
20,101 |
Let complex numbers $\omega_{1}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $\omega_{2}=\cos\frac{\pi}{12}+\sin\frac{\pi}{12}i$. If $z=\omega_{1}\cdot\omega_{2}$, find the imaginary part of the complex number $z$.
|
\frac { \sqrt {2}}{2}
| 0 |
20,102 |
Given that the sum of the first n terms of a geometric sequence {a_n} (where all terms are real numbers) is S_n, if S_10=10 and S_30=70, determine the value of S_40.
|
150
| 76.5625 |
20,103 |
Given the line $x - 3y + m = 0$ ($m \neq 0$) and the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), let points $A$ and $B$ be the intersections of the line with the two asymptotes of the hyperbola. If point $P(m, 0)$ satisfies $|PA| = |PB|$, find the eccentricity of the hyperbola.
|
\frac{\sqrt{5}}{2}
| 33.59375 |
20,104 |
Given the coordinates of the three vertices of $\triangle ABC$ are $A(0,1)$, $B(1,0)$, $C(0,-2)$, and $O$ is the origin, if a moving point $M$ satisfies $|\overrightarrow{CM}|=1$, calculate the maximum value of $|\overrightarrow{OA}+ \overrightarrow{OB}+ \overrightarrow{OM}|$.
|
\sqrt{2}+1
| 21.09375 |
20,105 |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than or equal to 10 and at least one die shows a number greater than 3?
|
\frac{2}{3}
| 0 |
20,106 |
Given the function $f(x)=(ax^{2}+bx+c)e^{x}$ $(a > 0)$, the derivative $y=f′(x)$ has two zeros at $-3$ and $0$.
(Ⅰ) Determine the intervals of monotonicity for $f(x)$.
(Ⅱ) If the minimum value of $f(x)$ is $-1$, find the maximum value of $f(x)$.
|
\dfrac {5}{e^{3}}
| 0 |
20,107 |
There are 4 male and 2 female volunteers, a total of 6 volunteers, and 2 elderly people standing in a row for a group photo. The photographer requests that the two elderly people stand next to each other in the very center, with the two female volunteers standing immediately to the left and right of the elderly people. The number of different ways they can stand is:
|
96
| 53.90625 |
20,108 |
Given a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, and $\frac{a}{b}=\frac{1+\cos A}{\cos C}$.
(1) Find angle $A$;
(2) If $a=1$, find the maximum area $S$ of $\triangle ABC$.
|
\frac{1}{4}
| 21.875 |
20,109 |
The function $f$ is defined on positive integers as follows:
\[f(n) = \left\{
\begin{array}{cl}
n + 15 & \text{if $n < 15$}, \\
f(n - 7) & \text{if $n \ge 15$}.
\end{array}
\right.\]
Find the maximum value of the function.
|
29
| 87.5 |
20,110 |
There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are from Spain. The gold, silver, and bronze medals are awarded to the top three finishers. In how many ways can the medals be awarded if at most two Spaniards get medals?
|
696
| 54.6875 |
20,111 |
Given $A=\{4, a^2\}$, $B=\{a-6, a+1, 9\}$, if $A \cap B = \{9\}$, find the value of $a$.
|
-3
| 85.9375 |
20,112 |
If for any real number $x$, the equation $(1-2x)^{10} = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots + a_{10}x^{10}$ holds, then the value of $(a_0 + a_1) + (a_0 + a_2) + (a_0 + a_3) + \ldots + (a_0 + a_{10})$ is _____. (Answer with a number)
|
10
| 89.84375 |
20,113 |
Solve the following equations using appropriate methods:<br/>$(1)2x^{2}-3x+1=0$;<br/>$(2)\left(y-2\right)^{2}=\left(2y+3\right)^{2}$.
|
-\frac{1}{3}
| 1.5625 |
20,114 |
Given that $α$ is an angle in the first quadrant, it satisfies $\sin α - \cos α = \frac{\sqrt{10}}{5}$. Find $\cos 2α$.
|
- \frac{4}{5}
| 55.46875 |
20,115 |
Find the largest three-digit integer that is divisible by each of its digits and the sum of the digits is divisible by 6.
|
936
| 0.78125 |
20,116 |
Given that $x$ is a multiple of $12600$, what is the greatest common divisor of $g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5)$ and $x$?
|
840
| 28.125 |
20,117 |
In a class of 50 students, it is decided to use systematic sampling to select 10 students. The 50 students are randomly assigned numbers from 1 to 50 and divided into groups, with the first group being 1-5, the second group 6-10, ..., and the tenth group 45-50. If a student with the number 12 is selected from the third group, then the student selected from the eighth group will have the number \_\_\_\_\_\_.
|
37
| 92.1875 |
20,118 |
What is the smallest positive integer $n$ such that $\frac{n}{n+50}$ is equal to a terminating decimal?
|
14
| 22.65625 |
20,119 |
Triangle $XYZ$ is a right, isosceles triangle. Angle $X$ measures 45 degrees. What is the number of degrees in the measure of angle $Y$?
|
45
| 79.6875 |
20,120 |
Given a geometric sequence {a_n} satisfies a_1 = 3, and a_1 + a_3 + a_5 = 21, find the value of a_3 + a_5 + a_7.
|
42
| 90.625 |
20,121 |
Ms. Johnson awards bonus points to students in her class whose test scores are above the median. The class consists of 81 students. What is the maximum number of students who could receive bonus points?
|
40
| 58.59375 |
20,122 |
In triangle $XYZ$, $XY = 4$, $XZ = 3$, and $YZ = 5$. The medians $XM$, $YN$, and $ZL$ of triangle $XYZ$ intersect at the centroid $G$. Let the projections of $G$ onto $YZ$, $XZ$, and $XY$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.
|
\frac{47}{15}
| 39.84375 |
20,123 |
If the function $f(x)$ satisfies $f(3x) = f\left(3x - \frac{3}{2}\right)$ for all $x \in \mathbb{R}$, then the smallest positive period of $f(x)$ is \_\_\_\_\_\_.
|
\frac{1}{2}
| 33.59375 |
20,124 |
For a four-digit natural number $A$, if the digit in the thousands place is $5$ more than the digit in the tens place, and the digit in the hundreds place is $3$ more than the digit in the units place, then $A$ is called a "five-three number." For example, for the four-digit number $6714$, since $6-1=5$ and $7-4=3$, therefore $6714$ is a "five-three number"; for the four-digit number $8821$, since $8-2\neq 5$, therefore $8421$ is not a "five-three number". The difference between the largest and smallest "five-three numbers" is ______; for a "five-three number" $A$ with the digit in the thousands place being $a$, the digit in the hundreds place being $b$, the digit in the tens place being $c$, and the digit in the units place being $d$, let $M(A)=a+c+2(b+d)$ and $N(A)=b-3$. If $\frac{M(A)}{N(A)}$ is divisible by $5$, then the value of $A$ that satisfies the condition is ______.
|
5401
| 0 |
20,125 |
Three faces of a right rectangular prism have areas of 54, 56, and 60 square units. Calculate the volume of the prism in cubic units and round it to the nearest whole number.
|
426
| 4.6875 |
20,126 |
By what common fraction does $0.\overline{06}$ exceed $0.06$?
|
\frac{1}{1650}
| 96.875 |
20,127 |
How many distinct four-digit numbers are divisible by 5 and end with 45?
|
90
| 90.625 |
20,128 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m}=(\cos A,\cos B)$ and $\overrightarrow{n}=(a,2c-b)$, and $\overrightarrow{m} \parallel \overrightarrow{n}$.
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) Find the maximum value of $\sin B+\sin C$ and determine the shape of $\triangle ABC$ at this value.
|
\sqrt {3}
| 0 |
20,129 |
A square has sides of length 8, and a circle centered at one of its vertices has a radius of 12. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$.
|
64 + 108\pi
| 75.78125 |
20,130 |
$(1)$ Solve the inequality: $3A_{x}^{3}≤2A_{x+1}^{2}+6A_{x}^{2}$;<br/>$(2)$ Find the value of $C_{n}^{5-n}+C_{n+1}^{9-n}$;<br/>$(3)$ Given $\frac{1}{C_{5}^{m}}-\frac{1}{C_{6}^{m}}=\frac{7}{10C_{7}^{m}}$, find $C_{8}^{m}$.
|
28
| 36.71875 |
20,131 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $P$ is a point on the ellipse such that $PF\_2$ is perpendicular to the $x$-axis. If $|F\_1F\_2| = 2|PF\_2|$, calculate the eccentricity of the ellipse.
|
\frac{\sqrt{5} - 1}{2}
| 81.25 |
20,132 |
If two sides of a triangle are 8 and 15 units, and the angle between them is 30 degrees, what is the length of the third side?
|
\sqrt{289 - 120\sqrt{3}}
| 0.78125 |
20,133 |
Andrew flips a fair coin $5$ times, and counts the number of heads that appear. Beth flips a fair coin $6$ times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth.
|
0.5
| 83.59375 |
20,134 |
How many distinct triangles can be drawn using three of the dots below as vertices, where the dots are arranged in a grid of 2 rows and 4 columns?
|
48
| 15.625 |
20,135 |
Let ${ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} $ , where there are $ b $ a's in total. That is $ a\uparrow\uparrow b $ is given by the recurrence \[ a\uparrow\uparrow b = \begin{cases} a & b=1 a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} \] What is the remainder of $ 3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) $ when divided by $ 60 $ ?
|
27
| 79.6875 |
20,136 |
Given $y=f(x)$ is a quadratic function, and $f(0)=-5$, $f(-1)=-4$, $f(2)=-5$,
(1) Find the analytical expression of this quadratic function.
(2) Find the maximum and minimum values of the function $f(x)$ when $x \in [0,5]$.
|
- \frac {16}{3}
| 39.0625 |
20,137 |
Given an isosceles trapezoid with \(AB = 24\) units, \(CD = 10\) units, and legs \(AD\) and \(BC\) each measuring \(13\) units. Find the length of diagonal \(AC\).
|
13
| 24.21875 |
20,138 |
Ted is solving the equation by completing the square: $$64x^2+48x-36 = 0.$$ He aims to write the equation in a form: $$(ax + b)^2 = c,$$ with \(a\), \(b\), and \(c\) as integers and \(a > 0\). Determine the value of \(a + b + c\).
|
56
| 84.375 |
20,139 |
At a math competition, a team of $8$ students has $2$ hours to solve $30$ problems. If each problem needs to be solved by $2$ students, on average how many minutes can a student spend on a problem?
|
16
| 7.03125 |
20,140 |
An iterative process is used to find an average of the numbers -1, 0, 5, 10, and 15. Arrange the five numbers in a certain sequence. Find the average of the first two numbers, then the average of the result with the third number, and so on until the fifth number is included. What is the difference between the largest and smallest possible final results of this iterative average process?
|
8.875
| 1.5625 |
20,141 |
Three boys played a "Word" game in which they each wrote down ten words. For each word a boy wrote, he scored three points if neither of the other boys had the same word; he scored one point if only one of the other boys had the same word. No points were awarded for words which all three boys had. When they added up their scores, they found that they each had different scores. Sam had the smallest score (19 points), and James scored the most. How many points did James score?
|
25
| 17.1875 |
20,142 |
Given the function $f(x)=\sin(\omega x+\varphi)$, which is monotonically increasing on the interval ($\frac{\pi}{6}$,$\frac{{2\pi}}{3}$), and the lines $x=\frac{\pi}{6}$ and $x=\frac{{2\pi}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, find $f(-\frac{{5\pi}}{{12}})$.
|
\frac{\sqrt{3}}{2}
| 19.53125 |
20,143 |
The volume of the solid formed by rotating an isosceles right triangle with legs of length 1 around its hypotenuse is __________.
|
\frac{\sqrt{2}}{6}\pi
| 3.90625 |
20,144 |
Given the function $f(x) = x^2 - 2\cos{\theta}x + 1$, where $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$.
(1) When $\theta = \frac{\pi}{3}$, find the maximum and minimum values of $f(x)$.
(2) If $f(x)$ is a monotonous function on $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$ and $\theta \in [0, 2\pi)$, find the range of $\theta$.
(3) If $\sin{\alpha}$ and $\cos{\alpha}$ are the two real roots of the equation $f(x) = \frac{1}{4} + \cos{\theta}$, find the value of $\frac{\tan^2{\alpha} + 1}{\tan{\alpha}}$.
|
\frac{16 + 4\sqrt{11}}{5}
| 12.5 |
20,145 |
Given 5 people stand in a row, and there is exactly 1 person between person A and person B, determine the total number of possible arrangements.
|
36
| 19.53125 |
20,146 |
$ABCDEFGH$ is a cube where each side has length $a$. Find $\sin \angle GAC$.
|
\frac{\sqrt{3}}{3}
| 64.84375 |
20,147 |
Given $\cos \left(40^{\circ}-\theta \right)+\cos \left(40^{\circ}+\theta \right)+\cos \left(80^{\circ}-\theta \right)=0$, calculate the value of $\tan \theta$.
|
-\sqrt{3}
| 23.4375 |
20,148 |
A circle has a radius of 3 units. There are many line segments of length 4 units that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
A) $3\pi$
B) $5\pi$
C) $4\pi$
D) $7\pi$
E) $6\pi$
|
4\pi
| 37.5 |
20,149 |
The $5G$ technology is very important to society and the country. From a strategic perspective, the industry generally defines it as the fourth industrial revolution after the steam engine revolution, the electrical revolution, and the computer revolution. A certain technology group produces two core components of $5G$ communication base stations, $A$ and $B$. The table below shows the data of the research and development investment $x$ (in billion yuan) and the revenue $y$ (in billion yuan) of the $A$ component in recent years by the technology group:
| Research Investment $x$ (billion yuan) | 1 | 2 | 3 | 4 | 5 |
|---------------------------------------|---|---|---|---|---|
| Revenue $y$ (billion yuan) | 3 | 7 | 9 | 10| 11|
$(1)$ Use the sample correlation coefficient $r$ to determine if a linear regression model can be used to fit the relationship between $y$ and $x$ (when $|r| \in [0.75, 1]$, it can be considered that the two variables have a strong linear correlation);
$(2)$ Find the empirical regression equation of $y$ with respect to $x$. If the revenue from producing the $A$ component is not less than 15 billion yuan, estimate how much research and development funding is needed at least? (Round to $0.001$ billion yuan)
Given: Sample correlation coefficient $r = \frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}^{2}}\sqrt{\sum_{i=1}^{n}{({y}_{i}-\overline{y})}^{2}}}$, slope of the regression line $\hat{b} = \frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}^{2}}$, intercept $\hat{a} = \overline{y} - \hat{b}\overline{x}$.
|
6.684
| 59.375 |
20,150 |
A bus arrives randomly sometime between 1:00 and 2:30, waits for 20 minutes, and then leaves. If Laura also arrives randomly between 1:00 and 2:30, what is the probability that the bus will be there when Laura arrives?
|
\frac{16}{81}
| 2.34375 |
20,151 |
A school wants to understand the psychological state of its senior high school students regarding their studies. They decide to use a systematic sampling method to select 40 students out of 800 for a certain test. The students are randomly numbered from 1 to 800. After grouping, the first group is selected through simple random sampling, and the number drawn is 18. Among the 40 selected students, those with numbers in the interval \[1, 200\] take test paper A, those in the interval \[201, 560\] take test paper B, and the rest take test paper C. The number of students taking test paper C is \_\_\_\_\_\_.
|
12
| 81.25 |
20,152 |
Let the function y=f(x) have the domain D. If for any x1, x2 ∈ D, when x1+x2=2a, it always holds that f(x1)+f(x2)=2b, then the point (a,b) is called the symmetry center of the graph of the function y=f(x). Investigate a symmetry center of the function f(x)=x+sinπx-3, and find the value of f(1/2016)+f(2/2016)+f(3/2016)+...+f(4030/2016)+f(4031/2016).
|
-8062
| 58.59375 |
20,153 |
How many distinct arrangements of the letters in the word "balloon" are there?
|
1260
| 39.84375 |
20,154 |
A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along a side of the square to that vertex. Determine the probability that the bug returns to its starting vertex on its eighth move and express this probability in lowest terms as $m/n$. Find $m+n$.
|
2734
| 0 |
20,155 |
If $(x-1)(x+3)(x-4)(x-8)+m$ is a perfect square, find the value of $m$.
|
196
| 47.65625 |
20,156 |
Given that $\frac{\cos 2\alpha}{\sqrt{2}\sin\left(\alpha+\frac{\pi}{4}\right)}=\frac{\sqrt{5}}{2}$, find the value of $\tan\alpha+\frac{1}{\tan\alpha}$.
|
-8
| 67.96875 |
20,157 |
When a student used a calculator to find the average of 30 data points, they mistakenly entered one of the data points, 105, as 15. Find the difference between the calculated average and the actual average.
|
-3
| 14.0625 |
20,158 |
The measures of angles $X$ and $Y$ are both positive, integer numbers of degrees. The measure of angle $X$ is a multiple of the measure of angle $Y$, and angles $X$ and $Y$ are supplementary angles. How many measures are possible for angle $X$?
|
17
| 32.03125 |
20,159 |
If $\sin \theta= \frac {3}{5}$ and $\frac {5\pi}{2} < \theta < 3\pi$, then $\sin \frac {\theta}{2}=$ ______.
|
-\frac {3 \sqrt {10}}{10}
| 0 |
20,160 |
Let the sequence $\{a_n\}$ satisfy that the sum of the first $n$ terms $S_n$ fulfills $S_n + a_1 = 2a_n$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence. Find the value of $a_1 + a_5$.
|
34
| 85.9375 |
20,161 |
A $6 \times 9$ rectangle can be rolled to form two different cylinders. Calculate the ratio of the larger volume to the smaller volume. Express your answer as a common fraction.
|
\frac{3}{2}
| 61.71875 |
20,162 |
Consider the following scenario where three stores offer different discounts on a television priced at $149.99$:
\begin{tabular}{|l|l|}
\hline
\textbf{Store} & \textbf{Sale Price for Television $Y$} \\
\hline
Value Market & $\$10$~off the list price~$\$149.99$ \\
Tech Bargains & $30\%$~off the list price~$\$149.99$ \\
The Gadget Hub & $20\%$~off the list price~$\$149.99$ \\
\hline
\end{tabular}
How much cheaper, in cents, is the cheapest store's price compared to the most expensive?
|
3500
| 89.0625 |
20,163 |
Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$ . Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$
|
$\pi/2$
| 0 |
20,164 |
The number $18!=6,402,373,705,728,000$ has many positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
A) $\frac{1}{16}$
B) $\frac{1}{18}$
C) $\frac{1}{15}$
D) $\frac{1}{20}$
E) $\frac{1}{21}$
|
\frac{1}{16}
| 16.40625 |
20,165 |
Given $\sin x_{1}=\sin x_{2}=\frac{1}{3}$ and $0 \lt x_{1} \lt x_{2} \lt 2\pi$, find $\cos |\overrightarrow{a}|$.
|
-\frac{7}{9}
| 10.15625 |
20,166 |
Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:
$\bullet$ Carolyn always has the first turn.
$\bullet$ Carolyn and Paul alternate turns.
$\bullet$ On each of her turns, Carolyn must remove one number from the list such that this number has at least one positive divisor other than itself remaining in the list.
$\bullet$ On each of his turns, Paul must remove from the list all of the positive divisors of the number that Carolyn has just removed, and he can optionally choose one additional number that is a multiple of any divisor he is removing.
$\bullet$ If Carolyn cannot remove any more numbers, then Paul removes the rest of the numbers.
For example, if $n=8,$ a possible sequence of moves could be considered.
Suppose that $n=8$ and Carolyn removes the integer $4$ on her first turn. Determine the sum of the numbers that Carolyn removes.
|
12
| 15.625 |
20,167 |
Add $5_7 + 16_7.$ Express your answer in base $7.$
|
24_7
| 99.21875 |
20,168 |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that fits inside the cylinder?
|
2\sqrt{61}
| 85.9375 |
20,169 |
Let $a, b \in \mathbb{R}^+$, and $a+b=1$. Find the minimum value of $\sqrt{a^2+1} + \sqrt{b^2+4}$.
|
\sqrt{10}
| 69.53125 |
20,170 |
Find the number of solutions to the equation
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,150 \pi)$.
|
75
| 17.1875 |
20,171 |
In a gymnastics competition, the position where Qiqi stands is the 6th from the front, the 12th from the back, the 15th from the left, and the 11th from the right. If the number of people in each row is the same and the number of people in each column is also the same, how many people are there in total participating in the gymnastics competition?
|
425
| 64.84375 |
20,172 |
A triangle has side lengths of $x,75,100$ where $x<75$ and altitudes of lengths $y,28,60$ where $y<28$ . What is the value of $x+y$ ?
*2019 CCA Math Bonanza Team Round #2*
|
56
| 10.15625 |
20,173 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sqrt{3}a=2b\sin A$.
$(1)$ Find angle $B$;
$(2)$ If $b=\sqrt{7}$, $c=3$, and $D$ is the midpoint of side $AC$, find $BD$.
|
\frac{\sqrt{19}}{2}
| 31.25 |
20,174 |
There are 52 students in a class. Now, using the systematic sampling method, a sample of size 4 is drawn. It is known that the seat numbers in the sample are 6, X, 30, and 42. What should be the seat number X?
|
18
| 21.875 |
20,175 |
Several cuboids with edge lengths of $2, 7, 13$ are arranged in the same direction to form a cube with an edge length of 2002. How many small cuboids does a diagonal of the cube pass through?
|
1210
| 14.84375 |
20,176 |
A rectangle has its length increased by $30\%$ and its width increased by $15\%$. Determine the percentage increase in the area of the rectangle.
|
49.5\%
| 78.125 |
20,177 |
Given that $α$ is an angle in the third quadrant, $f(α)= \frac {\sin (π-α)\cdot \cos (2π-α)\cdot \tan (-α-π)}{\tan (-α )\cdot \sin (-π -α)}$.
(1) Simplify $f(α)$;
(2) If $\cos (α- \frac {3}{2}π)= \frac {1}{5}$, find the value of $f(α)$;
(3) If $α=-1860^{\circ}$, find the value of $f(α)$.
|
\frac{1}{2}
| 61.71875 |
20,178 |
A parabola, given by the equation $y^{2}=2px (p > 0)$, has a focus that lies on the line $l$. This line intersects the parabola at two points, $A$ and $B$. A circle with the chord $AB$ as its diameter has the equation $(x-3)^{2}+(y-2)^{2}=16$. Find the value of $p$.
|
p = 2
| 24.21875 |
20,179 |
On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$. Meaning if the length of the object was originally $x$, then after the first day, it is $x + \frac{1}{3} x.$ On the second day, he increased the object's new length from the previous day by $\frac{1}{4}$; on the third day by $\frac{1}{5}$, and so on, with each day increasing the object's length by the next increment in the series $\frac{1}{n+3}$ for the $n^{\text{th}}$ day. If by the $n^{\text{th}}$ day Barry wants the object's length to be exactly 50 times its original length, what is the value of $n$?
|
147
| 59.375 |
20,180 |
For the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ to be a decreasing function on the interval $(0, +\infty)$, then $m = \boxed{\text{answer}}$.
|
-1
| 16.40625 |
20,181 |
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$ , with $i = 1,2, ..., 2015$ .
|
2016
| 0.78125 |
20,182 |
Given a triangle \( \triangle ABC \) with \(\angle B = 90^\circ\). The incircle touches sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Line \(AD\) intersects the incircle at another point \(P\), and \(PF \perp PC\). Find the ratio of the side lengths of \(\triangle ABC\).
|
3:4:5
| 30.46875 |
20,183 |
Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected to serve. Find the number of ways to select exactly one person to serve for both days.
|
60
| 40.625 |
20,184 |
Simplify first, then find the value of the algebraic expression $\frac{a}{{{a^2}-2a+1}}÷({1+\frac{1}{{a-1}}})$, where $a=\sqrt{2}$.
|
\sqrt{2}+1
| 82.8125 |
20,185 |
Given the rectangular coordinate system xOy, establish a polar coordinate system with O as the pole and the non-negative semi-axis of the x-axis as the polar axis. The line l passes through point P(-1, 2) with an inclination angle of $\frac{2π}{3}$, and the polar coordinate equation of circle C is $ρ = 2\cos(θ + \frac{π}{3})$.
(I) Find the general equation of circle C and the parametric equation of line l;
(II) Suppose line l intersects circle C at points M and N. Find the value of |PM|•|PN|.
|
6 + 2\sqrt{3}
| 26.5625 |
20,186 |
Given a hexagon \( A B C D E F \) with an area of 60 that is inscribed in a circle \( \odot O \), where \( AB = BC, CD = DE, \) and \( EF = AF \). What is the area of \( \triangle B D F \)?
|
30
| 90.625 |
20,187 |
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$?
|
16.67\%
| 92.1875 |
20,188 |
In a pot, there are 6 sesame-filled dumplings, 5 peanut-filled dumplings, and 4 red bean paste-filled dumplings. These three types of dumplings look exactly the same from the outside. If 4 dumplings are randomly selected from the pot, the probability that at least one dumpling of each type is selected is.
|
\dfrac{48}{91}
| 17.1875 |
20,189 |
In a multiplication error involving two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the three-digit number $a$. The erroneous product obtained was $396$. Determine the correct value of the product $ab$.
|
693
| 2.34375 |
20,190 |
The number 119 has the following properties:
(a) Division by 2 leaves a remainder of 1;
(b) Division by 3 leaves a remainder of 2;
(c) Division by 4 leaves a remainder of 3;
(d) Division by 5 leaves a remainder of 4;
(e) Division by 6 leaves a remainder of 5.
How many positive integers less than 2007 satisfy these properties?
|
33
| 96.09375 |
20,191 |
Given rectangle ABCD where E is the midpoint of diagonal BD, point E is connected to point F on segment DA such that DF = 1/4 DA. Find the ratio of the area of triangle DFE to the area of quadrilateral ABEF.
|
\frac{1}{7}
| 52.34375 |
20,192 |
What is the probability that Hannah gets fewer than 4 heads if she flips 12 coins?
|
\frac{299}{4096}
| 0.78125 |
20,193 |
In Mr. Jacob's class, $12$ of the $20$ students received a 'B' on the latest exam. If the same proportion of students received a 'B' in Mrs. Cecilia's latest exam, and Mrs. Cecilia originally had $30$ students, but $6$ were absent during the exam, how many students present for Mrs. Cecilia’s exam received a 'B'?
|
14
| 97.65625 |
20,194 |
Determine all four-digit numbers $\overline{abcd}$ which are perfect squares and for which the equality holds: $\overline{ab}=3 \cdot \overline{cd} + 1$ .
|
2809
| 95.3125 |
20,195 |
Parallelogram $ABCD$ has vertices $A(3,4)$, $B(-2,1)$, $C(-5,-2)$, and $D(0,1)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is left of the $y$-axis? Express your answer as a common fraction.
|
\frac{1}{2}
| 50.78125 |
20,196 |
Assume a deck of 27 cards where each card features one of three symbols (star, circle, or square), each symbol painted in one of three colors (red, yellow, or blue), and each color applied in one of three intensities (light, medium, or dark). Each symbol-color-intensity combination is unique across the cards. A set of three cards is defined as complementary if:
i. Each card has a different symbol or all have the same symbol.
ii. Each card has a different color or all have the same color.
iii. Each card has a different intensity or all have the same intensity.
Determine the number of different complementary three-card sets available.
|
117
| 93.75 |
20,197 |
Given two plane vectors, the angle between them is $120^\circ$, and $a=1$, $|b|=2$. If the plane vector $m$ satisfies $m\cdot a=m\cdot b=1$, then $|m|=$ ______.
|
\frac{ \sqrt{21}}{3}
| 71.09375 |
20,198 |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\]
|
1001
| 61.71875 |
20,199 |
Let $f\left(x\right)=ax^{2}-1$ and $g\left(x\right)=\ln \left(ax\right)$ have an "$S$ point", then find the value of $a$.
|
\frac{2}{e}
| 69.53125 |
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