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40.3k
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5.15k
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100
|
|---|---|---|---|
18,800 |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-3,2)$.
(1) If vector $k\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to vector $\overrightarrow{a}-3\overrightarrow{b}$, find the value of the real number $k$;
(2) For what value of $k$ are vectors $k\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-3\overrightarrow{b}$ parallel? And determine whether they are in the same or opposite direction.
|
-\dfrac{1}{3}
| 70.3125 |
18,801 |
Let \( a \) and \( b \) be nonnegative real numbers such that
\[ \cos (ax + b) = \cos 31x \]
for all integers \( x \). Find the smallest possible value of \( a \).
|
31
| 80.46875 |
18,802 |
In a table tennis singles match between players A and B, the match is played in a best-of-seven format (i.e., the first player to win four games wins the match, and the match ends). Assume that both players have an equal chance of winning each game.
(1) Calculate the probability that player B wins by a score of 4 to 1;
(2) Calculate the probability that player A wins and the number of games played is more than 5.
|
\frac{5}{16}
| 60.15625 |
18,803 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $a\cos B=(3c-b)\cos A$.
$(1)$ If $a\sin B=2\sqrt{2}$, find $b$;
$(2)$ If $a=2\sqrt{2}$ and the area of $\triangle ABC$ is $\sqrt{2}$, find the perimeter of $\triangle ABC$.
|
4+2\sqrt{2}
| 13.28125 |
18,804 |
Mark's cousin has $10$ identical stickers and $5$ identical sheets of paper. How many ways are there for him to distribute all of the stickers on the sheets of paper, given that each sheet must have at least one sticker, and only the number of stickers on each sheet matters?
|
126
| 14.0625 |
18,805 |
Let $\{a_n\}$ be a geometric sequence with a common ratio not equal to 1, and $a_4=16$. The sum of the first $n$ terms is denoted as $S_n$, and $5S_1$, $2S_2$, $S_3$ form an arithmetic sequence.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $b_n = \frac{1}{\log_{2}a_n \cdot \log_{2}a_{n+1}}$, and let $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Find the minimum value of $T_n$.
|
\frac{1}{2}
| 83.59375 |
18,806 |
For how many integer values of $n$ between 1 and 990 inclusive does the decimal representation of $\frac{n}{1000}$ terminate?
|
990
| 94.53125 |
18,807 |
Given the function $f(x)=4-x^{2}+a\ln x$, if $f(x)\leqslant 3$ for all $x > 0$, determine the range of the real number $a$.
|
[2]
| 0 |
18,808 |
Given the points $(2, 3)$, $(10, 9)$, and $(6, m)$, where $m$ is an integer, determine the sum of all possible values of $m$ for which the area of the triangle formed by these points is a maximum.
|
12
| 46.09375 |
18,809 |
Given that $F_1$ and $F_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), an isosceles right triangle $MF_1F_2$ is constructed with $F_1$ as the right-angle vertex. If the midpoint of the side $MF_1$ lies on the hyperbola, calculate the eccentricity of the hyperbola.
|
\frac{\sqrt{5} + 1}{2}
| 14.0625 |
18,810 |
In $\triangle ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. The lengths of sides $AC$ and $BC$ are 6 and 7 respectively. What is the length of side $AB$?
|
$\sqrt{17}$
| 0 |
18,811 |
A mini soccer team has 12 members. We want to choose a starting lineup of 5 players, which includes one goalkeeper and four outfield players (order of outfield players does not matter). In how many ways can we choose this starting lineup?
|
3960
| 72.65625 |
18,812 |
Given that the sequence $\{a_n\}$ is a geometric sequence, with $a_1=2$, common ratio $q>0$, and $a_2$, $6$, $a_3$ forming an arithmetic sequence.
(I) Find the general term formula for the sequence $\{a_n\}$;
(II) Let $b_n=\log_2{a_n}$, and $$T_{n}= \frac {1}{b_{1}b_{2}}+ \frac {1}{b_{2}b_{3}}+ \frac {1}{b_{3}b_{4}}+…+ \frac {1}{b_{n}b_{n+1}}$$, find the maximum value of $n$ such that $$T_{n}< \frac {99}{100}$$.
|
98
| 92.96875 |
18,813 |
Given the function $f(x)=x^2+x+b\ (b\in\mathbb{R})$ with a value range of $[0,+\infty)$, the solution to the equation $f(x) < c$ is $m+8$. Determine the value of $c$.
|
16
| 84.375 |
18,814 |
Scenario: In a math activity class, the teacher presented a set of questions and asked the students to explore the pattern by reading the following solution process:
$\sqrt{1+\frac{5}{4}}=\sqrt{\frac{9}{4}}=\sqrt{{(\frac{3}{2})}^{2}}=\frac{3}{2}$;
$\sqrt{1+\frac{7}{9}}=\sqrt{\frac{16}{9}}=\sqrt{{(\frac{4}{3})}^{2}}=\frac{4}{3}$;
$\sqrt{1+\frac{9}{16}}=\sqrt{\frac{25}{16}}=\sqrt{{(\frac{5}{4})}^{2}}=\frac{5}{4}$;
$\ldots$
Practice and exploration:
$(1)$ According to this pattern, calculate: $\sqrt{1+\frac{17}{64}}=$______;
$(2)$ Calculate: $\sqrt{1+\frac{5}{4}}×\sqrt{1+\frac{7}{9}}×\sqrt{1+\frac{9}{16}}×⋅⋅⋅×\sqrt{1+\frac{21}{100}}$;
Transfer and application:
$(3)$ If $\sqrt{1+\frac{2023}{{n}^{2}}}=x$ follows the above pattern, please directly write down the value of $x$.
|
\frac{1012}{1011}
| 76.5625 |
18,815 |
Given that for triangle $ABC$, the internal angles $A$ and $B$ satisfy $$\frac {\sin B}{\sin A} = \cos(A + B),$$ find the maximum value of $\tan B$.
|
\frac{\sqrt{2}}{4}
| 32.03125 |
18,816 |
Xiaoying goes home at noon to cook noodles by herself, which involves the following steps: ① Wash the pot and fill it with water, taking 2 minutes; ② Wash the vegetables, taking 3 minutes; ③ Prepare the noodles and seasonings, taking 2 minutes; ④ Boil the water in the pot, taking 7 minutes; ⑤ Use the boiling water to cook the noodles and vegetables, taking 3 minutes. Except for step ④, each step can only be performed one at a time. The minimum time Xiaoying needs to cook the noodles is minutes.
|
12
| 87.5 |
18,817 |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=\frac{20}{3}-x$.
|
16
| 26.5625 |
18,818 |
Points $A$, $B$, $C$, $D$ are on the same sphere, with $AB=BC=\sqrt{2}$, $AC=2$. If the circumscribed sphere of tetrahedron $ABCD$ has its center exactly on edge $DA$, and $DC=2\sqrt{3}$, then the surface area of this sphere equals \_\_\_\_\_\_\_\_\_\_
|
16\pi
| 14.84375 |
18,819 |
Find the greatest common factor of 8! and 9!.
|
40320
| 100 |
18,820 |
The stem-and-leaf plot displays the lengths of songs on an album in minutes and seconds. There are 18 songs on the album. In the plot, $3\ 45$ represents $3$ minutes, $45$ seconds, which is equivalent to $225$ seconds. What is the median length of the songs? Express your answer in seconds.
\begin{tabular}{c|ccccc}
0&32&43&58&&\\
1&05&10&12&15&20\\
2&25&30&55&&\\
3&00&15&30&35&45\\
4&10&12&&&\\
\end{tabular}
|
147.5
| 12.5 |
18,821 |
What is the smallest positive integer \( n \) such that all the roots of \( z^6 - z^3 + 1 = 0 \) are \( n^{\text{th}} \) roots of unity?
|
18
| 93.75 |
18,822 |
Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is
|
-1
| 91.40625 |
18,823 |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$ with a minimum positive period of 3, and for $x \in \left(-\frac{3}{2}, 0\right)$, $f(x)=\log_{2}(-3x+1)$. Find $f(2011)$.
|
-2
| 75 |
18,824 |
Given $\theta∈(0,\frac{π}{2}$, $\sin \theta - \cos \theta = \frac{\sqrt{5}}{5}$, find the value of $\tan 2\theta$.
|
-\frac{4}{3}
| 48.4375 |
18,825 |
In the numbers from 100 to 999, how many numbers have digits in strictly increasing or strictly decreasing order?
(From the 41st American High School Mathematics Exam, 1990)
|
204
| 60.15625 |
18,826 |
Given triangle $ABC$ with midpoint $D$ on side $BC$, and point $G$ satisfies $\overrightarrow{GA}+ \overrightarrow{BG}+ \overrightarrow{CG}= \overrightarrow{0}$, and $\overrightarrow{AG}=\lambda \overrightarrow{GD}$, determine the value of $\lambda$.
|
-2
| 9.375 |
18,827 |
Given that the function $y=f(x)$ is an odd function defined on $\mathbb{R}$ and $f(-1)=2$, and the period of the function is $4$, calculate the values of $f(2012)$ and $f(2013)$.
|
-2
| 55.46875 |
18,828 |
One of the mascots for the 2012 Olympic Games is called 'Wenlock' because the town of Wenlock in Shropshire first held the Wenlock Olympian Games in 1850. How many years ago was that?
A) 62
B) 152
C) 158
D) 162
E) 172
|
162
| 92.96875 |
18,829 |
There exist constants $b_1, b_2, b_3, b_4, b_5, b_6, b_7$ such that
\[
\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta
\]
for all angles $\theta$. Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2$.
|
\frac{429}{1024}
| 35.15625 |
18,830 |
Given that the year 2010 corresponds to the Geng-Yin year, determine the year of the previous Geng-Yin year.
|
1950
| 47.65625 |
18,831 |
There are two lathes processing parts of the same model. The yield rate of the first lathe is $15\%$, and the yield rate of the second lathe is $10\%$. Assuming that the yield rates of the two lathes do not affect each other, the probability of both lathes producing excellent parts simultaneously is ______; if the processed parts are mixed together, knowing that the number of parts processed by the first lathe accounts for $60\%$ of the total, and the number of parts processed by the second lathe accounts for $40\%$, then randomly selecting a part, the probability of it being an excellent part is ______.
|
13\%
| 0 |
18,832 |
Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}=k+2( \frac {1}{3})^{n}$, find the value of the constant $k$.
|
-2
| 57.03125 |
18,833 |
Let $f(x) = 5x^2 - 4$ and $g(f(x)) = x^2 + x + x/3 + 1$. Find the sum of all possible values of $g(49)$.
|
\frac{116}{5}
| 89.0625 |
18,834 |
Triangle $ABC$ has sidelengths $AB=1$ , $BC=\sqrt{3}$ , and $AC=2$ . Points $D,E$ , and $F$ are chosen on $AB, BC$ , and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$ . Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a, b) = 1$ , find $a + b$ . (Here $[DEF]$ denotes the area of triangle $DEF$ .)
*Proposed by Vismay Sharan*
|
67
| 27.34375 |
18,835 |
Compute $\sin 12^\circ \sin 36^\circ \sin 72^\circ \sin 84^\circ.$
|
\frac{1}{16}
| 51.5625 |
18,836 |
In how many ways can 10 people be seated in a row of chairs if four of the people, Alice, Bob, Charlie, and Dave, refuse to sit in four consecutive seats?
|
3507840
| 89.84375 |
18,837 |
In recent years, the awareness of traffic safety among citizens has gradually increased, leading to a greater demand for helmets. A certain store purchased two types of helmets, type A and type B. It is known that they bought 20 type A helmets and 30 type B helmets, spending a total of 2920 yuan. The unit price of type A helmets is 11 yuan higher than the unit price of type B helmets.
$(1)$ What are the unit prices of type A and type B helmets, respectively?
$(2)$ The store decides to purchase 40 more helmets of type A and type B. They coincide with a promotional event by the manufacturer: type A helmets are sold at 20% off the unit price, and type B helmets are discounted by 6 yuan each. If the number of type A helmets purchased this time is not less than half the number of type B helmets, how many type A helmets should be purchased to minimize the total cost of this purchase? What is the minimum cost?
|
1976
| 65.625 |
18,838 |
Consider a unit cube in a coordinate system with vertices $A(0,0,0)$, $A'(1,1,1)$, and other vertices placed accordingly. A regular octahedron has vertices placed at fractions $\frac{1}{3}$ and $\frac{2}{3}$ along the segments connecting $A$ with $A'$'s adjacent vertices and vice versa. Determine the side length of this octahedron.
|
\frac{\sqrt{2}}{3}
| 15.625 |
18,839 |
Given $sinα+cosα=-\frac{{\sqrt{10}}}{5}, α∈(-\frac{π}{2},\frac{π}{2})$.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Find the value of $2\sin ^{2}\alpha +\sin \alpha \cos \alpha -1$.
|
\frac{1}{2}
| 66.40625 |
18,840 |
In daily life, specific times are usually expressed using the 24-hour clock system. There are a total of 24 time zones globally, with adjacent time zones differing by 1 hour. With the Prime Meridian located in Greenwich, England as the reference point, in areas east of Greenwich, the time difference is marked with a "+", while in areas west of Greenwich, the time difference is marked with a "-". The table below shows the time differences of various cities with respect to Greenwich:
| City | Beijing | New York | Sydney | Moscow |
|--------|---------|----------|--------|--------|
| Time Difference with Greenwich (hours) | +8 | -4 | +11 | +3 |
For example, when it is 12:00 in Greenwich, it is 20:00 in Beijing and 15:00 in Moscow.
$(1)$ What is the time difference between Beijing and New York?
$(2)$ If Xiao Ming in Sydney calls Xiao Liang in New York at 21:00, what time is it in New York?
$(3)$ Xiao Ming takes a direct flight from Beijing to Sydney at 23:00 on October 27th. After 12 hours, he arrives. What is the local time in Sydney when he arrives?
$(4)$ Xiao Hong went on a study tour to Moscow. After arriving in Moscow, he calls his father in Beijing at an exact hour. At that moment, his father's time in Beijing is exactly twice his time in Moscow. What is the specific time in Beijing when the call is connected?
|
10:00
| 40.625 |
18,841 |
For the ellipse $25x^2 - 100x + 4y^2 + 8y + 16 = 0,$ find the distance between the foci.
|
\frac{2\sqrt{462}}{5}
| 83.59375 |
18,842 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $a\sin C= \sqrt {3}c\cos A$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $a= \sqrt {13}$ and $c=3$, find the area of $\triangle ABC$.
|
3 \sqrt {3}
| 0 |
18,843 |
Given acute angles $\alpha$ and $\beta$ such that $\sin \alpha= \frac { \sqrt {5}}{5}$ and $\sin(\alpha-\beta)=- \frac { \sqrt {10}}{10}$, determine the value of $\beta$.
|
\frac{\pi}{4}
| 80.46875 |
18,844 |
A certain item is always sold with a 30% discount, and the profit margin is 47%. During the shopping festival, the item is sold at the original price, and there is a "buy one get one free" offer. Calculate the profit margin at this time. (Note: Profit margin = (selling price - cost) ÷ cost)
|
5\%
| 64.0625 |
18,845 |
Find $5273_{8} - 3614_{8}$. Express your answer in base $8$.
|
1457_8
| 53.90625 |
18,846 |
At a national competition, 31 participants are accommodated in a hotel where each participant gets his/her own room. The rooms are numbered from 1 to 31. When all participants have arrived, except those assigned to rooms 15, 16, and 17, what is the median room number of the remaining 28 participants?
|
16
| 60.15625 |
18,847 |
A frustum of a right circular cone is formed by cutting a small cone off of the top of a larger cone. If this frustum has a lower base radius of 8 inches, an upper base radius of 5 inches, and a height of 6 inches, what is its lateral surface area? Additionally, there is a cylindrical section of height 2 inches and radius equal to the upper base of the frustum attached to the top of the frustum. Calculate the total surface area excluding the bases.
|
39\pi\sqrt{5} + 20\pi
| 28.125 |
18,848 |
Given a circle $C$: $x^{2}+y^{2}-2x-2ay+a^{2}-24=0$ ($a\in\mathbb{R}$) whose center lies on the line $2x-y=0$.
$(1)$ Find the value of the real number $a$;
$(2)$ Find the minimum length of the chord formed by the intersection of circle $C$ and line $l$: $(2m+1)x+(m+1)y-7m-4=0$ ($m\in\mathbb{R}$).
|
4\sqrt{5}
| 28.125 |
18,849 |
In right triangle $DEF$, $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?
|
7.5
| 18.75 |
18,850 |
Let the number of elements in the set \( S \) be denoted by \( |S| \), and the number of subsets of the set \( S \) be denoted by \( n(S) \). Given three non-empty finite sets \( A, B, C \) that satisfy the following conditions:
$$
\begin{array}{l}
|A| = |B| = 2019, \\
n(A) + n(B) + n(C) = n(A \cup B \cup C).
\end{array}
$$
Determine the maximum value of \( |A \cap B \cap C| \) and briefly describe the reasoning process.
|
2018
| 40.625 |
18,851 |
Connecting the right-angled vertex of a right triangle and the two trisection points on the hypotenuse, the lengths of the two resulting line segments are $\sin \alpha$ and $\cos \alpha$ (where $0 < \alpha < \frac{\pi}{2}$). What is the length of the hypotenuse?
|
$\frac{3}{\sqrt{5}}$
| 0 |
18,852 |
Alice and Bob play a similar game with a basketball. On each turn, if Alice has the ball, there is a 2/3 chance that she will toss it to Bob and a 1/3 chance that she will keep the ball. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance that he keeps it. Alice starts with the ball. What is the probability that Alice has the ball again after two turns?
|
\frac{5}{18}
| 58.59375 |
18,853 |
Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$.
|
110.8333
| 5.46875 |
18,854 |
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?
|
\frac{1}{2}
| 35.15625 |
18,855 |
There are 12 sprinters in an international track event final, including 5 Americans. Gold, silver, and bronze medals are awarded to the first, second, and third place finishers respectively. Determine the number of ways the medals can be awarded if no more than two Americans receive medals.
|
1260
| 21.875 |
18,856 |
In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, find the distance from $F$ to the midpoint of segment $DE$.
|
7.5
| 18.75 |
18,857 |
Given that $P$ is a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. If $\frac{{\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}}}{{|\overrightarrow{PF_1}| \cdot |\overrightarrow{PF_2}|}}=\frac{1}{2}$, then the area of $\triangle F_{1}PF_{2}$ is ______.
|
3\sqrt{3}
| 34.375 |
18,858 |
The graph of $y = ax^2 + bx + c$ has a maximum value of 72, and passes through the points $(0, -1)$ and $(6, -1)$. Find $a + b + c$.
|
\frac{356}{9}
| 66.40625 |
18,859 |
Given the lines $l_{1}$: $x+\left(m-3\right)y+m=0$ and $l_{2}$: $mx-2y+4=0$.
$(1)$ If line $l_{1}$ is perpendicular to line $l_{2}$, find the value of $m$.
$(2)$ If line $l_{1}$ is parallel to line $l_{2}$, find the distance between $l_{1}$ and $l_{2}$.
|
\frac{3\sqrt{5}}{5}
| 61.71875 |
18,860 |
If $9:y^3 = y:81$, what is the value of $y$?
|
3\sqrt{3}
| 21.875 |
18,861 |
The average of the numbers $1, 2, 3,\dots, 148, 149,$ and $x$ is $50x$. What is $x$?
|
\frac{11175}{7499}
| 71.09375 |
18,862 |
The increasing sequence of positive integers $b_1,$ $b_2,$ $b_3,$ $\dots$ has the property that
\[b_{n + 2} = b_{n + 1} + b_n\]for all $n \ge 1.$ If $b_6 = 60,$ then find $b_7.$
|
97
| 69.53125 |
18,863 |
Distribute 5 students into 3 groups: Group A, Group B, and Group C, with Group A having at least two people, and Groups B and C having at least one person each, and calculate the number of different distribution schemes.
|
80
| 8.59375 |
18,864 |
Points \(A = (2,8)\), \(B = (2,2)\), and \(C = (6,2)\) lie in the first quadrant and are vertices of triangle \(ABC\). Point \(D=(a,b)\) is also in the first quadrant, and together with \(A\), \(B\), and \(C\), forms quadrilateral \(ABCD\). The quadrilateral formed by joining the midpoints of \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), and \(\overline{DA}\) is a square. Additionally, the diagonal of this square has the same length as the side \(\overline{AB}\) of triangle \(ABC\). Find the sum of the coordinates of point \(D\).
A) 12
B) 14
C) 15
D) 16
E) 18
|
14
| 17.96875 |
18,865 |
Let vertices $A, B, C$, and $D$ form a regular tetrahedron with each edge of length 1 unit. Define point $P$ on edge $AB$ such that $P = tA + (1-t)B$ for some $t$ in the range $0 \leq t \leq 1$ and point $Q$ on edge $CD$ such that $Q = sC + (1-s)D$ for some $s$ in the range $0 \leq s \leq 1$. Determine the minimum possible distance between $P$ and $Q$.
|
\frac{\sqrt{2}}{2}
| 28.90625 |
18,866 |
Given that $\frac{5+7+9}{3} = \frac{4020+4021+4022}{M}$, find $M$.
|
1723
| 73.4375 |
18,867 |
In the sequence $\{a_{n}\}$, $a_{1}=1$, $\sqrt{{a}_{n+1}}-\sqrt{{a}_{n}}=1$ ($n\in N^{*}$); the sum of the first $n$ terms of a geometric sequence $\{b_{n}\}$ is $S_{n}=2^{n}-m$. For $n\in N^{*}$, the smallest value of the real number $\lambda$ that satisfies $\lambda b_{n}\geqslant a_{n}$ for all $n$ is ______.
|
\frac{9}{4}
| 47.65625 |
18,868 |
Given vectors $\overrightarrow{a}=(\cos x, \sin x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x, 2\cos x-\sqrt{3}\sin x)$, let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.
$(2)$ If the maximum value of the function $g(x)=f(x-\frac{\pi}{6})+af(\frac{x}{2}-\frac{\pi}{6})-af(\frac{x}{2}+\frac{\pi}{12})$ on the interval $[0,\pi]$ is $6$, determine the value of the real number $a$.
|
2\sqrt{2}
| 0.78125 |
18,869 |
Let positive integers $a$, $b$, $c$ satisfy $ab + bc = 518$ and $ab - ac = 360$. The maximum value of $abc$ is ____.
|
1008
| 42.96875 |
18,870 |
A literary and art team went to a nursing home for a performance. Originally, there were 6 programs planned, but at the request of the elderly, they decided to add 3 more programs. However, the order of the original six programs remained unchanged, and the added 3 programs were neither at the beginning nor at the end. Thus, there are a total of different orders for this performance.
|
210
| 15.625 |
18,871 |
The area of a square inscribed in a semicircle compared to the area of a square inscribed in a full circle.
|
2:5
| 0 |
18,872 |
In a class of 50 students, it is decided to use systematic sampling to select 10 students out of these 50. The students are randomly assigned numbers from 1 to 50 and grouped, with the first group being numbers 1 to 5, the second group 6 to 10, and so on, up to the tenth group which is 46 to 50. If a student with the number 12 is selected from the third group, then the student with the number $\_\_\_$ will be selected from the eighth group.
|
37
| 85.9375 |
18,873 |
How many even numbers are greater than 300 and less than 600?
|
149
| 60.15625 |
18,874 |
The witch Gingema cast a spell on a wall clock so that the minute hand moves in the correct direction for five minutes, then three minutes in the opposite direction, then five minutes in the correct direction again, and so on. How many minutes will the hand show after 2022 minutes, given that it pointed exactly to 12 o'clock at the beginning of the five-minute interval of correct movement?
|
28
| 6.25 |
18,875 |
The real numbers $x$ and $y$ satisfy the equation $2\cos ^{2}(x+y-1)= \frac {(x+1)^{2}+(y-1)^{2}-2xy}{x-y+1}$. Find the minimum value of $xy$.
|
\frac{1}{4}
| 58.59375 |
18,876 |
Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).
|
65
| 86.71875 |
18,877 |
Let $b_1, b_2, \dots$ be a sequence defined by $b_1 = b_2 = 2$ and $b_{n+2} = b_{n+1} + b_n$ for $n \geq 1$. Find
\[
\sum_{n=1}^\infty \frac{b_n}{3^{n+1}}.
\]
|
\frac{2}{5}
| 50 |
18,878 |
The average of the seven numbers in a list is 62. The average of the first four numbers is 58. What is the average of the last three numbers?
|
67.\overline{3}
| 0 |
18,879 |
If \( g(x) = \frac{x^5 - 1}{4} \), find \( g^{-1}(-7/64) \).
|
\left(\frac{9}{16}\right)^{\frac{1}{5}}
| 0 |
18,880 |
The mean of the numbers 3, 7, 10, and 15 is twice the mean of $x$, 20, and 6. What is the value of $x$?
|
-12.875
| 1.5625 |
18,881 |
Simplify the expression $\frac{{2x+4}}{{{x^2}-1}}÷\frac{{x+2}}{{{x^2}-2x+1}}-\frac{{2x}}{{x+1}}$, then substitute an appropriate number from $-2$, $-1$, $0$, $1$ to evaluate.
|
-2
| 69.53125 |
18,882 |
Let $a$ and $b$ be integers such that $ab = 72.$ Find the minimum value of $a + b.$
|
-73
| 94.53125 |
18,883 |
A chord PQ of the left branch of the hyperbola $x^2 - y^2 = 4$ passes through its left focus $F_1$, and the length of $|PQ|$ is 7. If $F_2$ is the right focus of the hyperbola, then the perimeter of $\triangle PF_2Q$ is.
|
22
| 50.78125 |
18,884 |
A square with a side length of 10 centimeters is rotated about its horizontal line of symmetry. Calculate the volume of the resulting cylinder in cubic centimeters and express your answer in terms of $\pi$.
|
250\pi
| 85.9375 |
18,885 |
Athletes A and B have probabilities of successfully jumping over a 2-meter high bar of 0.7 and 0.6, respectively. The outcomes of their jumps are independent of each other. Find:
(Ⅰ) The probability that A succeeds on the third attempt.
(Ⅱ) The probability that at least one of A or B succeeds on the first attempt.
(Ⅲ) The probability that A succeeds exactly one more time than B in two attempts for each.
|
0.3024
| 75.78125 |
18,886 |
Given that three numbers are randomly selected from the set {1, 2, 3, 4, 5}, find the probability that the sum of the remaining two numbers is odd.
|
0.6
| 7.03125 |
18,887 |
$(1)$ Calculate: $tan60°×{({-2})^{-1}}-({\sqrt{\frac{3}{4}}-\sqrt[3]{8}})+|{-\frac{1}{2}\sqrt{12}}|$;<br/>$(2)$ Simplify and find the value: $({\frac{{x+2}}{{{x^2}-2x}}-\frac{{x-1}}{{{x^2}-4x+4}}})÷\frac{{x-4}}{x}$, where $x=\sqrt{2}+2$.
|
\frac{1}{2}
| 56.25 |
18,888 |
Calculate \(3 \cdot 15 + 20 \div 4 + 1\).
Then add parentheses to the expression so that the result is:
1. The largest possible integer,
2. The smallest possible integer.
|
13
| 10.15625 |
18,889 |
Given $f\left( \alpha \right)=\frac{\cos \left( \frac{\pi }{2}+\alpha \right)\cdot \cos \left( 2\pi -\alpha \right)\cdot \sin \left( -\alpha +\frac{3}{2}\pi \right)}{\sin \left( -\pi -\alpha \right)\sin \left( \frac{3}{2}\pi +\alpha \right)}$.
$(1)$ Simplify $f\left( \alpha \right)$; $(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos \left( \alpha -\frac{3}{2}\pi \right)=\frac{1}{5}$, find the value of $f\left( \alpha \right)$.
|
\frac{2 \sqrt{6}}{5}
| 89.0625 |
18,890 |
A solid right prism $PQRSTU$ has a height of 20, as shown. Its bases are equilateral triangles with side length 15. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RS$, respectively. Determine the perimeter of triangle $MNO$.
|
32.5
| 11.71875 |
18,891 |
In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product.
|
14
| 53.125 |
18,892 |
Given the equation of the Monge circle of the ellipse $\Gamma$ as $C: x^{2}+y^{2}=3b^{2}$, calculate the eccentricity of the ellipse $\Gamma$.
|
\frac{{\sqrt{2}}}{2}
| 0 |
18,893 |
Simplify $\frac{{1+\cos{20}°}}{{2\sin{20}°}}-\sin{10°}\left(\frac{1}{{\tan{5°}}}-\tan{5°}\right)=\_\_\_\_\_\_$.
|
\frac{\sqrt{3}}{2}
| 44.53125 |
18,894 |
Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \]
|
14
| 72.65625 |
18,895 |
Let $S_{n}$ and $T_{n}$ represent the sum of the first $n$ terms of the arithmetic sequences ${a_{n}}$ and ${b_{n}}$, respectively. Given that $\frac{S_{n}}{T_{n}} = \frac{2n+1}{4n-2}$ for all positive integers $n$, find the value of $\frac{a_{10}}{b_{3}+b_{18}} + \frac{a_{11}}{b_{6}+b_{15}}$.
|
\frac{41}{78}
| 35.9375 |
18,896 |
What is the largest value of $x$ that satisfies the equation $\sqrt{3x} = 6x^2$? Express your answer in simplest fractional form.
|
\frac{1}{\sqrt[3]{12}}
| 62.5 |
18,897 |
Given circle $C: (x-2)^{2} + (y-2)^{2} = 8-m$, if circle $C$ has three common tangents with circle $D: (x+1)^{2} + (y+2)^{2} = 1$, then the value of $m$ is ______.
|
-8
| 95.3125 |
18,898 |
Given a four-digit number $\overline{ABCD}$ such that $\overline{ABCD} + \overline{AB} \times \overline{CD}$ is a multiple of 1111, what is the minimum value of $\overline{ABCD}$?
|
1729
| 91.40625 |
18,899 |
Given $3\sin \left(-3\pi +\theta \right)+\cos \left(\pi -\theta \right)=0$, then the value of $\frac{sinθcosθ}{cos2θ}$ is ____.
|
-\frac{3}{8}
| 78.90625 |
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