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40.3k
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100
|
|---|---|---|---|
19,700 |
What integer value will satisfy the equation $$ 14^2 \times 35^2 = 10^2 \times (M - 10)^2 \ ? $$
|
59
| 36.71875 |
19,701 |
Given the lengths of the three sides of $\triangle ABC$ are $AB=7$, $BC=5$, and $CA=6$, the value of $\overrightarrow {AB}\cdot \overrightarrow {BC}$ is \_\_\_\_\_\_.
|
-19
| 72.65625 |
19,702 |
Chelsea goes to La Verde's at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among \( n \) bags, so that each bag contains at most 13 pounds of fruit. What is the minimum \( n \) for which this is possible?
|
75
| 0.78125 |
19,703 |
On bookshelf A, there are 4 English books and 2 Chinese books, while on bookshelf B, there are 2 English books and 3 Chinese books.
$(Ⅰ)$ Without replacement, 2 books are taken from bookshelf A, one at a time. Find the probability of getting an English book on the first draw and still getting an English book on the second draw.
$(Ⅱ)$ First, 2 books are randomly taken from bookshelf B and placed on bookshelf A. Then, 2 books are randomly taken from bookshelf A. Find the probability of getting 2 English books from bookshelf A.
|
\frac{93}{280}
| 80.46875 |
19,704 |
Compute $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$ . (The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$ .)
*Proposed by Tristan Shin*
|
2019
| 51.5625 |
19,705 |
The product of the first three terms of a geometric sequence is 2, the product of the last three terms is 4, and the product of all terms is 64. Calculate the number of terms in this sequence.
|
12
| 34.375 |
19,706 |
Consider a $3 \times 3$ array where each row and each column is an arithmetic sequence with three terms. The first term of the first row is $3$, and the last term of the first row is $15$. Similarly, the first term of the last row is $9$, and the last term of the last row is $33$. Determine the value of the center square, labeled $Y$.
|
15
| 68.75 |
19,707 |
Given that in triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively, and b = 3, c = 1, and A = 2B.
(1) Find the value of a;
(2) Find the value of sin(A + $\frac{\pi}{4}$).
|
\frac{4 - \sqrt{2}}{6}
| 64.0625 |
19,708 |
Observe the sequence: (1), (4, 7), (10, 13, 16), (19, 22, 25, 28), ..., then 2008 is in the $\boxed{\text{th}}$ group.
|
37
| 10.9375 |
19,709 |
Given that $[x]$ is the greatest integer less than or equal to $x$, calculate $\sum_{N=1}^{1024}\left[\log _{2} N\right]$.
|
8204
| 97.65625 |
19,710 |
Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \epsilon$, the area of $WXYZ$ can be expressed as the quadratic polynomial \[Area(WXYZ) = \gamma \epsilon - \delta \epsilon^2.\]
Determine the coefficient $\delta = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
|
17
| 38.28125 |
19,711 |
A square has a side length of $40\sqrt{3}$ cm. Calculate the length of the diagonal of the square and the area of a circle inscribed within it.
|
1200\pi
| 87.5 |
19,712 |
A right circular cone is sliced into five pieces by planes parallel to its base. Each slice has the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece?
|
\frac{37}{61}
| 38.28125 |
19,713 |
In square \(R S T U\), a quarter-circle arc with center \(S\) is drawn from \(T\) to \(R\). A point \(P\) on this arc is 1 unit from \(TU\) and 8 units from \(RU\). What is the length of the side of square \(RSTU\)?
|
13
| 14.0625 |
19,714 |
If $\frac{2013 \times 2013}{2014 \times 2014 + 2012} = \frac{n}{m}$ (where $m$ and $n$ are coprime natural numbers), then what is the value of $m + n$?
|
1343
| 77.34375 |
19,715 |
Given a complex number $z=3+bi\left(b=R\right)$, and $\left(1+3i\right)\cdot z$ is an imaginary number.<br/>$(1)$ Find the complex number $z$;<br/>$(2)$ If $ω=\frac{z}{{2+i}}$, find the complex number $\omega$ and its modulus $|\omega|$.
|
\sqrt{2}
| 82.8125 |
19,716 |
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$ , we color red the foot of the perpendicular from $C$ to $\ell$ . The set of red points enclose a bounded region of area $\mathcal{A}$ . Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exceeding $\mathcal A$ ).
*Proposed by Yang Liu*
|
157
| 63.28125 |
19,717 |
What is the sum of the last three digits of each term in the following part of the Fibonacci Factorial Series: $1!+2!+3!+5!+8!+13!+21!$?
|
249
| 3.125 |
19,718 |
Given vectors $\overrightarrow{a}=(\sqrt{2}\cos \omega x,1)$ and $\overrightarrow{b}=(2\sin (\omega x+ \frac{\pi}{4}),-1)$ where $\frac{1}{4}\leqslant \omega\leqslant \frac{3}{2}$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, and the graph of $f(x)$ has an axis of symmetry at $x= \frac{5\pi}{8}$.
$(1)$ Find the value of $f( \frac{3}{4}\pi)$;
$(2)$ If $f( \frac{\alpha}{2}- \frac{\pi}{8})= \frac{\sqrt{2}}{3}$ and $f( \frac{\beta}{2}- \frac{\pi}{8})= \frac{2\sqrt{2}}{3}$, and $\alpha,\beta\in(-\frac{\pi}{2}, \frac{\pi}{2})$, find the value of $\cos (\alpha-\beta)$.
|
\frac{2\sqrt{10}+2}{9}
| 63.28125 |
19,719 |
Given that there are 25 cities in the County of Maplewood, and the average population per city lies between $6,200$ and $6,800$, estimate the total population of all the cities in the County of Maplewood.
|
162,500
| 0 |
19,720 |
(This question has a total of 10 points)
From a group consisting of 5 male doctors and 4 female doctors, select 3 doctors to form a medical team. The requirement is that the team must include both male and female doctors. How many different team formation plans are there?
|
70
| 26.5625 |
19,721 |
For real numbers $a$ and $b$ , define $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$ Find the smallest possible value of the expression $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$
|
4 \sqrt{2018}
| 50.78125 |
19,722 |
In a local government meeting, leaders from five different companies are present. It is known that two representatives are from Company A, and each of the remaining four companies has one representative attending. If three individuals give a speech at the meeting, how many possible combinations are there where these three speakers come from three different companies?
|
16
| 38.28125 |
19,723 |
Select 4 individuals from a group of 6 to visit Paris, London, Sydney, and Moscow, with the requirement that each city is visited by one person, each individual visits only one city, and among these 6 individuals, individuals A and B shall not visit Paris. The total number of different selection schemes is __________. (Answer with a number)
|
240
| 57.03125 |
19,724 |
When point P moves on the circle $C: x^2 - 4x + y^2 = 0$, there exist two fixed points $A(1, 0)$ and $B(a, 0)$, such that $|PB| = 2|PA|$, then $a = \ $.
|
-2
| 89.0625 |
19,725 |
Consider the equation $x^2 + 16x = 100$. The positive solution can also be written in the form $\sqrt{a} - b$ for positive natural numbers $a$ and $b$. Find $a + b$.
|
172
| 64.0625 |
19,726 |
Given that a triangular corner with side lengths DB=EB=1.5 is cut from an equilateral triangle ABC of side length 4.5, determine the perimeter of the remaining quadrilateral.
|
12
| 25 |
19,727 |
Given the parabola $C$: $y^{2}=4x$ with focus $F$, two lines $l_{1}$ and $l_{2}$ are drawn through point $F$. Line $l_{1}$ intersects the parabola $C$ at points $A$ and $B$, and line $l_{2}$ intersects the parabola $C$ at points $D$ and $E$. If the sum of the squares of the slopes of $l_{1}$ and $l_{2}$ is $1$, then find the minimum value of $|AB|+|DE|$.
|
24
| 14.84375 |
19,728 |
In triangle ABC, where AB = 24 and BC = 18, find the largest possible value of $\tan A$.
|
\frac{3\sqrt{7}}{7}
| 11.71875 |
19,729 |
The number of 4-digit integers with distinct digits, whose first and last digits' absolute difference is 2, is between 1000 and 9999.
|
840
| 2.34375 |
19,730 |
(In the coordinate system and parametric equations optional question) In the polar coordinate system, it is known that the line $l: p(\sin\theta - \cos\theta) = a$ divides the region enclosed by the curve $C: p = 2\cos\theta$ into two parts with equal area. Find the value of the constant $a$.
|
-1
| 41.40625 |
19,731 |
How many 12 step paths are there from point $A$ to point $C$ which pass through point $B$ on a grid, where $A$ is at the top left corner, $B$ is 5 steps to the right and 2 steps down from $A$, and $C$ is 7 steps to the right and 4 steps down from $A$?
|
126
| 72.65625 |
19,732 |
Given that a certain product requires $6$ processing steps, where $2$ of these steps must be consecutive and another $2$ steps cannot be consecutive, calculate the number of possible processing sequences.
|
144
| 47.65625 |
19,733 |
A summer camp organizes 5 high school students to visit five universities, including Peking University and Tsinghua University. Determine the number of different ways in which exactly 2 students choose Peking University.
|
640
| 63.28125 |
19,734 |
Given that tetrahedron PQRS has edge lengths PQ = 3, PR = 4, PS = 5, QR = 5, QS = √34, and RS = √41, calculate the volume of tetrahedron PQRS.
|
10
| 2.34375 |
19,735 |
Simplify $(2 \times 10^9) - (6 \times 10^7) \div (2 \times 10^2)$.
|
1999700000
| 34.375 |
19,736 |
Let $O$ be the origin. There exists a scalar $m$ such that for any points $E,$ $F,$ $G,$ and $H$ satisfying the vector equation:
\[4 \overrightarrow{OE} - 3 \overrightarrow{OF} + 2 \overrightarrow{OG} + m \overrightarrow{OH} = \mathbf{0},\]
the four points $E,$ $F,$ $G,$ and $H$ are coplanar. Find the value of $m.$
|
-3
| 83.59375 |
19,737 |
A zoo houses five different pairs of animals, each pair consisting of one male and one female. To maintain a feeding order by gender alternation, if the initial animal fed is a male lion, how many distinct sequences can the zookeeper follow to feed all the animals?
|
2880
| 55.46875 |
19,738 |
Let $\mathbf{p}$ be the projection of vector $\mathbf{v}$ onto vector $\mathbf{u},$ and let $\mathbf{q}$ be the projection of $\mathbf{p}$ onto $\mathbf{u}.$ If $\frac{\|\mathbf{p}\|}{\|\mathbf{v}\|} = \frac{3}{4},$ then find $\frac{\|\mathbf{q}\|}{\|\mathbf{u}\|}.$
|
\frac{9}{16}
| 8.59375 |
19,739 |
The distance between location A and location B originally required a utility pole to be installed every 45m, including the two poles at both ends, making a total of 53 poles. Now, the plan has been changed to install a pole every 60m. Excluding the two poles at both ends, how many poles in between do not need to be moved?
|
12
| 79.6875 |
19,740 |
An eight-sided die is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$?
|
48
| 0.78125 |
19,741 |
If $\tan \alpha = -\frac{4}{3}$, then the value of $\sin^2\alpha + 2\sin \alpha \cos \alpha$ is ______.
|
-\frac{8}{25}
| 90.625 |
19,742 |
Given the function f(x) = $\sqrt{|x+2|+|6-x|-m}$, whose domain is R,
(I) Find the range of the real number m;
(II) If the maximum value of the real number m is n, and the positive numbers a and b satisfy $\frac{8}{3a+b}$ + $\frac{2}{a+2b}$ = n, find the minimum value of 2a + $\frac{3}{2}$b.
|
\frac{9}{8}
| 3.90625 |
19,743 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $A \neq B$.
1. Find the value of $\frac{a \sin A - b \sin B}{\sin (A-B)}$.
2. If the area of $\triangle ABC$ is $1$ and $\tan C = 2$, find the value of $a+b$.
|
\sqrt{5} + 1
| 80.46875 |
19,744 |
Take a standard set of dominoes and remove all duplicates and blanks. Then consider the remaining 15 dominoes as fractions. The dominoes are arranged in such a way that the sum of all fractions in each row equals 10. However, unlike proper fractions, you are allowed to use as many improper fractions (such as \(\frac{4}{3}, \frac{5}{2}, \frac{6}{1}\)) as you wish, as long as the sum in each row equals 10.
|
10
| 4.6875 |
19,745 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=3$, $c=2\sqrt{3}$, and $A=30^{\circ}$, find the values of angles $B$, $C$, and side $a$.
|
\sqrt{3}
| 0 |
19,746 |
Given the function $f(x)=3\sin(2x-\frac{π}{3})-2\cos^{2}(x-\frac{π}{6})+1$, the graph of function $f(x)$ is shifted to the left by $\frac{π}{6}$ units, resulting in the graph of function $g(x)$. Find $\sin (2x_{1}+2x_{2})$, where $x_{1}$ and $x_{2}$ are the two roots of the equation $g(x)=a$ in the interval $[0,\frac{π}{2}]$.
|
-\frac{3}{5}
| 38.28125 |
19,747 |
Find the integer $n$, $12 \le n \le 18$, such that \[n \equiv 9001 \pmod{7}.\]
|
13
| 64.0625 |
19,748 |
Reading material: After studying square roots, Kang Kang found that some expressions containing square roots can be written as the square of another expression, such as $3+2\sqrt{2}=({1+\sqrt{2}})^2$. With his good thinking skills, Kang Kang made the following exploration: Let $a+b\sqrt{2}=({m+n\sqrt{2}})^2$ (where $a$, $b$, $m$, $n$ are all positive integers), then $a+b\sqrt{2}=m^2+2n^2+2mn\sqrt{2}$ (rational and irrational numbers correspondingly equal), therefore $a=m^{2}+2n^{2}$, $b=2mn$. In this way, Kang Kang found a method to transform the expression $a+b\sqrt{2}$ into a square form. Please follow Kang Kang's method to explore and solve the following problems:
$(1)$ When $a$, $b$, $m$, $n$ are all positive integers, if $a+b\sqrt{3}=({c+d\sqrt{3}})^2$, express $a$ and $b$ in terms of $c$ and $d$: $a=$______, $b=$______;
$(2)$ If $7-4\sqrt{3}=({e-f\sqrt{3}})^2$, and $e$, $f$ are both positive integers, simplify $7-4\sqrt{3}$;
$(3)$ Simplify: $\sqrt{7+\sqrt{21-\sqrt{80}}}$.
|
1+\sqrt{5}
| 5.46875 |
19,749 |
In the hexagon $ABCDEF$ with vertices $A, B, C, D, E, F$, all internal triangles dividing the hexagon are similar to isosceles triangle $ABC$, where $AB = AC$. Among these triangles, there are $10$ smallest triangles each with area $2$, and the area of triangle $ABC$ is $80$. Determine the area of the quadrilateral $DBCE$.
A) 60
B) 65
C) 70
D) 75
E) 80
|
70
| 27.34375 |
19,750 |
A sphere is inscribed in a right circular cylinder. The height of the cylinder is 12 inches, and the diameter of its base is 10 inches. Find the volume of the inscribed sphere. Express your answer in terms of $\pi$.
|
\frac{500}{3} \pi
| 71.09375 |
19,751 |
Given a sequence $\{a_n\}$ that satisfies $a_na_{n+1}a_{n+2}a_{n+3}=24$, and $a_1=1$, $a_2=2$, $a_3=3$, find the sum $a_1+a_2+a_3+\ldots+a_{2013}$.
|
5031
| 89.84375 |
19,752 |
Any six points are taken inside or on a rectangle with dimensions $2 \times 1$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Determine the value of $b$.
|
\frac{\sqrt{5}}{2}
| 91.40625 |
19,753 |
Given that \\(\alpha\\) and \\(\beta\\) are acute angles, and \\(\cos \alpha= \frac{\sqrt{5}}{5}\\), \\(\sin (\alpha+\beta)= \frac{3}{5}\\), find the value of \\(\cos \beta\\.
|
\frac{2\sqrt{5}}{25}
| 14.84375 |
19,754 |
How many squares are shown in the drawing?
|
30
| 17.96875 |
19,755 |
Consider a $5 \times 5$ grid of squares, where each square is either colored blue or left blank. The design on the grid is considered symmetric if it remains unchanged under a 90° rotation around the center. How many symmetric designs can be created if there must be at least one blue square but not all squares can be blue?
|
30
| 13.28125 |
19,756 |
Evaluate $97 \times 97$ in your head.
|
9409
| 96.875 |
19,757 |
Use the method of random simulation to estimate the probability that it will rain on exactly two of the three days. Using a calculator, generate random integer values between 0 and 9, where 1, 2, 3, and 4 represent raining days and 5, 6, 7, 8, 9, and 0 signify non-raining days. Then, group every three random numbers to represent the weather for these three days. After conducting the random simulation, the following 20 groups of random numbers were produced:
907 966 191 925 271 932 812 458 569 683
431 257 393 027 556 488 730 113 537 989
Estimate the probability that exactly two days out of three will have rain.
|
0.25
| 87.5 |
19,758 |
Each of the numbers 1, 2, 3, and 4 is substituted, in some order, for \( p, q, r \), and \( s \). Find the greatest possible value of \( p^q + r^s \).
|
83
| 2.34375 |
19,759 |
Josh writes the numbers $2,4,6,\dots,198,200$. He marks out $2$, skips $4$, marks out $6$ and continues this pattern of skipping one number and marking the next until he reaches the end of the list. He then returns to the beginning and repeats this pattern on the new list of remaining numbers, continuing until only one number remains. What is that number?
|
128
| 23.4375 |
19,760 |
(1) Given $0 < x < \frac{1}{2}$, find the maximum value of $y= \frac{1}{2}x(1-2x)$;
(2) Given $x > 0$, find the maximum value of $y=2-x- \frac{4}{x}$;
(3) Given $x$, $y\in\mathbb{R}_{+}$, and $x+y=4$, find the minimum value of $\frac{1}{x}+ \frac{3}{y}$.
|
1+ \frac{ \sqrt{3}}{2}
| 53.125 |
19,761 |
If the distance from the foci of the hyperbola $C$ to its asymptotes is equal to the length of $C$'s real semi-axis, then the eccentricity of $C$ is \_\_\_\_\_\_.
|
\sqrt{2}
| 86.71875 |
19,762 |
Given a right prism with all vertices on the same sphere, with a height of $4$ and a volume of $32$, the surface area of this sphere is ______.
|
32\pi
| 62.5 |
19,763 |
The eccentricity of the ellipse given that the slope of line $l$ is $2$, and it intersects the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ at two different points, where the projections of these two intersection points on the $x$-axis are exactly the two foci of the ellipse.
|
\sqrt{2}-1
| 30.46875 |
19,764 |
Given that the sum of the polynomials $A$ and $B$ is $12x^{2}y+2xy+5$, where $B=3x^{2}y-5xy+x+7$. Find:<br/>$(1)$ The polynomial $A$;<br/>$(2)$ When $x$ takes any value, the value of the expression $2A-\left(A+3B\right)$ is a constant. Find the value of $y$.
|
\frac{2}{11}
| 84.375 |
19,765 |
12 real numbers x and y satisfy \( 1 + \cos^2(2x + 3y - 1) = \frac{x^2 + y^2 + 2(x+1)(1-y)}{x-y+1} \). Find the minimum value of xy.
|
\frac{1}{25}
| 3.125 |
19,766 |
Under the call for the development of the western region by the country, a certain western enterprise received a $4$ million yuan interest-free loan for equipment renewal. It is predicted that after the equipment renewal, the income of the enterprise in the first month is $200,000$. In the following $5$ months, the income of each month increases by $20\%$ compared to the previous month. Starting from the $7$th month, the income of each month increases by $20,000$ more than the previous month. Then, starting from the use of the new equipment, the enterprise needs ______ months to repay the $4$ million interest-free loan with the income obtained. (Round the result to the nearest whole number)
|
10
| 84.375 |
19,767 |
What is the area of a quadrilateral with vertices at $(0,0)$, $(4,3)$, $(7,0)$, and $(4,4)$?
|
3.5
| 0.78125 |
19,768 |
Two circles, both with the same radius $r$ , are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$ , so that $|AB|=|BC|=|CD|=14\text{cm}$ . Another line intersects the circles at $E,F$ , respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$ . Find the radius $r$ .
|
13
| 0.78125 |
19,769 |
Increasing the radius of a cylinder by $4$ units results in an increase in volume by $z$ cubic units. Increasing the height of the cylinder by $10$ units also results in an increase in volume by the same $z$ cubic units. If the original radius is $3$ units, what is the original height of the cylinder?
|
2.25
| 0 |
19,770 |
A pentagon is obtained by joining, in order, the points \((0,0)\), \((1,2)\), \((3,3)\), \((4,1)\), \((2,0)\), and back to \((0,0)\). The perimeter of the pentagon can be written in the form \(a + b\sqrt{c} + d\sqrt{e}\), where \(a\), \(b\), \(c\), \(d\), and \(e\) are whole numbers. Find \(a+b+c+d+e\).
|
11
| 90.625 |
19,771 |
If $a = \log 8$ and $b = \log 25,$ compute
\[5^{a/b} + 2^{b/a}.\]
|
2 \sqrt{2} + 5^{2/3}
| 0 |
19,772 |
Given an ellipse C centered at the origin with its left focus F($-\sqrt{3}$, 0) and right vertex A(2, 0).
(1) Find the standard equation of ellipse C;
(2) A line l with a slope of $\frac{1}{2}$ intersects ellipse C at points A and B. Find the maximum value of the chord length |AB| and the equation of line l at this time.
|
\sqrt{10}
| 0 |
19,773 |
Let set $I=\{1,2,3,4,5,6\}$, and sets $A, B \subseteq I$. If set $A$ contains 3 elements, set $B$ contains at least 2 elements, and all elements in $B$ are not less than the largest element in $A$, calculate the number of pairs of sets $A$ and $B$ that satisfy these conditions.
|
29
| 4.6875 |
19,774 |
Let $P$, $Q$, and $R$ be points on a circle of radius $24$. If $\angle PRQ = 40^\circ$, what is the circumference of the minor arc $PQ$? Express your answer in terms of $\pi$.
|
\frac{32\pi}{3}
| 93.75 |
19,775 |
Given that the terminal side of angle $\alpha$ passes through point $P\left(\sin \frac{7\pi }{6},\cos \frac{11\pi }{6}\right)$, find the value of $\frac{1}{3\sin ^{2}\alpha -\cos ^{2}\alpha }=\_\_\_\_\_\_\_\_\_\_.$
|
\frac{1}{2}
| 80.46875 |
19,776 |
Determine the number of ways to arrange the letters of the word "SUCCESS".
|
420
| 12.5 |
19,777 |
In an obtuse triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a=7$, $b=3$, and $\cos C=\frac{11}{14}$,
$(1)$ Find the length of side $c$ and the measure of angle $A$;
$(2)$ Calculate the value of $\sin \left(2C-\frac{\pi }{6}\right)$.
|
\frac{71}{98}
| 61.71875 |
19,778 |
Let \( S = \left\{\left(s_{1}, s_{2}, \cdots, s_{6}\right) \mid s_{i} \in \{0, 1\}\right\} \). For any \( x, y \in S \) where \( x = \left(x_{1}, x_{2}, \cdots, x_{6}\right) \) and \( y = \left(y_{1}, y_{2}, \cdots, y_{6}\right) \), define:
(1) \( x = y \) if and only if \( \sum_{i=1}^{6}\left(x_{i} - y_{i}\right)^{2} = 0 \);
(2) \( x y = x_{1} y_{1} + x_{2} y_{2} + \cdots + x_{6} y_{6} \).
If a non-empty set \( T \subseteq S \) satisfies \( u v \neq 0 \) for any \( u, v \in T \) where \( u \neq v \), then the maximum number of elements in set \( T \) is:
|
32
| 48.4375 |
19,779 |
A line passes through point $Q(\frac{1}{3}, \frac{4}{3})$ and intersects the hyperbola $x^{2}- \frac{y^{2}}{4}=1$ at points $A$ and $B$. Point $Q$ is the midpoint of chord $AB$.
1. Find the equation of the line containing $AB$.
2. Find the length of $|AB|$.
|
\frac{8\sqrt{2}}{3}
| 60.15625 |
19,780 |
Given $$x \in \left(- \frac{\pi}{2}, \frac{\pi}{2}\right)$$ and $$\sin x + \cos x = \frac{1}{5}$$, calculate the value of $\tan 2x$.
|
-\frac{24}{7}
| 53.125 |
19,781 |
Given that $\theta$ is an angle in the second quadrant, and $\tan 2\theta = -2\sqrt{2}$.
(1) Find the value of $\tan \theta$.
(2) Calculate the value of $\frac {2\cos^{2} \frac {\theta}{2}-\sin\theta-\tan \frac {5\pi}{4}}{\sqrt {2}\sin(\theta + \frac {\pi}{4})}$.
|
3 + 2\sqrt{2}
| 53.90625 |
19,782 |
Let $ABCDEF$ be a regular hexagon with each side length $s$. Points $G$, $H$, $I$, $J$, $K$, and $L$ are the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The segments $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ form another hexagon inside $ABCDEF$. Find the ratio of the area of this inner hexagon to the area of hexagon $ABCDEF$, expressed as a fraction in its simplest form.
|
\frac{3}{4}
| 19.53125 |
19,783 |
Given the parabola $y^{2}=4x$, let $AB$ and $CD$ be two chords perpendicular to each other and passing through its focus. Find the value of $\frac{1}{|AB|}+\frac{1}{|CD|}$.
|
\frac{1}{4}
| 79.6875 |
19,784 |
(In the preliminaries of optimal method and experimental design) When using the 0.618 method to find the optimal amount to add in an experiment, if the current range of excellence is $[628, 774]$ and the good point is 718, then the value of the addition point for the current experiment is ________.
|
684
| 41.40625 |
19,785 |
Let line $l_1: x + my + 6 = 0$ and line $l_2: (m - 2)x + 3y + 2m = 0$. When $m = \_\_\_\_\_\_$, $l_1 \parallel l_2$.
|
-1
| 42.1875 |
19,786 |
Given that the equation $2kx+2m=6-2x+nk$ has a solution independent of $k$, the value of $4m+2n$ is ______.
|
12
| 93.75 |
19,787 |
Let the function \( f(x) = x^3 + a x^2 + b x + c \), where \( a \), \( b \), and \( c \) are non-zero integers. If \( f(a) = a^3 \) and \( f(b) = b^3 \), find the value of \( c \).
|
16
| 22.65625 |
19,788 |
If parallelogram ABCD has an area of 100 square meters, and E and G are the midpoints of sides AD and CD, respectively, while F is the midpoint of side BC, find the area of quadrilateral DEFG.
|
25
| 53.125 |
19,789 |
Convert the following radians to degrees and degrees to radians:
(1) $$\frac {\pi}{12}$$ = \_\_\_\_\_\_ ; (2) $$\frac {13\pi}{6}$$ = \_\_\_\_\_\_ ; (3) -$$\frac {5}{12}$$π = \_\_\_\_\_\_ .
(4) 36° = \_\_\_\_\_\_ rad; (5) -105° = \_\_\_\_\_\_ rad.
|
- \frac {7}{12}\pi
| 0 |
19,790 |
Given the function $f(x)=x^{3}-x^{2}+1$.
$(1)$ Find the equation of the tangent line to the function $f(x)$ at the point $(1,f(1))$;
$(2)$ Find the extreme values of the function $f(x)$.
|
\dfrac {23}{27}
| 50.78125 |
19,791 |
Xiao Ming throws a die with uniform density three times and observes the number of points on the upper face each time. It is known that the numbers of points in the three throws are all different. Calculate the probability that the sum of the three numbers of points does not exceed $8$.
|
\frac{1}{5}
| 20.3125 |
19,792 |
Given that $\alpha$ and $\beta$ are acute angles, and $\cos(\alpha + \beta) = \frac{3}{5}$, $\sin\alpha = \frac{5}{13}$, find the value of $\cos\beta$.
|
\frac{56}{65}
| 59.375 |
19,793 |
Let the complex number $z=-3\cos \theta + i\sin \theta$ (where $i$ is the imaginary unit).
(1) When $\theta= \frac {4}{3}\pi$, find the value of $|z|$;
(2) When $\theta\in\left[ \frac {\pi}{2},\pi\right]$, the complex number $z_{1}=\cos \theta - i\sin \theta$, and $z_{1}z$ is a pure imaginary number, find the value of $\theta$.
|
\frac {2\pi}{3}
| 92.96875 |
19,794 |
Find the product of $0.5$ and $0.8$.
|
0.4
| 97.65625 |
19,795 |
Define $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$ . Determine the smallest positive integer $a$ such that $f\left(a\right)=f\left(a+2\right)$ .
*2017 CCA Math Bonanza Lightning Round #2.4*
|
13
| 7.8125 |
19,796 |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $\sqrt{3}c\sin A = a\cos C$.
$(I)$ Find the value of $C$;
$(II)$ If $c=2a$ and $b=2\sqrt{3}$, find the area of $\triangle ABC$.
|
\frac{\sqrt{15} - \sqrt{3}}{2}
| 30.46875 |
19,797 |
Consider a modified octahedron with an additional ring of vertices. There are 4 vertices on the top ring, 8 on the middle ring, and 4 on the bottom ring. An ant starts at the highest top vertex and walks down to one of four vertices on the next level down (the middle ring). From there, without returning to the previous vertex, the ant selects one of the 3 adjacent vertices (excluding the one it came from) and continues to the next level. What is the probability that the ant reaches the bottom central vertex?
|
\frac{1}{3}
| 30.46875 |
19,798 |
The real numbers $a$, $b$, and $c$ satisfy the equation $({a}^{2}+\frac{{b}^{2}}{4}+\frac{{c}^{2}}{9}=1)$. Find the maximum value of $a+b+c$.
|
\sqrt{14}
| 69.53125 |
19,799 |
John borrows $2000$ from Mary, who charges an interest rate of $6\%$ per month (which compounds monthly). What is the least integer number of months after which John will owe more than triple what he borrowed?
|
19
| 6.25 |
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