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|---|---|---|---|
19,800 |
In $\triangle DEF$ with sides $5$, $12$, and $13$, a circle with center $Q$ and radius $2$ rolls around inside the triangle, always keeping tangency to at least one side of the triangle. When $Q$ first returns to its original position, through what distance has $Q$ traveled?
|
18
| 43.75 |
19,801 |
Simplify $(2^8 + 4^5)(2^3 - (-2)^2)^{11}$.
|
5368709120
| 31.25 |
19,802 |
The sides opposite to the internal angles $A$, $B$, $C$ of $\triangle ABC$ are $a$, $b$, $c$ respectively. Given $\frac{a-b+c}{c}= \frac{b}{a+b-c}$ and $a=2$, the maximum area of $\triangle ABC$ is __________.
|
\sqrt{3}
| 40.625 |
19,803 |
Find four positive integers that are divisors of each number in the list $$45, 90, -15, 135, 180.$$ Calculate the sum of these four integers.
|
24
| 91.40625 |
19,804 |
Cagney can frost a cupcake every 15 seconds, Lacey can frost a cupcake every 25 seconds, and Hardy can frost a cupcake every 50 seconds. Calculate the number of cupcakes that Cagney, Lacey, and Hardy can frost together in 6 minutes.
|
45
| 53.125 |
19,805 |
Given two lines $l_{1}: ax-y+a=0$ and $l_{2}: (2a-3)x+ay-a=0$ are parallel, determine the value of $a$.
|
-3
| 19.53125 |
19,806 |
Let $\omega \in \mathbb{C}$ , and $\left | \omega \right | = 1$ . Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$ .
|
108
| 11.71875 |
19,807 |
Given a positive integer \( n \geqslant 2 \), positive real numbers \( a_1, a_2, \ldots, a_n \), and non-negative real numbers \( b_1, b_2, \ldots, b_n \), which satisfy the following conditions:
(a) \( a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n = n \);
(b) \( a_1 a_2 \cdots a_n + b_1 b_2 \cdots b_n = \frac{1}{2} \).
Find the maximum value of \( a_1 a_2 \cdots a_n \left( \frac{b_1}{a_1} + \frac{b_2}{a_2} + \cdots + \frac{b_n}{a_n} \right) \).
|
\frac{1}{2}
| 53.90625 |
19,808 |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there?
|
8748
| 82.03125 |
19,809 |
A car license plate contains three letters and three digits, for example, A123BE. The allowable letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Katya considers a plate number lucky if the second letter is a consonant, the first digit is odd, and the third digit is even (there are no restrictions on the other characters). How many license plates does Katya consider lucky?
|
288000
| 27.34375 |
19,810 |
Find the sum of the distances from one vertex of a rectangle with length $3$ and width $4$ to the centers of the opposite sides.
|
\sqrt{13} + 2
| 0 |
19,811 |
In a table tennis team of 5 players, which includes 2 veteran players and 3 new players, we need to select 3 players to be ranked as No. 1, No. 2, and No. 3 for a team competition. The selection must ensure that among the 3 chosen players, there is at least 1 veteran player and among players No. 1 and No. 2, there is at least 1 new player. There are $\boxed{\text{number}}$ ways to arrange this (answer with a number).
|
48
| 53.90625 |
19,812 |
In triangle $PQR$, $\cos(2P-Q) + \sin(P+Q) = 2$ and $PQ = 5$. What is $QR$?
|
5\sqrt{3}
| 3.125 |
19,813 |
In a new shooting competition, ten clay targets are set up in four hanging columns with four targets in column $A$, three in column $B$, two in column $C$, and one in column $D$. A shooter must continue following the sequence:
1) The shooter selects one of the columns.
2) The shooter must then hit the lowest remaining target in that chosen column.
What are the total possible sequences in which the shooter can break all the targets, assuming they adhere to the above rules?
|
12600
| 96.875 |
19,814 |
Given the equation of a circle $(x-1)^{2}+(y-1)^{2}=9$, point $P(2,2)$ lies inside the circle. The longest and shortest chords passing through point $P$ are $AC$ and $BD$ respectively. Determine the product $AC \cdot BD$.
|
12\sqrt{7}
| 100 |
19,815 |
Find the square root of $\dfrac{10!}{210}$.
|
72\sqrt{5}
| 0.78125 |
19,816 |
In triangle $\triangle ABC$, the length of the side opposite angle $A$ is equal to 2, the vector $\overrightarrow {m} = (2, 2\cos^2 \frac {B+C}{2}-1)$, and the vector $\overrightarrow {n} = (\sin \frac {A}{2}, -1)$.
(1) Find the size of angle $A$ when the dot product $\overrightarrow {m} \cdot \overrightarrow {n}$ reaches its maximum value;
(2) Under the condition of (1), find the maximum area of $\triangle ABC$.
|
\sqrt{3}
| 89.84375 |
19,817 |
A square sheet of paper that measures $18$ cm on a side has corners labeled $A$ , $B$ , $C$ , and $D$ in clockwise order. Point $B$ is folded over to a point $E$ on $\overline{AD}$ with $DE=6$ cm and the paper is creased. When the paper is unfolded, the crease intersects side $\overline{AB}$ at $F$ . Find the number of centimeters in $FB$ .
|
13
| 48.4375 |
19,818 |
Oil, as an important strategic reserve commodity, has always been of concern to countries. According to reports from relevant departments, it is estimated that the demand for oil in China in 2022 will be 735,000,000 tons. Express 735,000,000 in scientific notation as ____.
|
7.35 \times 10^{8}
| 0 |
19,819 |
Moe has a new, larger rectangular lawn measuring 120 feet by 180 feet. He uses a mower with a swath width of 30 inches. However, he overlaps each cut by 6 inches to ensure no grass is missed. Moe walks at a rate of 6000 feet per hour while pushing the mower. What is the closest estimate of the number of hours it will take Moe to mow the lawn?
|
1.8
| 35.9375 |
19,820 |
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. Given that \( a^2 - (b - c)^2 = (2 - \sqrt{3})bc \) and \( \sin A \sin B = \cos^2 \frac{C}{2} \), and the length of the median \( AM \) from \( A \) to side \( BC \) is \( \sqrt{7} \):
1. Find the measures of angles \( A \) and \( B \);
2. Find the area of \( \triangle ABC \).
|
\sqrt{3}
| 21.875 |
19,821 |
Given: $$\frac { A_{ n }^{ 3 }}{6}=n$$ (where $n\in\mathbb{N}^{*}$), and $(2-x)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}$
Find the value of $a_{0}-a_{1}+a_{2}-\ldots+(-1)^{n}a_{n}$.
|
81
| 95.3125 |
19,822 |
The increasing sequence \( T = 2, 3, 5, 6, 7, 8, 10, 11, \ldots \) consists of all positive integers which are not perfect squares. What is the 2012th term of \( T \)?
|
2057
| 57.03125 |
19,823 |
For all composite integers $n$, what is the largest integer that always divides into the difference between $n^4 - n^2$?
|
12
| 58.59375 |
19,824 |
The opposite of $-23$ is ______; the reciprocal is ______; the absolute value is ______.
|
23
| 5.46875 |
19,825 |
Mrs. Carter's algebra class consists of 48 students. Due to a schedule conflict, 40 students took the Chapter 5 test, averaging 75%, while the remaining 8 students took it the following day, achieving an average score of 82%. What is the new overall mean score of the class on the Chapter 5 test? Express the answer as a percent.
|
76.17\%
| 34.375 |
19,826 |
A class has 54 students, and there are 4 tickets for the Shanghai World Expo to be distributed among the students using a systematic sampling method. If it is known that students with numbers 3, 29, and 42 have already been selected, then the student number of the fourth selected student is ▲.
|
16
| 1.5625 |
19,827 |
Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, calculate the value of $f\left(-\frac{{5π}}{{12}}\right)$.
|
\frac{\sqrt{3}}{2}
| 23.4375 |
19,828 |
A truck travels due west at $\frac{3}{4}$ mile per minute on a straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\frac{1}{2}\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $130$ miles due north of the truck. Determine the average time $\frac{1}{2}(t_1 + t_2)$ during which the truck is within the storm circle, where $t_1$ is the time the truck enters and $t_2$ is the time the truck exits the storm circle.
|
208
| 66.40625 |
19,829 |
Sixteen 6-inch wide square posts are evenly spaced with 4 feet between them to enclose a square field. What is the outer perimeter, in feet, of the fence?
|
56
| 54.6875 |
19,830 |
How many three-digit whole numbers have at least one 8 or at least one 9 as digits?
|
452
| 89.84375 |
19,831 |
Given that $\cos 78^{\circ}$ is approximately $\frac{1}{5}$, $\sin 66^{\circ}$ is approximately:
|
0.92
| 3.90625 |
19,832 |
Let \(\mathbf{v}\) be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of \(\|\mathbf{v}\|\).
|
10 - 2\sqrt{5}
| 78.90625 |
19,833 |
Four consecutive even integers have a product of 6720. What is the largest of these four integers?
|
14
| 16.40625 |
19,834 |
Find the sum of all distinct possible values of $x^2-4x+100$ , where $x$ is an integer between 1 and 100, inclusive.
*Proposed by Robin Park*
|
328053
| 64.0625 |
19,835 |
The probability that a randomly chosen divisor of $25!$ is odd.
|
\frac{1}{23}
| 75 |
19,836 |
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $a \neq b$, $c = \sqrt{3}$, and $\sqrt{3} \cos^2 A - \sqrt{3} \cos^2 B = \sin A \cos A - \sin B \cos B$.
(I) Find the measure of angle $C$;
(II) If $\sin A = \frac{4}{5}$, find the area of $\triangle ABC$.
|
\frac{24\sqrt{3} + 18}{25}
| 25.78125 |
19,837 |
Given two similar triangles $\triangle ABC\sim\triangle FGH$, where $BC = 24 \text{ cm}$ and $FG = 15 \text{ cm}$. If the length of $AC$ is $18 \text{ cm}$, find the length of $GH$. Express your answer as a decimal to the nearest tenth.
|
11.3
| 32.03125 |
19,838 |
Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.
a) Write down any five-digit palindromic number that is divisible by 5.
b) How many five-digit palindromic numbers are there that are divisible by 5?
|
100
| 86.71875 |
19,839 |
Given that $| \overrightarrow{a}|=| \overrightarrow{b}|=| \overrightarrow{c}|=1$, and $ \overrightarrow{a}+ \overrightarrow{b}+ \sqrt {3} \overrightarrow{c}=0$, find the value of $ \overrightarrow{a} \overrightarrow{b}+ \overrightarrow{b} \overrightarrow{c}+ \overrightarrow{c} \overrightarrow{a}$.
|
\dfrac {1}{2}- \sqrt {3}
| 0 |
19,840 |
Find the value of $c$ if the roots of the quadratic $9x^2 - 5x + c$ are $\frac{-5\pm i\sqrt{415}}{18}$.
|
\frac{110}{9}
| 98.4375 |
19,841 |
At the beginning of school year in one of the first grade classes: $i)$ every student had exatly $20$ acquaintances $ii)$ every two students knowing each other had exactly $13$ mutual acquaintances $iii)$ every two students not knowing each other had exactly $12$ mutual acquaintances
Find number of students in this class
|
31
| 58.59375 |
19,842 |
Michael read on average 30 pages each day for the first two days, then increased his average to 50 pages each day for the next four days, and finally read 70 pages on the last day. Calculate the total number of pages in the book.
|
330
| 46.875 |
19,843 |
If the complex number $Z=(1+ai)i$ is an "equal parts complex number", determine the value of the real number $a$.
|
-1
| 76.5625 |
19,844 |
In square ABCD, point E is on AB and point F is on BC such that AE=3EB and BF=FC. Find the ratio of the area of triangle DEF to the area of square ABCD.
|
\frac{5}{16}
| 53.125 |
19,845 |
Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to the two radii and the arc at three points. Determine the radius of the inscribed circle, expressed in simplest radical form.
|
6\sqrt{2} - 6
| 1.5625 |
19,846 |
If Fang Fang cuts a piece of paper into 9 pieces, then selects one of the resulting pieces to cut into 9 pieces again, and so on, determine the number of cuts made to achieve a total of 2009 paper pieces.
|
251
| 56.25 |
19,847 |
In triangle $ABC$, $\angle A$ is a right angle, and $\sin B$ is given as $\frac{3}{5}$. Calculate $\cos C$.
|
\frac{3}{5}
| 88.28125 |
19,848 |
Using the vertices of a cube as vertices, how many triangular pyramids can you form?
|
58
| 90.625 |
19,849 |
Given $\tan (\alpha-\beta)= \frac {1}{2}$, $\tan \beta=- \frac {1}{7}$, and $\alpha$, $\beta\in(0,\pi)$, find the value of $2\alpha-\beta$.
|
- \frac {3\pi}{4}
| 0 |
19,850 |
Given a complex number $Z = x + yi$ ($x, y \in \mathbb{R}$) such that $|Z - 4i| = |Z + 2|$, find the minimum value of $2^x + 4^y$.
|
4\sqrt{2}
| 82.03125 |
19,851 |
Sixty cards are placed into a box, each bearing a number 1 through 15, with each number represented on four cards. Four cards are drawn from the box at random without replacement. Let \(p\) be the probability that all four cards bear the same number. Let \(q\) be the probability that three of the cards bear a number \(a\) and the other bears a number \(b\) that is not equal to \(a\). What is the value of \(q/p\)?
|
224
| 93.75 |
19,852 |
If the square roots of a positive number are $x+1$ and $4-2x$, then the positive number is ______.
|
36
| 21.875 |
19,853 |
Given that the price of a gallon of gasoline initially increased by $30\%$ in January, then decreased by $10\%$ in February, increased by $15\%$ in March, and returned to its original value at the end of April, find the value of $x\%$ that represents the percentage decrease in April to the nearest integer.
|
26
| 49.21875 |
19,854 |
In $\triangle ABC$, the sides corresponding to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $b = a \sin C + c \cos A$,
(1) Find the value of $A + B$;
(2) If $c = \sqrt{2}$, find the maximum area of $\triangle ABC$.
|
\frac{1 + \sqrt{2}}{2}
| 11.71875 |
19,855 |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $50$. Determine how many values of $n$ make $q+r$ divisible by $13$.
A) 7000
B) 7200
C) 7400
D) 7600
|
7200
| 11.71875 |
19,856 |
We color some cells in $10000 \times 10000$ square, such that every $10 \times 10$ square and every $1 \times 100$ line have at least one coloring cell. What minimum number of cells we should color ?
|
10000
| 17.1875 |
19,857 |
Determine by how many times the number \((2014)^{2^{2014}} - 1\) is greater than the number written in the following form:
\[
\left(\left((2014)^{2^0} + 1\right) \cdot \left((2014)^{2^1} + 1\right) \cdot \left((2014)^{2^2} + 1\right) \ldots \cdot \left((2014)^{2^{2013}} + 1\right)\right) + 1.
\]
|
2013
| 52.34375 |
19,858 |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=t \\ y= \sqrt {2}+2t \end{cases}$ (where $t$ is the parameter), with point $O$ as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve $C$ is $\rho=4\cos\theta$.
(1) Find the Cartesian coordinate equation of curve $C$ and the general equation of line $l$;
(2) If the $x$-coordinates of all points on curve $C$ are shortened to $\frac {1}{2}$ of their original length, and then the resulting curve is translated 1 unit to the left, obtaining curve $C_1$, find the maximum distance from the points on curve $C_1$ to line $l$.
|
\frac {3 \sqrt {10}}{5}
| 0 |
19,859 |
Given that $\cos(\frac{\pi}{6} - \alpha) = \frac{\sqrt{3}}{3}$, find the value of $\sin(\frac{5\pi}{6} - 2\alpha)$.
|
-\frac{1}{3}
| 43.75 |
19,860 |
Grace has $\$4.80$ in U.S. coins. She has the same number of dimes and pennies. What is the greatest number of dimes she could have?
|
43
| 45.3125 |
19,861 |
A trapezoid $ABCD$ has bases $AD$ and $BC$. If $BC = 60$ units, and altitudes from $B$ and $C$ to line $AD$ divide it into segments of lengths $AP = 20$ units and $DQ = 19$ units, with the length of the altitude itself being $30$ units, what is the perimeter of trapezoid $ABCD$?
**A)** $\sqrt{1300} + 159$
**B)** $\sqrt{1261} + 159$
**C)** $\sqrt{1300} + \sqrt{1261} + 159$
**D)** $259$
**E)** $\sqrt{1300} + 60 + \sqrt{1261}$
|
\sqrt{1300} + \sqrt{1261} + 159
| 32.03125 |
19,862 |
A list of $3042$ positive integers has a unique mode, which occurs exactly $15$ times. Calculate the least number of distinct values that can occur in the list.
|
218
| 22.65625 |
19,863 |
For each positive integer $n$, let $f(n)$ denote the last digit of the sum $1+2+3+\ldots+n$.
For example: $f(1)=1$, $f(2)=3$ (the last digit of $1+2$), $f(5)=5$ (the last digit of $1+2+3+4+5$), $f(7)=8$ (the last digit of $1+2+3+4+5+6+7$)
Then, the value of $f(1)+f(2)+f(3)+\ldots+f(2005)$ is .
|
7015
| 49.21875 |
19,864 |
Given that $\cos \alpha + \sin \alpha = \frac{2}{3}$, find the value of $\frac{\sqrt{2}\sin(2\alpha - \frac{\pi}{4}) + 1}{1 + \tan \alpha}$.
|
-\frac{5}{9}
| 24.21875 |
19,865 |
Two cards are dealt at random from two standard decks of 104 cards mixed together. What is the probability that the first card drawn is an ace and the second card drawn is also an ace?
|
\dfrac{7}{1339}
| 73.4375 |
19,866 |
A factory must filter its emissions before discharging them. The relationship between the concentration of pollutants $p$ (in milligrams per liter) and the filtration time $t$ (in hours) during the filtration process is given by the equation $p(t) = p_0e^{-kt}$. Here, $e$ is the base of the natural logarithm, and $p_0$ is the initial pollutant concentration. After filtering for one hour, it is observed that the pollutant concentration has decreased by $\frac{1}{5}$.
(Ⅰ) Determine the function $p(t)$.
(Ⅱ) To ensure that the pollutant concentration does not exceed $\frac{1}{1000}$ of the initial value, for how many additional hours must the filtration process be continued? (Given that $\lg 2 \approx 0.3$)
|
30
| 57.8125 |
19,867 |
A student, Ellie, was supposed to calculate $x-y-z$, but due to a misunderstanding, she computed $x-(y+z)$ and obtained 18. The actual answer should have been 6. What is the value of $x-y$?
|
12
| 20.3125 |
19,868 |
Two books of different subjects are taken from three shelves, each having 10 Chinese books, 9 math books, and 8 English books. Calculate the total number of different ways to do this.
|
242
| 47.65625 |
19,869 |
Solve for $X$ and $Y$ such that
\[\frac{Yx + 8}{x^2 - 11x + 30} = \frac{X}{x - 5} + \frac{7}{x - 6}\].
Find $X+Y$.
|
-\frac{22}{3}
| 95.3125 |
19,870 |
Given a regular decagon $ABCDEFGHIJ$ with area $n$, calculate the area $m$ of pentagon $ACEGI$, which is defined using every second vertex of the decagon, and then determine the value of $\frac{m}{n}$.
|
\frac{1}{2}
| 71.875 |
19,871 |
Find the volume of the region in space defined by
\[ |x + 2y + z| + |x + 2y - z| \le 12 \]
and $x, y \geq 0$, $z \geq -2$.
|
72
| 80.46875 |
19,872 |
Consider the following diagram showing a rectangular grid of dots consisting of 3 rows and 4 columns. How many rectangles can be formed in this grid?
|
60
| 62.5 |
19,873 |
Given the function $f(x)=- \sqrt {3}\sin ^{2}x+\sin x\cos x$.
(1) Find the value of $f( \dfrac {25π}{6})$;
(2) Let $α∈(0,π)$, $f( \dfrac {α}{2})= \dfrac {1}{4}- \dfrac { \sqrt {3}}{2}$, find the value of $\sin α$.
|
\dfrac {1+3 \sqrt {5}}{8}
| 0 |
19,874 |
A sequence consists of $2020$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2020$ terms is $5410$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?
|
2200
| 1.5625 |
19,875 |
Let the circles $S_1$ and $S_2$ meet at the points $A$ and $B$. A line through $B$ meets $S_1$ at a point $D$ other than $B$ and meets $S_2$ at a point $C$ other than $B$. The tangent to $S_1$ through $D$ and the tangent to $S_2$ through $C$ meet at $E$. If $|AD|=15$, $|AC|=16$, $|AB|=10$, what is $|AE|$?
|
24
| 20.3125 |
19,876 |
Define the determinant operation $\begin{vmatrix} a_{1} & a_{2} \\ b_{1} & b_{2}\end{vmatrix} =a_{1}b_{2}-a_{2}b_{1}$, and consider the function $f(x)= \begin{vmatrix} \sqrt {3} & \sin x \\ 1 & \cos x\end{vmatrix}$. If the graph of this function is translated to the left by $t(t > 0)$ units, and the resulting graph corresponds to an even function, then find the minimum value of $t$.
|
\dfrac{5\pi}{6}
| 63.28125 |
19,877 |
Given the complex number $z=a^{2}-1+(a+1)i (a \in \mathbb{R})$ is a purely imaginary number, find the imaginary part of $\dfrac{1}{z+a}$.
|
-\dfrac{2}{5}
| 82.03125 |
19,878 |
In Tranquility Town, the streets are all $30$ feet wide and the blocks they enclose are rectangles with lengths of side $500$ feet and $300$ feet. Alice walks around the rectangle on the $500$-foot side of the street, while Bob walks on the opposite side of the street. How many more feet than Alice does Bob walk for every lap around the rectangle?
|
240
| 32.8125 |
19,879 |
Let the focus of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ be $F_{1}$, $F_{2}$, and $P$ be a point on the ellipse with $\angle F_1PF_2=\frac{π}{3}$. If the radii of the circumcircle and incircle of $\triangle F_{1}PF_{2}$ are $R$ and $r$ respectively, and when $R=3r$, the eccentricity of the ellipse is ______.
|
\frac{3}{5}
| 17.96875 |
19,880 |
Given externally tangent circles with centers at points $A$ and $B$ and radii of lengths $6$ and $4$, respectively, a line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate $BC$.
|
20
| 77.34375 |
19,881 |
I have modified my walking game. On move 1, I still do nothing, but for each move $n$ where $2 \le n \le 30$, I walk one step backward if $n$ is prime and three steps forward if $n$ is composite. After all 30 moves, I assess how far I am from my starting point. How many total steps would I need to walk to return?
|
47
| 0.78125 |
19,882 |
Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that?
|
14
| 34.375 |
19,883 |
The diagram shows a shape made from ten squares of side-length \(1 \mathrm{~cm}\), joined edge to edge. What is the length of its perimeter, in centimetres?
A) 14
B) 18
C) 30
D) 32
E) 40
|
18
| 9.375 |
19,884 |
Calculate the value of the following expressions:
(1) $(2 \frac {7}{9})^{0.5}+0.1^{-2}+(2 \frac {10}{27})^{- \frac {2}{3}}-3\pi^{0}+ \frac {37}{48}$;
(2) $(-3 \frac {3}{8})^{- \frac {2}{3}}+(0.002)^{- \frac {1}{2}}-10(\sqrt {5}-2)^{-1}+(\sqrt {2}- \sqrt {3})^{0}$.
|
- \frac {167}{9}
| 55.46875 |
19,885 |
Given that both $α$ and $β$ are acute angles, $\cos α= \frac {1}{7}$, and $\cos (α+β)=- \frac {11}{14}$, find the value of $\cos β$.
|
\frac {1}{2}
| 85.9375 |
19,886 |
Given that $\tan(\alpha+ \frac {\pi}{4})= \frac {3}{4}$, calculate the value of $\cos ^{2}(\frac {\pi}{4}-\alpha)$.
|
\frac{9}{25}
| 89.84375 |
19,887 |
From the numbers $1,2,3, \cdots, 2014$, select 315 different numbers (order does not matter) to form an arithmetic sequence. Among these, the number of ways to form an arithmetic sequence that includes the number 1 is ___. The total number of ways to form an arithmetic sequence is ___.
|
5490
| 8.59375 |
19,888 |
For how many three-digit numbers can you subtract 297 and obtain a second three-digit number which is the original three-digit number reversed?
|
60
| 35.15625 |
19,889 |
Given the function $f(x)=\sin x+a\cos x(x∈R)$ whose one symmetric axis is $x=- \frac {π}{4}$.
(I) Find the value of $a$ and the monotonically increasing interval of the function $f(x)$;
(II) If $α$, $β∈(0, \frac {π}{2})$, and $f(α+ \frac {π}{4})= \frac { \sqrt {10}}{5}$, $f(β+ \frac {3π}{4})= \frac {3 \sqrt {5}}{5}$, find $\sin (α+β)$
|
\frac { \sqrt {2}}{2}
| 0 |
19,890 |
A high school math team received 5 college students for a teaching internship, who are about to graduate. They need to be assigned to three freshman classes: 1, 2, and 3, with at least one and at most two interns per class. Calculate the number of different allocation schemes.
|
90
| 25.78125 |
19,891 |
Given the polar equation of curve $C$ is $\rho \sin^2\theta = 4\cos\theta$, and the lines $l_1: \theta= \frac{\pi}{3}$, $l_2: \rho\sin\theta=4\sqrt{3}$ intersect curve $C$ at points $A$ and $B$ (with $A$ not being the pole),
(Ⅰ) Find the polar coordinates of points $A$ and $B$;
(Ⅱ) If $O$ is the pole, find the area of $\Delta AOB$.
|
\frac{16}{3}\sqrt{3}
| 0 |
19,892 |
In a regional frisbee league, teams have 7 members and each of the 5 teams takes turns hosting matches. At each match, each team selects three members of that team to be on the match committee, except the host team, which selects four members. How many possible 13-member match committees are there?
|
262,609,375
| 0 |
19,893 |
Find the number of integers $n$ that satisfy:
\[50 < n^2 < 200.\]
|
14
| 35.9375 |
19,894 |
A school has between 150 and 250 students enrolled. Each day, all the students split into eight different sections for a special workshop. If two students are absent, each section can contain an equal number of students. Find the sum of all possible values of student enrollment at the school.
|
2626
| 1.5625 |
19,895 |
A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?
|
$\sqrt{13}$
| 0 |
19,896 |
Quadrilateral $ABCD$ has right angles at $A$ and $C$, with diagonal $AC = 5$. If $AB = BC$ and sides $AD$ and $DC$ are of distinct integer lengths, what is the area of quadrilateral $ABCD$? Express your answer in simplest radical form.
|
12.25
| 0 |
19,897 |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate?
|
20
| 51.5625 |
19,898 |
Given an increasing geometric sequence $\{a_{n}\}$ with a common ratio greater than $1$ such that $a_{2}+a_{4}=20$, $a_{3}=8$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let $b_{m}$ be the number of terms of $\{a_{n}\}$ in the interval $\left(0,m\right]\left(m\in N*\right)$. Find the sum of the first $100$ terms of the sequence $\{b_{m}\}$, denoted as $S_{100}$.
|
480
| 16.40625 |
19,899 |
Given the function f(x) = sinωx + cosωx, if there exists a real number x₁ such that for any real number x, f(x₁) ≤ f(x) ≤ f(x₁ + 2018) holds true, find the minimum positive value of ω.
|
\frac{\pi}{2018}
| 3.90625 |
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