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40.3k
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100
22,700
Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^3 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(81) = 81 \), compute \( [g(3)]^4 \).
81
68.75
22,701
There is a question: "If the value of the algebraic expression $5a+3b$ is $-4$, then what is the value of the algebraic expression $2\left(a+b\right)+4\left(2a+b\right)$?" Tang, who loves to use his brain, solved the problem as follows: Original expression $=2a+2b+8a+4b=10a+6b=2\left(5a+3b\right)=2\times \left(-4\right)=-8$ Tang treated $5a+3b$ as a whole to solve it. Treating expressions as a whole is an important method in solving problems in high school mathematics. Please follow the method above to complete the following questions: $(1)$ Given $a^{2}+a=3$, then $2a^{2}+2a+2023=$______; $(2)$ Given $a-2b=-3$, find the value of $3\left(a-b\right)-7a+11b+2$; $(3)$ Given $a^{2}+2ab=-5$, $ab-2b^{2}=-3$, find the value of the algebraic expression $a^{2}+ab+2b^{2}$.
-2
71.875
22,702
I planned to work 25 hours a week for 15 weeks to earn $3750$ for a vacation. However, due to a family emergency, I couldn't work for the first three weeks. How many hours per week must I work for the remaining weeks to still afford the vacation?
31.25
71.09375
22,703
Define an operation between sets A and B: $A*B = \{x | x = x_1 + x_2, \text{ where } x_1 \in A, x_2 \in B\}$. If $A = \{1, 2, 3\}$ and $B = \{1, 2\}$, then the sum of all elements in $A*B$ is ____.
14
100
22,704
In triangle $ABC$ with $\overline{AB}=\overline{AC}=3.6$ , a point $D$ is taken on $AB$ at a distance $1.2$ from $A$ . Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$ . Then $\overline{AE}$ is:
10.8
36.71875
22,705
For the set $E=\{a_1, a_2, \ldots, a_{100}\}$, define a subset $X=\{a_1, a_2, \ldots, a_n\}$, and its "characteristic sequence" as $x_1, x_2, \ldots, x_{100}$, where $x_1=x_{10}=\ldots=x_n=1$. The rest of the items are 0. For example, the "characteristic sequence" of the subset $\{a_2, a_3\}$ is $0, 1, 0, 0, \ldots, 0$ (1) The sum of the first three items of the "characteristic sequence" of the subset $\{a_1, a_3, a_5\}$ is     ; (2) If the "characteristic sequence" $P_1, P_2, \ldots, P_{100}$ of a subset $P$ of $E$ satisfies $p_1=1$, $p_i+p_{i+1}=1$, $1\leq i\leq 99$; and the "characteristic sequence" $q_1, q_2, \ldots, q_{100}$ of a subset $Q$ of $E$ satisfies $q_1=1$, $q_j+q_{j+1}+q_{j+2}=1$, $1\leq j\leq 98$, then the number of elements in $P\cap Q$ is     .
17
43.75
22,706
In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 15$, $\tan D = 2$, and $\tan B = 3$, then what is $BC$?
10\sqrt{10}
27.34375
22,707
Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and *bigness* $N$ and another one with integer side lengths and *bigness* $N + 1$ .
55
78.90625
22,708
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=3\cos \alpha, \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter), in the polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\rho\sin \left( \theta- \frac{\pi}{4} \right)= \sqrt{2}$. $(1)$ Find the general equation of $C$ and the inclination angle of $l$; $(2)$ Let point $P(0,2)$, $l$ and $C$ intersect at points $A$ and $B$, find the value of $|PA|+|PB|$.
\frac{18 \sqrt{2}}{5}
34.375
22,709
For all positive numbers $a,b \in \mathbb{R}$ such that $a+b=1$, find the supremum of the expression $-\frac{1}{2a}-\frac{2}{b}$.
-\frac{9}{2}
59.375
22,710
Given that \\(\alpha\\) is an angle in the second quadrant, and \\(\sin (π+α)=- \frac {3}{5}\\), find the value of \\(\tan 2α\\).
- \frac {24}{7}
88.28125
22,711
Given the set $\{a, \frac{b}{a}, 1\} = \{a^2, a+b, 0\}$, find the value of $a^{2015} + b^{2016}$.
-1
73.4375
22,712
In a rectangular configuration $ABCD$, there are three squares with non-overlapping interiors. One of them, which is shaded, has an area of 4 square inches. The side length of the larger square is twice the side length of the shaded square. What is the area of rectangle $ABCD$, given that all squares fit exactly within $ABCD$ with no other spaces?
24
38.28125
22,713
Given that 10 spots for the finals of the 2009 National High School Mathematics Competition are to be distributed to four different schools in a certain district, with the requirement that one school gets 1 spot, another gets 2 spots, a third gets 3 spots, and the last one gets 4 spots, calculate the total number of different distribution schemes.
24
16.40625
22,714
Mia buys 10 pencils and 5 erasers for a total of $2.00. Both a pencil and an eraser cost at least 3 cents each, and a pencil costs more than an eraser. Determine the total cost, in cents, of one pencil and one eraser.
22
7.8125
22,715
Given the planar vectors $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ that satisfy $|\overrightarrow {e_{1}}| = |3\overrightarrow {e_{1}} + \overrightarrow {e_{2}}| = 2$, determine the maximum value of the projection of $\overrightarrow {e_{1}}$ onto $\overrightarrow {e_{2}}$.
-\frac{4\sqrt{2}}{3}
0.78125
22,716
Express $326_{13} + 4C9_{14}$ as a base 10 integer, where $C = 12$ in base 14.
1500
11.71875
22,717
Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, and $g(x) \neq 0$, $f''(x)g(x) < f(x)g''(x)$, $f(x)=a^{x}g(x)$, $\frac{f(1)}{g(1)}+ \frac{f(-1)}{g(-1)}= \frac{5}{2}$, determine the probability that the sum of the first $k$ terms of the sequence $\left\{ \frac{f(n)}{g(n)}\right\} (n=1,2,…,10)$ is greater than $\frac{15}{16}$.
\frac{3}{5}
10.9375
22,718
A square is inscribed in the ellipse \[\frac{x^2}{4} + \frac{y^2}{8} = 1,\] such that its sides are parallel to the coordinate axes. Find the area of this square.
\frac{32}{3}
78.125
22,719
Triangle $GHI$ has sides of length 7, 24, and 25 units, and triangle $JKL$ has sides of length 9, 40, and 41 units. Both triangles have an altitude to the hypotenuse such that for $GHI$, the altitude splits the triangle into two triangles whose areas have a ratio of 2:3. For $JKL$, the altitude splits the triangle into two triangles with areas in the ratio of 4:5. What is the ratio of the area of triangle $GHI$ to the area of triangle $JKL$? Express your answer as a common fraction.
\dfrac{7}{15}
100
22,720
In the three-dimensional Cartesian coordinate system, given points $A(2,a,-1)$, $B(-2,3,b)$, $C(1,2,-2)$.<br/>$(1)$ If points $A$, $B$, and $C$ are collinear, find the values of $a$ and $b$;<br/>$(2)$ Given $b=-3$, $D(-1,3,-3)$, and points $A$, $B$, $C$, and $D$ are coplanar, find the value of $a$.
a=1
26.5625
22,721
If the complex number $z$ satisfies $z(1-i)=|1-i|+i$, then the imaginary part of $\overline{z}$ is ______.
-\dfrac{\sqrt{2}+1}{2}
83.59375
22,722
A basketball team consists of 18 players, including a set of 3 triplets: Bob, Bill, and Ben; and a set of twins: Tim and Tom. In how many ways can we choose 7 starters if exactly two of the triplets and one of the twins must be in the starting lineup?
4290
0
22,723
Let \\(f(x)=ax^{2}-b\sin x\\) and \\(f′(0)=1\\), \\(f′\left( \dfrac {π}{3}\right)= \dfrac {1}{2}\\). Find the values of \\(a\\) and \\(b\\).
-1
0.78125
22,724
Given the function $$f(x)=\sin(x+ \frac {\pi}{6})+2\sin^{2} \frac {x}{2}$$. (1) Find the equation of the axis of symmetry and the coordinates of the center of symmetry for the function $f(x)$. (2) Determine the intervals of monotonicity for the function $f(x)$. (3) In triangle $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively, and $a= \sqrt {3}$, $f(A)= \frac {3}{2}$, the area of triangle $ABC$ is $\frac { \sqrt {3}}{2}$. Find the value of $\sin B + \sin C$.
\frac {3}{2}
59.375
22,725
Given three rays $AB$, $BC$, $BB_{1}$ are not coplanar, and the diagonals of quadrilaterals $BB_{1}A_{1}A$ and $BB_{1}C_{1}C$ bisect each other, and $\overrightarrow{AC_{1}}=x\overrightarrow{AB}+2y\overrightarrow{BC}+3z\overrightarrow{CC_{1}}$, find the value of $x+y+z$.
\frac{11}{6}
50.78125
22,726
Given the hyperbola $x^{2}- \frac{y^{2}}{24}=1$, let the focal points be F<sub>1</sub> and F<sub>2</sub>, respectively. If P is a point on the left branch of the hyperbola such that $|PF_{1}|=\frac{3}{5}|F_{1}F_{2}|$, find the area of triangle $\triangle PF_{1}F_{2}$.
24
83.59375
22,727
Find the minimum value of the function $f(x)=27x-x^{3}$ in the interval $[-4,2]$.
-54
79.6875
22,728
Let $\mathbf{v} = \begin{pmatrix} 4 \\ -5 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 13 \\ 2 \end{pmatrix}$. Find the area of the parallelogram formed by vectors $\mathbf{v}$ and $2\mathbf{w}$.
146
97.65625
22,729
The increasing sequence \(1, 3, 4, 9, 10, 12, 13, \cdots\) consists of some positive integers that are either powers of 3 or sums of distinct powers of 3. Find the value of the 2014th term.
88329
0
22,730
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a(4-2 \sqrt {7}\cos B)=b(2 \sqrt {7}\cos A-5)$, find the minimum value of $\cos C$.
-\frac{1}{2}
37.5
22,731
Find the maximum value of the function \( f(x) \), which is defined as the minimum of the three functions \( 4x + 1 \), \( x + 2 \), and \( -2x + 4 \) for each real number \( x \).
\frac{8}{3}
45.3125
22,732
A line passes through the vectors $\mathbf{a}$ and $\mathbf{b}$. For a certain value of $k$, the vector \[ k \mathbf{a} + \frac{5}{8} \mathbf{b} \] must also lie on the line. Find $k$.
\frac{3}{8}
98.4375
22,733
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?
115
28.90625
22,734
An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. What is the area of this triangle?
1.5
0
22,735
The sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$. Given that $S_{10}=0$ and $S_{15}=25$, find the minimum value of $nS_n$.
-49
78.125
22,736
Ben is throwing darts at a circular target with diameter 10. Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target. Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target. What is the expected number of points that Ben can earn from throwing a single dart? (Note that $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$ .)
11/5
36.71875
22,737
Given the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{12} = 1$ with eccentricity $e$, and the parabola $x=2py^{2}$ with focus at $(e,0)$, find the value of the real number $p$.
\frac{1}{16}
67.1875
22,738
Given the function $f\left(x\right)=x^{3}+ax^{2}+x+1$ achieves an extremum at $x=-1$. Find:<br/>$(1)$ The equation of the tangent line to $f\left(x\right)$ at $\left(0,f\left(0\right)\right)$;<br/>$(2)$ The maximum and minimum values of $f\left(x\right)$ on the interval $\left[-2,0\right]$.
-1
95.3125
22,739
Given that $| \overrightarrow{a}|=6$, $| \overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$.
-4
3.125
22,740
Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be distinct non-zero vectors such that no two are parallel. The vectors are related via: \[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{2} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\] Let \(\phi\) be the angle between \(\mathbf{v}\) and \(\mathbf{w}\). Determine \(\sin \phi.\)
\frac{\sqrt{3}}{2}
92.1875
22,741
Given the function $f(x) = 2\sin\omega x\cos\omega x - 2\sqrt{3}\sin^2\omega x + \sqrt{3}(\_\omega (\_ > 0)), the lines $x = \_x\_{1}$ and $x = \_x\_{2}$ are any two symmetry axes of the graph of the function $y = f(x)$, and the minimum value of $|x\_1 - x\_2|$ is $\frac{\pi}{2}$. 1. Find the value of $\omega$; 2. Find the intervals where the function $f(x)$ is increasing; 3. If $f(\alpha) = \frac{2}{3}$, find the value of $\sin(\frac{5}{6}\pi - 4\alpha)$.
-\frac{7}{9}
31.25
22,742
The function defined on the set of real numbers, \(f(x)\), satisfies \(f(x-1) = \frac{1 + f(x+1)}{1 - f(x+1)}\). Find the value of \(f(1) \cdot f(2) \cdot f(3) \cdots f(2008) + 2008\).
2009
55.46875
22,743
How many distinct trees with exactly 7 vertices exist?
11
15.625
22,744
Simplify:<br/>$(1)(-\frac{1}{2}+\frac{2}{3}-\frac{1}{4})÷(-\frac{1}{24})$;<br/>$(2)3\frac{1}{2}×(-\frac{5}{7})-(-\frac{5}{7})×2\frac{1}{2}-\frac{5}{7}×(-\frac{1}{2})$.
-\frac{5}{14}
36.71875
22,745
Simplify first, then evaluate: $(\frac{{x-1}}{{x-3}}-\frac{{x+1}}{x})÷\frac{{{x^2}+3x}}{{{x^2}-6x+9}}$, where $x$ satisfies $x^{2}+2x-6=0$.
-\frac{1}{2}
54.6875
22,746
Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$. (1) Find the smallest positive period and the interval where the function is decreasing; (2) In triangle $ABC$, the lengths of the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, where $a=7$. If acute angle $A$ satisfies $f(\frac{A}{2}-\frac{\pi}{6})=\sqrt{3}$, and $\sin B+\sin C=\frac{13\sqrt{3}}{14}$, find the area of triangle $ABC$.
10\sqrt{3}
85.15625
22,747
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be arithmetic progressions such that $a_1 = 50, b_1 = 100$, and $a_{50} + b_{50} = 850$. Find the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$
25000
24.21875
22,748
Given the curve $x^{2}-y-2\ln \sqrt{x}=0$ and the line $4x+4y+1=0$, find the shortest distance from any point $P$ on the curve to the line.
\dfrac{\sqrt{2}(1+\ln2)}{2}
12.5
22,749
In $\triangle PQR$, where $PQ=7$, $PR=9$, $QR=12$, and $S$ is the midpoint of $\overline{QR}$. What is the sum of the radii of the circles inscribed in $\triangle PQS$ and $\triangle PRS$? A) $\frac{14\sqrt{5}}{13}$ B) $\frac{14\sqrt{5}}{6.5 + \sqrt{29}}$ C) $\frac{12\sqrt{4}}{8.5}$ D) $\frac{10\sqrt{3}}{7 + \sqrt{24}}$
\frac{14\sqrt{5}}{6.5 + \sqrt{29}}
16.40625
22,750
Given: $\because 4 \lt 7 \lt 9$, $\therefore 2 \lt \sqrt{7} \lt 3$, $\therefore$ the integer part of $\sqrt{7}$ is $2$, and the decimal part is $\sqrt{7}-2$. The integer part of $\sqrt{51}$ is ______, and the decimal part of $9-\sqrt{51}$ is ______.
8-\sqrt{51}
61.71875
22,751
Given $\sqrt{99225}=315$, $\sqrt{x}=3.15$, then $x=(\ )$.
9.9225
99.21875
22,752
Evaluate $\cos \frac {\pi}{7}\cos \frac {2\pi}{7}\cos \frac {4\pi}{7}=$ ______.
- \frac {1}{8}
18.75
22,753
In the diagram, \(\triangle ABC\) is right-angled at \(C\). Point \(D\) is on \(AC\) so that \(\angle ABC = 2 \angle DBC\). If \(DC = 1\) and \(BD = 3\), determine the length of \(AD\).
\frac{9}{7}
5.46875
22,754
Find the sum of the squares of the solutions to \[\left| x^2 - x + \frac{1}{2010} \right| = \frac{1}{2010}.\]
\frac{2008}{1005}
39.0625
22,755
Find the length of \(PQ\) in the triangle below, where \(PQR\) is a right triangle with \( \angle RPQ = 45^\circ \) and the length \(PR\) is \(10\).
10\sqrt{2}
3.90625
22,756
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$. Find the sum of the first $60$ terms of $\{a_n\}$.
1830
37.5
22,757
Determine the number of revolutions a wheel, with a fixed center and with an outside diameter of 8 feet, would require to cause a point on the rim to travel one mile.
\frac{660}{\pi}
6.25
22,758
Given that $\sin(a + \frac{\pi}{4}) = \sqrt{2}(\sin \alpha + 2\cos \alpha)$, determine the value of $\sin 2\alpha$.
-\frac{3}{5}
21.875
22,759
Given seven positive integers from a list of eleven positive integers are \(3, 5, 6, 9, 10, 4, 7\). What is the largest possible value of the median of this list of eleven positive integers if no additional number in the list can exceed 10?
10
7.8125
22,760
Given that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, and ${S}_{n}=\frac{{n}^{2}+3n}{2}$. $(1)$ Find the general formula for the sequence $\{a_{n}\}$; $(2)$ Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{{a}_{n}{a}_{n+1}}\}$. If $\lambda T_{n}\leqslant a_{n+1}$ holds for all $n\in \mathbb{N}^{*}$, find the maximum value of the real number $\lambda$.
16
42.96875
22,761
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $(\sqrt{3}b-c)\cos A=a\cos C$, find the value of $\cos A$.
\frac{\sqrt{3}}{3}
87.5
22,762
If $0 < \alpha < \frac{\pi}{2}$ and $\tan \alpha = 2$, then $\frac{\sin 2\alpha + 1}{\cos^4 \alpha - \sin^4 \alpha} = \_\_\_\_\_\_$.
-3
92.96875
22,763
How many $6$ -digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit.
3003
92.1875
22,764
Given the function $f(x)=a\ln(x+1)+bx+1$ $(1)$ If the function $y=f(x)$ has an extremum at $x=1$, and the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$ is parallel to the line $2x+y-3=0$, find the value of $a$; $(2)$ If $b= \frac{1}{2}$, discuss the monotonicity of the function $y=f(x)$.
-4
74.21875
22,765
Shuffle the 12 cards of the four suits of A, K, Q in a deck of playing cards. (1) If a person draws 2 cards at random, what is the probability that both cards are Aces? (2) If the person has already drawn 2 Kings without replacement, what is the probability that another person draws 2 Aces?
\frac{2}{15}
79.6875
22,766
In triangle $ABC$, we have $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 3\cos B$. What is $AB$?
\frac{40\sqrt{2}}{3}
47.65625
22,767
Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital $\textit{every}$ month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs. Hannah comes up with a plan that the committee likes. Beginning in October, the comittee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that $\textit{must}$ go by (including October when the first set of five teachers is selected, but not September) such that some five-member comittee $\textit{must have}$ taught together twice (all five members are the same during two different months)?
1261
88.28125
22,768
Larry can swim from Harvard to MIT (with the current of the Charles River) in $40$ minutes, or back (against the current) in $45$ minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in $15$ minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.
14:24
77.34375
22,769
A shooter's probabilities of hitting the 10, 9, 8, 7 rings, and below 7 rings in a shooting are 0.24, 0.28, 0.19, 0.16, and 0.13, respectively. Calculate the probability that the shooter in a single shot: (1) Hits the 10 or 9 rings, (2) Hits at least the 7 ring, (3) Hits less than 8 rings.
0.29
74.21875
22,770
Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation $$ 4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1. $$
\frac{19}{12}
89.0625
22,771
Let the solution set of the inequality about $x$, $|x-2| < a$ ($a \in \mathbb{R}$), be $A$, and $\frac{3}{2} \in A$, $-\frac{1}{2} \notin A$. (1) For any $x \in \mathbb{R}$, the inequality $|x-1| + |x-3| \geq a^2 + a$ always holds true, and $a \in \mathbb{N}$. Find the value of $a$. (2) If $a + b = 1$, and $a, b \in \mathbb{R}^+$, find the minimum value of $\frac{1}{3b} + \frac{b}{a}$, and indicate the value of $a$ when the minimum is attained.
\frac{1 + 2\sqrt{3}}{3}
1.5625
22,772
If \(\lceil{\sqrt{x}}\rceil=20\), how many possible integer values of \(x\) are there?
39
88.28125
22,773
Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$
\frac{\sqrt{7}}{5}
76.5625
22,774
A triangle $\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\overline{BC}$ is parallel to the $x$-axis. The area of $\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. Determine the length of $\overline{BC}$.
10
56.25
22,775
Simplify first, then evaluate. $(1) 3x^{3}-[x^{3}+(6x^{2}-7x)]-2(x^{3}-3x^{2}-4x)$, where $x=-1$; $(2) 2(ab^{2}-2a^{2}b)-3(ab^{2}-a^{2}b)+(2ab^{2}-2a^{2}b)$, where $a=2$, $b=1$.
-10
80.46875
22,776
A regular polygon has interior angles of 160 degrees, and each side is 4 units long. How many sides does the polygon have, and what is its perimeter?
72
37.5
22,777
Simplify first, then evaluate: $(1- \frac {2}{x+1})÷ \frac {x^{2}-x}{x^{2}-1}$, where $x=-2$.
\frac {3}{2}
67.96875
22,778
Given that the random variable $\xi$ follows a normal distribution $N(4, 6^2)$, and $P(\xi \leq 5) = 0.89$, find the probability $P(\xi \leq 3)$.
0.11
64.84375
22,779
Set $S = \{1, 2, 3, ..., 2005\}$ . If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$ .
15
45.3125
22,780
Randomly split 2.5 into the sum of two non-negative numbers, and round each number to its nearest integer. What is the probability that the sum of the two resulting integers is 3?
\frac{2}{5}
5.46875
22,781
A group consists of 4 male students and 3 female students. From this group, 4 people are selected to complete three different tasks, with the condition that at least two of the selected individuals must be female, and each task must have at least one person assigned to it. The number of different ways to select and assign these individuals is ____.
792
53.90625
22,782
The sampling group size is 10.
1000
0.78125
22,783
From the numbers 0, 1, 2, 3, and 4, select three different digits to form a three-digit number and calculate the total number of such numbers that are odd.
18
78.125
22,784
Given a complex number $z$ satisfying $z+ \bar{z}=6$ and $|z|=5$. $(1)$ Find the imaginary part of the complex number $z$; $(2)$ Find the real part of the complex number $\dfrac{z}{1-i}$.
\dfrac{7}{2}
52.34375
22,785
Observe the following set of equations, and based on the equations above, guess that ${{S}_{2n-1}}=\left( 4n-3 \right)\left( an+b \right)$, then ${{a}^{2}}+{{b}^{2}}=$ ____. ${{S}_{1}}=1$ ${{S}_{2}}=2+3+4=9$, ${{S}_{3}}=3+4+5+6+7=25$, ${{S}_{4}}=4+5+6+7+8+9+10=49$, (......)
25
74.21875
22,786
A farmer contracted several acres of fruit trees. This year, he invested 13,800 yuan, and the total fruit yield was 18,000 kilograms. The fruit sells for a yuan per kilogram in the market and b yuan per kilogram when sold directly from the orchard (b < a). The farmer transports the fruit to the market for sale, selling an average of 1,000 kilograms per day, requiring the help of 2 people, paying each 100 yuan per day, and the transportation cost of the agricultural vehicle and other taxes and fees average 200 yuan per day. (1) Use algebraic expressions involving a and b to represent the income from selling the fruit in both ways. (2) If a = 4.5 yuan, b = 4 yuan, and all the fruit is sold out within the same period using both methods, calculate which method of selling is better. (3) If the farmer strengthens orchard management, aiming for a net income of 72,000 yuan next year, and uses the better selling method from (2), what is the growth rate of the net income (Net income = Total income - Total expenses)?
20\%
45.3125
22,787
Given non-negative numbers $x$, $y$, and $z$ such that $x+y+z=2$, determine the minimum value of $\frac{1}{3}x^3+y^2+z$.
\frac{13}{12}
50.78125
22,788
Calculate the sum of the first five prime numbers that have a units digit of 3.
135
91.40625
22,789
Given the variance of a sample is $$s^{2}= \frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \_\_\_\_\_\_.
60
3.90625
22,790
In trapezoid EFGH, sides EF and GH are equal. It is known that EF = 12 units and GH = 10 units. Additionally, each of the non-parallel sides forms a right-angled triangle with half of the difference in lengths of EF and GH and a given leg of 6 units. Determine the perimeter of trapezoid EFGH.
22 + 2\sqrt{37}
91.40625
22,791
Consider a sequence $\{a_n\}$ where the sum of the first $n$ terms, $S_n$, satisfies $S_n = 2a_n - a_1$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence. (1) Find the general formula for the sequence $\{a_n\}$. (2) Let $b_n = \log_2 a_n$ and $c_n = \frac{3}{b_nb_{n+1}}$. Denote the sum of the first $n$ terms of the sequence $\{c_n\}$ as $T_n$. If $T_n < \frac{m}{3}$ holds for all positive integers $n$, determine the smallest possible positive integer value of $m$.
10
6.25
22,792
Bonnie constructs the frame of a cuboid using wire pieces. She uses eight pieces each of 8 inches for the length, and four pieces each of 10 inches for the width and height. Roark, on the other hand, uses 2-inch-long wire pieces to construct frames of smaller cuboids, all of the same size with dimensions 1 inch by 2 inches by 2 inches. The total volume of Roark's cuboids is the same as Bonnie's cuboid. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire?
\frac{9}{250}
31.25
22,793
Given a vertical wooden pillar, a rope is tied to its top, with 4 feet of the rope hanging down to the ground. Additionally, when pulling the rope, it runs out when 8 feet away from the base of the pillar. Determine the length of the rope.
10
72.65625
22,794
If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$ . Define a new positive real number, called $\phi_d$ , where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$ ). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$ , $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$ .
4038096
53.90625
22,795
You roll a fair 12-sided die repeatedly. The probability that all the prime numbers show up at least once before seeing any of the other numbers can be expressed as a fraction \( \frac{p}{q} \) in lowest terms. What is \( p+q \)?
793
47.65625
22,796
The value of \( 333 + 33 + 3 \) is:
369
91.40625
22,797
Given vectors $\overrightarrow{a}=(\sin x, \frac{3}{2})$ and $\overrightarrow{b}=(\cos x,-1)$. (1) When $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $2\cos ^{2}x-\sin 2x$. (2) Find the maximum value of $f(x)=( \overrightarrow{a}+ \overrightarrow{b}) \cdot \overrightarrow{b}$ on $\left[-\frac{\pi}{2},0\right]$.
\frac{1}{2}
50.78125
22,798
Calculate:<br/>$(1)(\frac{5}{7})×(-4\frac{2}{3})÷1\frac{2}{3}$;<br/>$(2)(-2\frac{1}{7})÷(-1.2)×(-1\frac{2}{5})$.
-\frac{5}{2}
48.4375
22,799
$$ \text{Consider the system of inequalities:} \begin{cases} x + 2y \leq 6 \\ 3x + y \geq 3 \\ x \leq 4 \\ y \geq 0 \end{cases} $$ Determine the number of units in the length of the longest side of the polygonal region formed by this system. Express your answer in simplest radical form.
2\sqrt{5}
50