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40.3k
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100
17,800
Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\angle BAC = 60^{\circ}$, $D$ is a point on side $BC$ such that $AD = \sqrt{7}$, and $BD:DC = 2c:b$, then the minimum value of the area of $\triangle ABC$ is ____.
2\sqrt{3}
0.78125
17,801
Keisha's basketball team must decide on a new uniform. The seventh-graders will pick the color of the shorts (black, gold, or red) and the eighth-graders will pick the color of the jersey (black, white, gold, or blue), and each group will not confer with the other. Additionally, the ninth-graders will choose whether to include a cap (yes or no). If for all items, each possible choice is equally likely to be chosen, what is the probability that the shorts will be a different color than the jersey, given that the team decides to include a cap?
\frac{5}{6}
4.6875
17,802
In an experiment, a certain constant \( c \) is measured to be 2.43865 with an error range of \(\pm 0.00312\). The experimenter wants to publish the value of \( c \), with each digit being significant. This means that regardless of how large \( c \) is, the announced value of \( c \) (with \( n \) digits) must match the first \( n \) digits of the true value of \( c \). What is the most precise value of \( c \) that the experimenter can publish?
2.44
16.40625
17,803
Select 3 distinct numbers from the set {1, 2, ..., 10} such that they do not form an arithmetic sequence. How many such selections are possible? (Answer with a number).
100
89.0625
17,804
Given that P and Q are points on the graphs of the functions $2x-y+6=0$ and $y=2\ln x+2$ respectively, find the minimum value of the line segment |PQ|.
\frac{6\sqrt{5}}{5}
18.75
17,805
Given $sin(x+\frac{π}{12})=-\frac{1}{4}$, find the value of $cos(\frac{5π}{6}-2x)$.
-\frac{7}{8}
70.3125
17,806
Given the function $y=a^{x+4}+2$ with $a \gt 0$ and $a \gt 1$, find the value of $\sin \alpha$ if the terminal side of angle $\alpha$ passes through a point on the graph of the function.
\frac{3}{5}
67.96875
17,807
The volume of the solid of revolution generated by rotating the region bounded by the curve $y= \sqrt{2x}$, the line $y=x-4$, and the x-axis around the x-axis is \_\_\_\_\_\_.
\frac{128\pi}{3}
10.9375
17,808
Given the general term formula of the sequence $\{a_n\}$ as $a_n= \frac{3^{-n}+2^{-n}+(-1)^{n}(3^{-n}-2^{-n})}{2}$, where $n=1$, $2$, $...$, determine the value of $\lim_{n \rightarrow \infty }(a_1+a_2+...+a_n)$.
\frac{19}{24}
71.09375
17,809
For some positive integer $n$ , the sum of all odd positive integers between $n^2-n$ and $n^2+n$ is a number between $9000$ and $10000$ , inclusive. Compute $n$ . *2020 CCA Math Bonanza Lightning Round #3.1*
21
91.40625
17,810
Arrange all the four-digit numbers formed using $1, 2, 3,$ and $4$, each used exactly once, in ascending order. What is the difference between the 23rd number and the 21st number?
99
6.25
17,811
Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \in [-3,-1)$, $f(x)=-(x+2)^{2}$, and when $x \in [-1,3)$, $f(x)=x$. Find the value of $f(1)+f(2)+f(3)+\ldots+f(2016)$.
336
61.71875
17,812
In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \sqrt{2}$, the spherical distance between points $A$ and $C$ is ( ).
$\frac{\pi}{2}$
0
17,813
What is the constant term of the expansion of $\left(5x + \frac{2}{5x}\right)^8$?
1120
97.65625
17,814
How many of the positive divisors of 3960 are multiples of 5?
24
96.875
17,815
Among the two-digit numbers less than 20, the largest prime number is ____, and the largest composite number is ____.
18
69.53125
17,816
There are 4 distinct balls and 4 distinct boxes. All the balls need to be placed into the boxes. (1) If exactly one box remains empty, how many different arrangements are possible? (2) If exactly two boxes remain empty, how many different arrangements are possible?
84
95.3125
17,817
Determine the product of all positive integer values of \( c \) such that \( 9x^2 + 24x + c = 0 \) has real roots.
20922789888000
73.4375
17,818
On a circular track, Alphonse is at point \( A \) and Beryl is diametrically opposite at point \( B \). Alphonse runs counterclockwise and Beryl runs clockwise. They run at constant, but different, speeds. After running for a while they notice that when they pass each other it is always at the same three places on the track. What is the ratio of their speeds?
2:1
10.9375
17,819
In January 1859, an eight-year-old boy dropped a newly-hatched eel into a well in Sweden. The eel, named Ale, finally died in August 2014. How many years old was Åle when it died?
155
75
17,820
For the fractional equation involving $x$, $\frac{x+m}{x-2}+\frac{1}{2-x}=3$, if it has a root with an increase, then $m=\_\_\_\_\_\_$.
-1
37.5
17,821
Given the line $l: \lambda x-y-\lambda +1=0$ and the circle $C: x^{2}+y^{2}-4y=0$, calculate the minimum value of $|AB|$.
2\sqrt{2}
39.84375
17,822
Using the property that in finding the limit of the ratio of two infinitesimals, they can be replaced with their equivalent infinitesimals (property II), find the following limits: 1) \(\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 3 x}\) 2) \(\lim _{x \rightarrow 0} \frac{\tan^{2} 2 x}{\sin ^{2} \frac{x}{3}}\) 3) \(\lim _{x \rightarrow 0} \frac{x \sin 2 x}{(\arctan 5 x)^{2}}\) 4) \(\lim _{n \rightarrow+\infty} \frac{\tan^{3} \frac{1}{n} \cdot \arctan \frac{3}{n \sqrt{n}}}{\frac{2}{n^{3}} \cdot \tan \frac{1}{\sqrt{n}} \cdot \arcsin \frac{5}{n}}\)
\frac{3}{10}
72.65625
17,823
Mia and Jake ordered a pizza cut into 12 equally-sized slices. Mia wanted a plain pizza but Jake wanted pepperoni on one-third of the pizza. The cost of a plain pizza was $12, and the additional cost for pepperoni on part of the pizza was $3. Jake ate all the pepperoni slices and three plain slices. Mia ate the rest. Each paid for what they ate. How much more did Jake pay than Mia?
2.5
17.1875
17,824
A box contains $3$ cards labeled with $1$, $2$, and $3$ respectively. A card is randomly drawn from the box, its number recorded, and then returned to the box. This process is repeated. The probability that at least one of the numbers drawn is even is _______.
\frac{5}{9}
45.3125
17,825
In $\triangle ABC$, the internal angles $A$, $B$, and $C$ satisfy the equation $$2(\tan B + \tan C) = \frac{\tan B}{\cos C} + \frac{\tan C}{\cos B}$$. Find the minimum value of $\cos A$.
\frac{1}{2}
54.6875
17,826
If $3 \in \{a, a^2 - 2a\}$, then the value of the real number $a$ is ______.
-1
8.59375
17,827
Point $P$ is on the circle $C_{1}: x^{2}+y^{2}-8x-4y+11=0$, and point $Q$ is on the circle $C_{2}: x^{2}+y^{2}+4x+2y+1=0$. What is the minimum value of $|PQ|$?
3\sqrt{5} - 5
89.0625
17,828
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the largest integer not exceeding $x$. Evaluate the expression $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \ldots + \lfloor \log_{2}1023 \rfloor + \lfloor \log_{2}1024 \rfloor$.
8204
97.65625
17,829
Given the real numbers \( x_1, x_2, \ldots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left|x_k - x_{k+1}\right| = 2001 \). Let \( y_k = \frac{1}{k} \left( x_1 + x_2 + \cdots + x_k \right) \) for \( k = 1, 2, \ldots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_k - y_{k+1} \right| \).
2000
34.375
17,830
What integer \( n \) satisfies \( 0 \leq n < 151 \) and $$150n \equiv 93 \pmod{151}~?$$
58
25
17,831
What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$?
16.67\%
94.53125
17,832
A cube with an edge length of 1 and its circumscribed sphere intersect with a plane to form a cross section that is a circle and an inscribed equilateral triangle. What is the distance from the center of the sphere to the plane of the cross section?
$\frac{\sqrt{3}}{6}$
0
17,833
Simplify first, then evaluate: $(1-\frac{2}{{m+1}})\div \frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\tan 60^{\circ}-1$.
\frac{3-\sqrt{3}}{3}
18.75
17,834
Two workers were assigned to produce a batch of identical parts; after the first worked for \(a\) hours and the second for \(0.6a\) hours, it turned out that they had completed \(\frac{5}{n}\) of the entire job. After working together for another \(0.6a\) hours, they found that they still had \(\frac{1}{n}\) of the batch left to produce. How many hours will it take for each of them, working separately, to complete the whole job? The number \(n\) is a natural number; find it.
10
1.5625
17,835
Let $O$ be the origin, and $F$ be the right focus of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. The line $l$ passing through $F$ intersects the ellipse $C$ at points $A$ and $B$. Two points $P$ and $Q$ on the ellipse satisfy $$ \overrightarrow{O P}+\overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O P}+\overrightarrow{O Q}=0, $$ and the points $P, A, Q,$ and $B$ are concyclic. Find the eccentricity of the ellipse $C$.
\frac{\sqrt{2}}{2}
49.21875
17,836
In how many ways can you arrange the digits of 11250 to get a five-digit number that is a multiple of 2?
24
41.40625
17,837
In a math interest class, the teacher gave a problem for everyone to discuss: "Given real numbers $a$, $b$, $c$ not all equal to zero satisfying $a+b+c=0$, find the maximum value of $\frac{|a+2b+3c|}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}."$ Jia quickly offered his opinion: Isn't this just the Cauchy inequality? We can directly solve it; Yi: I am not very clear about the Cauchy inequality, but I think we can solve the problem by constructing the dot product of vectors; Bing: I am willing to try elimination, to see if it will be easier with fewer variables; Ding: This is similar to the distance formula in analytic geometry, can we try to generalize it to space. Smart you can try to use their methods, or design your own approach to find the correct maximum value as ______.
\sqrt{2}
1.5625
17,838
Given triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, satisfying $\frac{a}{{2\cos A}}=\frac{b}{{3\cos B}}=\frac{c}{{6\cos C}}$, then $\sin 2A=$____.
\frac{3\sqrt{11}}{10}
47.65625
17,839
Given the polar equation of curve $C$ is $\rho\sin^2\theta-8\cos\theta=0$, with the pole as the origin of the Cartesian coordinate system $xOy$, and the polar axis as the positive half-axis of $x$. In the Cartesian coordinate system, a line $l$ with an inclination angle $\alpha$ passes through point $P(2,0)$. $(1)$ Write the Cartesian equation of curve $C$ and the parametric equation of line $l$; $(2)$ Suppose the polar coordinates of points $Q$ and $G$ are $(2, \frac{3\pi}{2})$ and $(2,\pi)$, respectively. If line $l$ passes through point $Q$ and intersects curve $C$ at points $A$ and $B$, find the area of $\triangle GAB$.
16\sqrt{2}
17.96875
17,840
Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$ , what is the largest possible area of the triangle?
25
97.65625
17,841
A cube with a side length of 10 is divided into 1000 smaller cubes with a side length of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (in any of the three directions) is zero. In one of the small cubes (denoted as \( A \)), the number one is written. Three layers pass through cube \( A \), each parallel to the faces of the larger cube (with each layer having a thickness of 1). Find the sum of all the numbers in the cubes that are not in these layers.
-1
27.34375
17,842
Given there are ten steps from the first floor to the second floor, calculate the total number of ways Xiao Ming can go from the first floor to the second floor.
89
57.03125
17,843
Given the set $A=\{x|1 < x < k\}$ and the set $B=\{y|y=2x-5, x \in A\}$, if $A \cap B = \{x|1 < x < 2\}$, find the value of the real number $k$.
3.5
0
17,844
Given the function $y=\cos(2x+\frac{\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\frac{\pi}{6}$ units.
\frac{11\pi}{24}
3.125
17,845
In the Cartesian coordinate plane, a polar coordinate system is established with the origin as the pole and the non-negative half of the x-axis as the polar axis. It is known that point A has polar coordinates $$( \sqrt{2}, \frac{\pi}{4})$$, and the parametric equation of line $l$ is: $$\begin{cases} x= \frac{3}{2} - \frac{\sqrt{2}}{2}t \\ y= \frac{1}{2} + \frac{\sqrt{2}}{2}t \end{cases}$$ (where $t$ is the parameter), and point A lies on line $l$. (Ⅰ) Find the corresponding parameter $t$ of point A; (Ⅱ) If the parametric equation of curve C is: $$\begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$$ (where $\theta$ is the parameter), and line $l$ intersects curve C at points M and N, find the length of line segment |MN|.
\frac{4\sqrt{2}}{5}
10.9375
17,846
The positive integer \( N \) is the smallest one whose digits add to 41. What is the sum of the digits of \( N + 2021 \)? A) 10 B) 12 C) 16 D) 2021 E) 4042
10
89.0625
17,847
Given 8 teams, of which 3 are weak teams, they are randomly divided into two groups, A and B, with 4 teams in each group. Find: (1) The probability that one of the groups A or B has exactly 2 weak teams. (2) The probability that group A has at least 2 weak teams.
\frac{1}{2}
33.59375
17,848
Given a circle $O$ with radius $1$, $PA$ and $PB$ are two tangents to the circle, and $A$ and $B$ are the points of tangency. The minimum value of $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is \_\_\_\_\_\_.
-3+2\sqrt{2}
6.25
17,849
Given that $\cos (α- \frac {π}{6})+ \sin α= \frac {4}{5} \sqrt {3}$, and $α \in (0, \frac {π}{3})$, find the value of $\sin (α+ \frac {5}{12}π)$.
\frac{7 \sqrt{2}}{10}
23.4375
17,850
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$, and $a_1=1$, $a_3$, $a_{13}$ form a geometric sequence. Find the minimum value of $\frac{2S_n+8}{a_n+3}$ for all positive integers $n$.
\frac{5}{2}
12.5
17,851
Given $$\frac {1}{3} \leq a \leq 1$$, if the function $f(x) = ax^2 - 2x + 1$ has a domain of $[1, 3]$. (1) Find the minimum value of $f(x)$ in its domain (expressed in terms of $a$); (2) Let the maximum value of $f(x)$ in its domain be $M(a)$, and the minimum value be $N(a)$. Find the minimum value of $M(a) - N(a)$.
\frac {1}{2}
10.15625
17,852
Given the integers from 1 to 25, Ajibola wants to remove the smallest possible number of integers so that the remaining integers can be split into two groups with equal products. What is the sum of the numbers which Ajibola removes?
79
17.96875
17,853
Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exactly enough cheese if each customer will buy a portion of cheese of weight exactly equal to the average weight of the previous purchases. Could it happen that the salesgirl can declare, after each of the first $10$ customers has made their purchase, that there just enough cheese for the next $10$ customers? If so, how much cheese will be left in the store after the first $10$ customers have made their purchases? (The average weight of a series of purchases is the total weight of the cheese sold divided by the number of purchases.)
10
57.8125
17,854
How many ways are there to distribute 7 balls into 4 boxes if the balls are not distinguishable and neither are the boxes?
11
22.65625
17,855
Given a cube with a side length of \(4\), if a solid cube of side length \(1\) is removed from each corner, calculate the total number of edges of the resulting structure.
36
64.0625
17,856
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{1}{2}t\\ y=\frac{\sqrt{3}}{2}t\end{array}\right.$ (where $t$ is a parameter). Taking $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $ρsin^{2}\frac{θ}{2}=1$. $(1)$ Find the rectangular coordinate equation of $C$ and the polar coordinates of the intersection points of $C$ with the $y$-axis. $(2)$ If the line $l$ intersects $C$ at points $A$ and $B$, and intersects the $x$-axis at point $P$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.
\frac{\sqrt{7}}{4}
26.5625
17,857
Calculate the values of: (1) $8^{\frac{2}{3}} - (0.5)^{-3} + \left(\frac{1}{\sqrt{3}}\right)^{-2} \times \left(\frac{81}{16}\right)^{-\frac{1}{4}}$; (2) $\log 5 \cdot \log 8000 + (\log 2^{\sqrt{3}})^2 + e^{\ln 1} + \ln(e \sqrt{e})$.
\frac{11}{2}
0.78125
17,858
Given vectors $\vec{m}=(2a\cos x,\sin x)$ and $\vec{n}=(\cos x,b\cos x)$, the function $f(x)=\vec{m}\cdot \vec{n}-\frac{\sqrt{3}}{2}$, and $f(x)$ has a y-intercept of $\frac{\sqrt{3}}{2}$, and the closest highest point to the y-axis has coordinates $\left(\frac{\pi}{12},1\right)$. $(1)$ Find the values of $a$ and $b$; $(2)$ Move the graph of the function $f(x)$ to the left by $\varphi (\varphi > 0)$ units, and then stretch the x-coordinates of the points on the graph by a factor of $2$ without changing the y-coordinates, to obtain the graph of the function $y=\sin x$. Find the minimum value of $\varphi$.
\frac{5\pi}{6}
43.75
17,859
In the arithmetic sequence $\{a_{n}\}$, $d=-2$, $a_{1}+a_{4}+a_{7}+…+a_{31}=50$. Find the value of $a_{2}+a_{6}+a_{10}+…+a_{42}$.
-82
71.875
17,860
Given that $a$ and $b$ are coefficients, and the difference between $ax^2+2xy-x$ and $3x^2-2bxy+3y$ does not contain a quadratic term, find the value of $a^2-4b$.
13
90.625
17,861
What is the product of all real numbers that are tripled when added to their reciprocals?
-\frac{1}{2}
81.25
17,862
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that \[g(g(x - y)) = g(x) g(y) - g(x) + g(y) - 2xy\]for all $x,$ $y.$ Find the sum of all possible values of $g(1).$
-\sqrt{2}
0
17,863
In △ABC, B = $$\frac{\pi}{3}$$, AB = 8, BC = 5, find the area of the circumcircle of △ABC.
\frac{49\pi}{3}
56.25
17,864
Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 24x^2 + 50x - 42 = 0.$ Find the value of $\frac{a}{\frac{1}{a}+bc} + \frac{b}{\frac{1}{b}+ca} + \frac{c}{\frac{1}{c}+ab}.$
\frac{476}{43}
90.625
17,865
Find the minimum value of \[(13 - x)(11 - x)(13 + x)(11 + x) + 1000.\]
424
9.375
17,866
Consider the function $f(x) = x^3 + 3\sqrt{x}$. Evaluate $3f(3) - 2f(9)$.
-1395 + 9\sqrt{3}
27.34375
17,867
The number $n$ is a four-digit positive integer and is the product of three distinct prime factors $x$, $y$ and $10y+x$, where $x$ and $y$ are each less than 10. What is the largest possible value of $n$?
1533
2.34375
17,868
The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$ , while its charge is $\frac12$ at pH $9.6$ . Charge increases linearly with pH. What is the isoelectric point of glycine?
5.97
46.09375
17,869
Let $g(x, y)$ be the function for the set of ordered pairs of positive coprime integers such that: \begin{align*} g(x, x) &= x, \\ g(x, y) &= g(y, x), \quad \text{and} \\ (x + y) g(x, y) &= y g(x, x + y). \end{align*} Calculate $g(15, 33)$.
165
1.5625
17,870
What is the smallest positive multiple of $17$ that is $3$ more than a multiple of $71$?
1139
8.59375
17,871
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a^2 = b(b + c)$. Find the value of $\frac{B}{A}$.
\frac{1}{2}
28.125
17,872
Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{2\sqrt{2}}{3}$, the line $y=\frac{1}{2}$ intersects $C$ at points $A$ and $B$, where $|AB|=3\sqrt{3}$. $(1)$ Find the equation of $C$; $(2)$ Let the left and right foci of $C$ be $F_{1}$ and $F_{2}$ respectively. The line passing through $F_{1}$ with a slope of $1$ intersects $C$ at points $G$ and $H$. Find the perimeter of $\triangle F_{2}GH$.
12
70.3125
17,873
The decimal representation of \(\dfrac{1}{25^{10}}\) consists of a string of zeros after the decimal point, followed by non-zero digits. Find the number of zeros in that initial string of zeros after the decimal point.
20
14.84375
17,874
There are \( n \) distinct lines in the plane. One of these lines intersects exactly 5 of the \( n \) lines, another one intersects exactly 9 of the \( n \) lines, and yet another one intersects exactly 11 of them. Which of the following is the smallest possible value of \( n \)?
12
65.625
17,875
Determine the smallest positive real number \(x\) such that \[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 7.\]
\frac{71}{8}
5.46875
17,876
Two real numbers $a$ and $b$ satisfy $a+b=8$ and $a^3+b^3=172$. Compute the value of $ab$.
\frac{85}{6}
96.875
17,877
Determine the slope of line $\overline{CD}$ where circles given by $x^2 + y^2 - 6x + 4y - 5 = 0$ and $x^2 + y^2 - 10x + 16y + 24 = 0$ intersect at points $C$ and $D$.
\frac{1}{3}
28.125
17,878
Rectangle $ABCD$ has area $4032$. An ellipse with area $4032\pi$ passes through points $A$ and $C$ and has foci at points $B$ and $D$. Determine the perimeter of the rectangle.
8\sqrt{2016}
0
17,879
A certain type of rice must comply with the normal distribution of weight $(kg)$, denoted as $\xi ~N(10,{σ}^{2})$, according to national regulations. Based on inspection results, $P(9.9≤\xi≤10.1)=0.96$. A company purchases a bag of this packaged rice as a welfare gift for each employee. If the company has $2000$ employees, the approximate number of employees receiving rice weighing less than $9.9kg$ is \_\_\_\_\_\_\_\_\_\_\_.
40
71.875
17,880
In Montana, 500 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the ``Soda'' sector of the graph is $200^\circ$, to the nearest whole degree. How many of the people surveyed chose ``Soda''? Express your answer as a whole number.
278
94.53125
17,881
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, the symmetric point $Q$ of the right focus $F(c, 0)$ with respect to the line $y = \dfrac{b}{c}x$ is on the ellipse. Find the eccentricity of the ellipse.
\dfrac{\sqrt{2}}{2}
51.5625
17,882
Suppose $x$ and $y$ are positive real numbers such that $x^2 - 3xy + 4y^2 = 12$. Find the maximum possible value of $x^2 + 3xy + 4y^2$.
84
14.84375
17,883
Let $g(x) = 2x^7 - 3x^3 + 4x - 8.$ If $g(6) = 12,$ find $g(-6).$
-28
22.65625
17,884
Let \(a\), \(b\), \(c\), and \(d\) be nonnegative numbers whose sum is 150. Find the largest possible value of \[ ab + bc + cd. \]
5625
96.875
17,885
Given the function $f(x)=e^{x}$, $g(x)=-x^{2}+2x+b(b\in\mathbb{R})$, denote $h(x)=f(x)- \frac {1}{f(x)}$ (I) Determine the parity of $h(x)$ and write down the monotonic interval of $h(x)$, no proof required; (II) For any $x\in[1,2]$, there exist $x_{1}$, $x_{2}\in[1,2]$ such that $f(x)\leqslant f(x_{1})$, $g(x)\leqslant g(x_{2})$. If $f(x_{1})=g(x_{2})$, find the value of the real number $b$.
e^{2}-1
0
17,886
Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form?
-30
64.84375
17,887
Rotate a square around a line that lies on one of its sides to form a cylinder. If the volume of the cylinder is $27\pi \text{cm}^3$, then the lateral surface area of the cylinder is _________ $\text{cm}^2$.
18\pi
88.28125
17,888
Given a function $f(x)$ defined on $\mathbb{R}$ that is an odd function, and the period of the function $f(2x+1)$ is 5, if $f(1) = 5$, calculate the value of $f(2009) + f(2010)$.
-5
96.09375
17,889
Using systematic sampling method to select 32 people from 960 for a questionnaire survey, they are randomly numbered from 1 to 960. After grouping, the number drawn by simple random sampling in the first group is 9. Among the 32 people drawn, those with numbers in the interval [1,450] will fill out questionnaire A, those in the interval [451,750] will fill out questionnaire B, and the rest will fill out questionnaire C. How many of the drawn people will fill out questionnaire B?
10
87.5
17,890
Given that $O$ is the coordinate origin, and vectors $\overrightarrow{OA}=(\sin α,1)$, $\overrightarrow{OB}=(\cos α,0)$, $\overrightarrow{OC}=(-\sin α,2)$, and point $P$ satisfies $\overrightarrow{AB}=\overrightarrow{BP}$. (I) Denote function $f(α)=\overrightarrow{PB} \cdot \overrightarrow{CA}$, find the minimum positive period of function $f(α)$; (II) If points $O$, $P$, and $C$ are collinear, find the value of $| \overrightarrow{OA}+ \overrightarrow{OB}|$.
\frac{\sqrt{74}}{5}
88.28125
17,891
Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7?
\frac{148}{590}
0
17,892
Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$ .
804
94.53125
17,893
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 9$ and $CP = 27.$ If $\tan \angle APD = 2,$ then find $AB.$
27
8.59375
17,894
What is the smallest number that can be written as a sum of $2$ squares in $3$ ways?
325
90.625
17,895
Given that $\sin(x+\pi) + \cos(x-\pi) = \frac{1}{2}, x \in (0, \pi)$. (1) Find the value of $\sin x \cos x$; (2) Find the value of $\sin x - \cos x$.
\frac{\sqrt{7}}{2}
92.1875
17,896
Let $g$ be a function satisfying $g(x^2y) = g(x)/y^2$ for all positive real numbers $x$ and $y$. If $g(800) = 4$, what is the value of $g(7200)$?
\frac{4}{81}
39.84375
17,897
Given the set $A=\{x|ax^{2}-x+1=0,a\in\mathbb{R},x\in\mathbb{R}\}$. If the proposition "Set $A$ contains only one element" is true, find the value of $a$.
\frac{1}{4}
96.875
17,898
What would the 25th number be in a numeric system where the base is five?
100
89.0625
17,899
Given that a group of students is sitting evenly spaced around a circular table and a bag containing 120 pieces of candy is circulated among them, determine the possible number of students if Sam picks both the first and a final piece after the bag has completed exactly two full rounds.
60
34.375