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21,000 | A game wheel is divided into six regions labeled $A$, $B$, $C$, $D$, $E$, and $F$. The probability of the wheel stopping on region $A$ is $\frac{1}{3}$, the probability it stops on region $B$ is $\frac{1}{6}$, and the probability of it stopping on regions $C$ and $D$ are equal, as are the probabilities for regions $E$ and $F$. What is the probability of the wheel stopping in region $C$? | \frac{1}{8} | 96.09375 |
21,001 | Given the function $f(x)=|x-1|+|x+1|$.
(I) Solve the inequality $f(x) < 3$;
(II) If the minimum value of $f(x)$ is $m$, let $a > 0$, $b > 0$, and $a+b=m$, find the minimum value of $\frac{1}{a}+ \frac{2}{b}$. | \frac{3}{2}+ \sqrt{2} | 12.5 |
21,002 | Given that the golden ratio $m = \frac{{\sqrt{5}-1}}{2}$, calculate the value of $\frac{{\sin{42}°+m}}{{\cos{42}°}}$. | \sqrt{3} | 0.78125 |
21,003 | In the diagram, \( B, C \) and \( D \) lie on a straight line, with \(\angle ACD=100^{\circ}\), \(\angle ADB=x^{\circ}\), \(\angle ABD=2x^{\circ}\), and \(\angle DAC=\angle BAC=y^{\circ}\). The value of \( x \) is: | 20 | 32.8125 |
21,004 | Compute $20\left(\frac{256}{4} + \frac{64}{16} + \frac{16}{64} + 2\right)$. | 1405 | 72.65625 |
21,005 | A store increased the price of a certain Super VCD by 40% and then advertised a "10% discount and a free 50 yuan taxi fare" promotion. As a result, each Super VCD still made a profit of 340 yuan. What was the cost price of each Super VCD? | 1500 | 71.875 |
21,006 | The equation $\sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 7x = 2$ is to be simplified to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\] for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$ | 14 | 92.96875 |
21,007 | Given a circle \\(O: x^2 + y^2 = 2\\) and a line \\(l: y = kx - 2\\).
\\((1)\\) If line \\(l\\) intersects circle \\(O\\) at two distinct points \\(A\\) and \\(B\\), and \\(\angle AOB = \frac{\pi}{2}\\), find the value of \\(k\\).
\\((2)\\) If \\(EF\\) and \\(GH\\) are two perpendicular chords of the circle \\(O: x^2 + y^2 = 2\\), with the foot of the perpendicular being \\(M(1, \frac{\sqrt{2}}{2})\\), find the maximum area of the quadrilateral \\(EGFH\\). | \frac{5}{2} | 1.5625 |
21,008 | For what real number \\(m\\) is the complex number \\(z=m^{2}+m-2+(m^{2}-1)i\\)
\\((1)\\) a real number; \\((2)\\) an imaginary number; \\((3)\\) a pure imaginary number? | -2 | 96.875 |
21,009 | A gardener plans to place potted plants along both sides of a 150-meter-long path (including at both ends), with one pot every 2 meters. In total, \_\_\_\_\_\_ pots are needed. | 152 | 96.875 |
21,010 | If \( a = \log 25 \) and \( b = \log 49 \), compute
\[
5^{a/b} + 7^{b/a}.
\] | 12 | 94.53125 |
21,011 | Calculate $[x]$, where $x = -3.7 + 1.5$. | -3 | 96.09375 |
21,012 | Given that $cos({\frac{π}{4}-α})=\frac{3}{5}$, $sin({\frac{{5π}}{4}+β})=-\frac{{12}}{{13}}$, $α∈({\frac{π}{4},\frac{{3π}}{4}})$, $β∈({0,\frac{π}{4}})$, calculate the value of $\sin \left(\alpha +\beta \right)$. | \frac{56}{65} | 17.96875 |
21,013 | Given \(\frac{x+ \sqrt{2}i}{i}=y+i\), where \(x\), \(y\in\mathbb{R}\), and \(i\) is the imaginary unit, find \(|x-yi|\). | \sqrt{3} | 96.875 |
21,014 | 1. Given $\tan \frac{\alpha}{2} = \frac{1}{2}$, find the value of $\sin\left(\alpha + \frac{\pi}{6}\right)$.
2. Given $\alpha \in \left(\pi, \frac{3\pi}{2}\right)$ and $\cos\alpha = -\frac{5}{13}$, $\tan \frac{\beta}{2} = \frac{1}{3}$, find the value of $\cos\left(\frac{\alpha}{2} + \beta\right)$. | -\frac{17\sqrt{13}}{65} | 46.875 |
21,015 | Given the angle $\frac {19\pi}{5}$, express it in the form of $2k\pi+\alpha$ ($k\in\mathbb{Z}$), then determine the angle $\alpha$ that makes $|\alpha|$ the smallest. | -\frac {\pi}{5} | 33.59375 |
21,016 | In \\(\triangle ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. Given that \\(a=2\\), \\(c=3\\), and \\(\cos B= \dfrac {1}{4}\\),
\\((1)\\) find the value of \\(b\\);
\\((2)\\) find the value of \\(\sin C\\). | \dfrac {3 \sqrt {6}}{8} | 0 |
21,017 | Given that $(a + \frac{1}{a})^3 = 3$, find the value of $a^4 + \frac{1}{a^4}$.
A) $9^{1/3} - 4 \cdot 3^{1/3} + 2$
B) $9^{1/3} - 2 \cdot 3^{1/3} + 2$
C) $9^{1/3} + 4 \cdot 3^{1/3} + 2$
D) $4 \cdot 3^{1/3} - 9^{1/3}$ | 9^{1/3} - 4 \cdot 3^{1/3} + 2 | 63.28125 |
21,018 | (1) Simplify: $\dfrac {\tan (3\pi-\alpha)\cos (2\pi-\alpha)\sin (-\alpha+ \dfrac {3\pi}{2})}{\cos (-\alpha-\pi)\sin (-\pi+\alpha)\cos (\alpha+ \dfrac {5\pi}{2})}$;
(2) Given $\tan \alpha= \dfrac {1}{4}$, find the value of $\dfrac {1}{2\cos ^{2}\alpha -3\sin \alpha \cos \alpha }$. | \dfrac {17}{20} | 83.59375 |
21,019 | The volume of a given sphere is \(72\pi\) cubic inches. Find the surface area of the sphere. Express your answer in terms of \(\pi\). | 36\pi 2^{2/3} | 0 |
21,020 | Add $36_7 + 274_7.$ Express your answer in base 7. | 343_7 | 68.75 |
21,021 | The area of the region in the $xy$ -plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$ , for some integer $k$ . Find $k$ .
*Proposed by Michael Tang* | 210 | 31.25 |
21,022 | In the sequence \(1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \cdots, 200, 200, \cdots, 200\), each number \(n\) appears \(n\) times consecutively, where \(n \in \mathbf{N}\) and \(1 \leq n \leq 200\). What is the median of this sequence? | 142 | 98.4375 |
21,023 | Calculate $f(x) = 3x^5 + 5x^4 + 6x^3 - 8x^2 + 35x + 12$ using the Horner's Rule when $x = -2$. Find the value of $v_4$. | 83 | 69.53125 |
21,024 | Two concentric circles have radii $1$ and $4$ . Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles are externally tangent to the circle with radius $1$ . The radius of the six congruent circles can be written $\textstyle\frac{k+\sqrt m}n$ , where $k,m,$ and $n$ are integers with $k$ and $n$ relatively prime. Find $k+m+n$ .
[asy]
size(150);
defaultpen(linewidth(0.8));
real r = (sqrt(133)-9)/2;
draw(circle(origin,1)^^circle(origin,4));
for(int i=0;i<=2;i=i+1)
{
filldraw(circle(dir(90 + i*120)*(4-r),r),gray);
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for(int j=0;j<=2;j=j+1)
{
filldraw(circle(dir(30+j*120)*(1+r),r),darkgray);
}
[/asy] | 126 | 0.78125 |
21,025 | How many positive integer multiples of $210$ can be expressed in the form $6^{j} - 6^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 49$? | 600 | 37.5 |
21,026 | Let $x, y$ be two positive integers, with $x> y$ , such that $2n = x + y$ , where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$ | 66 | 31.25 |
21,027 | Let \( A \) be a point on the parabola \( y = x^2 - 4x \), and let \( B \) be a point on the line \( y = 2x - 3 \). Find the shortest possible distance \( AB \). | \frac{6\sqrt{5}}{5} | 3.90625 |
21,028 | Given triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $\frac{\sqrt{3}c}{\cos C} = \frac{a}{\cos(\frac{3\pi}{2} + A)}$.
(I) Find the value of $C$;
(II) If $\frac{c}{a} = 2$, $b = 4\sqrt{3}$, find the area of $\triangle ABC$. | 2\sqrt{15} - 2\sqrt{3} | 71.875 |
21,029 | The equations of the asymptotes of the hyperbola \\( \frac {x^{2}}{3}- \frac {y^{2}}{6}=1 \\) are \_\_\_\_\_\_, and its eccentricity is \_\_\_\_\_\_. | \sqrt {3} | 0 |
21,030 | Given that $a = \sin (2015\pi - \frac {\pi}{6})$ and the function $f(x) = \begin{cases} a^{x}, & x > 0 \\ f(-x), & x < 0 \end{cases}$, calculate the value of $f(\log_{2} \frac {1}{6})$. | \frac {1}{6} | 71.875 |
21,031 | In the xy-plane, what is the length of the shortest path from $(0,0)$ to $(15,20)$ that does not go inside the circle $(x-7)^2 + (y-9)^2 = 36$?
A) $2\sqrt{94} + 3\pi$
B) $2\sqrt{130} + 3\pi$
C) $2\sqrt{94} + 6\pi$
D) $2\sqrt{130} + 6\pi$
E) $94 + 3\pi$ | 2\sqrt{94} + 3\pi | 19.53125 |
21,032 | How many five-character license plates consist of two consonants, followed by two vowels, and ending with a digit? (For this problem, consider Y is not a vowel.) | 110,250 | 0 |
21,033 | Calculate:<br/>$(1)-3+5-\left(-2\right)$;<br/>$(2)-6÷\frac{1}{4}×(-4)$;<br/>$(3)(\frac{5}{6}-\frac{3}{4}+\frac{1}{3})×(-24)$;<br/>$(4)-1^{2023}-[4-(-3)^2]÷(\frac{2}{7}-1)$. | -8 | 21.875 |
21,034 | For what value of the parameter \( p \) will the sum of the squares of the roots of the equation
\[
x^{2}+(3 p-2) x-7 p-1=0
\]
be minimized? What is this minimum value? | \frac{53}{9} | 83.59375 |
21,035 | Determine the number of perfect cubic divisors in the product $1! \cdot 2! \cdot 3! \cdot \ldots \cdot 6!$. | 10 | 42.96875 |
21,036 | A school has eight identical copies of a particular book. At any given time, some of these copies are in the school library and some are with students. How many different ways are there for some of the books to be in the library and the rest to be with students if at least one book is in the library and at least one is with students? | 254 | 16.40625 |
21,037 | What is the smallest \( n > 1 \) for which the average of the first \( n \) (non-zero) squares is a square? | 337 | 100 |
21,038 | Given that the sequence $\{a_n\}$ is an arithmetic sequence, and if $\frac{a_{12}}{a_{11}} < -1$, find the maximum value of $n$ for which the sum of its first $n$ terms, $s_n$, is greater than $0$. | 21 | 55.46875 |
21,039 | For two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are given by $S_n$ and $T_n$ respectively, and $\frac{S_{n}}{T_{n}} = \frac{3n - 1}{2n + 3}$. Determine the ratio $\frac{a_{7}}{b_{7}}$. | \frac{38}{29} | 32.8125 |
21,040 | A list of five positive integers has a median of 4 and a mean of 15. What is the maximum possible value of the list's largest element? | 65 | 67.96875 |
21,041 | For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 | 63.28125 |
21,042 | Given that point $P$ moves on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, find the minimum distance from point $P$ to line $l$: $x+y-2\sqrt{5}=0$. | \frac{\sqrt{10}}{2} | 44.53125 |
21,043 | Ket $f(x) = x^{2} +ax + b$ . If for all nonzero real $x$ $$ f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right) $$ and the roots of $f(x) = 0$ are integers, what is the value of $a^{2}+b^{2}$ ? | 13 | 89.84375 |
21,044 | On a line \( l \) in space, points \( A \), \( B \), and \( C \) are sequentially located such that \( AB = 18 \) and \( BC = 14 \). Find the distance between lines \( l \) and \( m \) if the distances from points \( A \), \( B \), and \( C \) to line \( m \) are 12, 15, and 20, respectively. | 12 | 20.3125 |
21,045 | Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. Given that $\frac{{a^2 + c^2 - b^2}}{{\cos B}} = 4$. Find:<br/>
$(1)$ $ac$;<br/>
$(2)$ If $\frac{{2b\cos C - 2c\cos B}}{{b\cos C + c\cos B}} - \frac{c}{a} = 2$, find the area of $\triangle ABC$. | \frac{\sqrt{15}}{4} | 71.875 |
21,046 | Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, whose sums of the first $n$ terms are $A_n$ and $B_n$ respectively, and $\frac {A_{n}}{B_{n}} = \frac {7n+1}{4n+27}$, determine $\frac {a_{6}}{b_{6}}$. | \frac{78}{71} | 42.1875 |
21,047 | A set of numbers $\{-3, 1, 5, 8, 10, 14\}$ needs to be rearranged with new rules:
1. The largest isn't in the last position, but it is in one of the last four places.
2. The smallest isn’t in the first position, but it is in one of the first four places.
3. The median isn't in the middle positions.
What is the product of the second and fifth numbers after rearrangement?
A) $-21$
B) $24$
C) $-24$
D) $30$
E) None of the above | -24 | 24.21875 |
21,048 | Given a sequence ${a_n}$ whose first $n$ terms have a sum of $S_n$, and the point $(n, \frac{S_n}{n})$ lies on the line $y = \frac{1}{2}x + \frac{11}{2}$. Another sequence ${b_n}$ satisfies $b_{n+2} - 2b_{n+1} + b_n = 0$ ($n \in \mathbb{N}^*$), and $b_3 = 11$, with the sum of the first 9 terms being 153.
(I) Find the general term formulas for the sequences ${a_n}$ and ${b_n}$;
(II) Let $c_n = \frac{3}{(2a_n - 11)(2b_n - 1)}$. The sum of the first $n$ terms of the sequence ${c_n}$ is $T_n$. Find the maximum positive integer value $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all $n \in \mathbb{N}^*$. | 18 | 6.25 |
21,049 | Given a geometric sequence $\{a_n\}$ with the sum of its first n terms denoted as $S_n$, if $S_5$, $S_4$, and $S_6$ form an arithmetic sequence, determine the common ratio $q$ of the sequence $\{a_n\}$. | -2 | 92.1875 |
21,050 | Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $C: \frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, if a point $P$ on the right branch of the hyperbola $C$ satisfies $|PF\_1|=3|PF\_2|$ and $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}=a^{2}$, calculate the eccentricity of the hyperbola $C$. | \sqrt{2} | 50.78125 |
21,051 | The product underwent a price reduction from 25 yuan to 16 yuan. Calculate the average percentage reduction for each price reduction. | 20\% | 18.75 |
21,052 | In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 800$ and $AD = 1800$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $PQ$. | 500 | 23.4375 |
21,053 | Given that the odd function $f(x)$ is an increasing function defined on $\mathbb{R}$, and the sequence $x_n$ is an arithmetic sequence with a common difference of 2, satisfying $f(x_8) + f(x_9) + f(x_{10}) + f(x_{11}) = 0$, then the value of $x_{2011}$ is equal to. | 4003 | 94.53125 |
21,054 | Given $\cos \left(a- \frac{\pi}{6}\right) + \sin a = \frac{4 \sqrt{3}}{5}$, find the value of $\sin \left(a+ \frac{7\pi}{6}\right)$. | -\frac{4}{5} | 75.78125 |
21,055 | Given that the sum of the first 10 terms of a geometric sequence $\{a_n\}$ is 32 and the sum of the first 20 terms is 56, find the sum of the first 30 terms. | 74 | 86.71875 |
21,056 | A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $ . Calculate the distance between $ A $ and $ B $ (in a straight line).
| 29 | 62.5 |
21,057 | We have $ 23^2 = 529 $ ordered pairs $ (x, y) $ with $ x $ and $ y $ positive integers from 1 to 23, inclusive. How many of them have the property that $ x^2 + y^2 + x + y $ is a multiple of 6? | 225 | 67.1875 |
21,058 | There are 4 male students and 2 female students taking the photo. The male student named "甲" cannot stand at either end, and the female students must stand next to each other. Find the number of ways to arrange the students. | 144 | 29.6875 |
21,059 | Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n$, and $S_n + \frac{1}{3}a_n = 1 (n \in \mathbb{N}^*)$.
(I) Find the general term formula of the sequence ${a_n}$;
(II) Let $b_n = \log_4(1 - S_{n+1}) (n \in \mathbb{N}^*)$, $T_n = \frac{1}{b_1b_2} + \frac{1}{b_2b_3} + \dots + \frac{1}{b_nb_{n+1}}$, find the smallest positive integer $n$ such that $T_n \geq \frac{1007}{2016}$. | 2014 | 78.90625 |
21,060 | Given the parametric equation of line C1 as $$\begin{cases} x=2+t \\ y=t \end{cases}$$ (where t is the parameter), and the polar coordinate equation of the ellipse C2 as ρ²cos²θ + 9ρ²sin²θ = 9. Establish a rectangular coordinate system with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis.
1. Find the general equation of line C1 and the standard equation of ellipse C2 in rectangular coordinates.
2. If line C1 intersects with ellipse C2 at points A and B, and intersects with the x-axis at point E, find the value of |EA + EB|. | \frac{6\sqrt{3}}{5} | 25.78125 |
21,061 | Investigate the function $f(x)=e^{x}-e^{-x}+\sin x+1$, and let $a=\frac{2023+2024}{2}=2023.5$. Determine the value of $f(-2023)+f(-2022)+\ldots +f(2022)+f(2023)$. | 4047 | 81.25 |
21,062 | Two concentric circles have the same center, labeled $C$. The larger circle has a radius of $12$ units while the smaller circle has a radius of $7$ units. Determine the area of the ring formed between these two circles and also calculate the circumference of the larger circle. | 24\pi | 80.46875 |
21,063 | In the geometric sequence ${a_n}$, there exists a positive integer $m$ such that $a_m = 3$ and $a_{m+6} = 24$. Find the value of $a_{m+18}$. | 1536 | 71.09375 |
21,064 | In \\(\triangle ABC\\), \\(AB=BC\\), \\(\cos B=-\dfrac{7}{18}\\). If an ellipse with foci at points \\(A\\) and \\(B\\) passes through point \\(C\\), find the eccentricity of the ellipse. | \dfrac{3}{8} | 67.96875 |
21,065 | Given that $y$ is a multiple of $3456$, what is the greatest common divisor of $g(y) = (5y+4)(9y+1)(12y+6)(3y+9)$ and $y$? | 216 | 66.40625 |
21,066 | Among the subsets of the set $\{1,2, \cdots, 100\}$, calculate the maximum number of elements a subset can have if it does not contain any pair of numbers where one number is exactly three times the other. | 67 | 28.90625 |
21,067 | The equation ${{a}^{2}}{{x}^{2}}+(a+2){{y}^{2}}+2ax+a=0$ represents a circle. Find the possible values of $a$. | -1 | 39.84375 |
21,068 | Determine the number of ways to place 7 identical balls into 4 distinct boxes such that each box contains at least one ball. | 20 | 96.09375 |
21,069 | Seven distinct integers are picked at random from $\{1,2,3,\ldots,12\}$. What is the probability that, among those selected, the third smallest is $4$? | \frac{7}{33} | 5.46875 |
21,070 | The parametric equation of line $l$ is
$$
\begin{cases}
x= \frac { \sqrt {2}}{2}t \\
y= \frac { \sqrt {2}}{2}t+4 \sqrt {2}
\end{cases}
$$
(where $t$ is the parameter), and the polar equation of circle $c$ is $\rho=2\cos(\theta+ \frac{\pi}{4})$. Tangent lines are drawn from the points on the line to the circle; find the minimum length of these tangent lines. | 2\sqrt{6} | 59.375 |
21,071 | On March 12, 2016, the fourth Beijing Agriculture Carnival opened in Changping. The event was divided into seven sections: "Three Pavilions, Two Gardens, One Belt, and One Valley." The "Three Pavilions" refer to the Boutique Agriculture Pavilion, the Creative Agriculture Pavilion, and the Smart Agriculture Pavilion; the "Two Gardens" refer to the Theme Carnival Park and the Agricultural Experience Park; the "One Belt" refers to the Strawberry Leisure Experience Belt; and the "One Valley" refers to the Yanshou Ecological Sightseeing Valley. Due to limited time, a group of students plans to visit "One Pavilion, One Garden, One Belt, and One Valley." How many different routes can they take for their visit? (Answer with a number) | 144 | 3.125 |
21,072 | A circle is inscribed in a regular hexagon. A smaller hexagon has two non-adjacent sides coinciding with the sides of the larger hexagon and the remaining vertices touching the circle. What percentage of the area of the larger hexagon is the area of the smaller hexagon? Assume the side of the larger hexagon is twice the radius of the circle. | 25\% | 53.125 |
21,073 | Given $f(x)=x^{3}+ax^{2}+bx+a^{2}$, the extreme value at $x=1$ is $10$. Find the value of $a+b$. | -7 | 47.65625 |
21,074 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $\sin (A-B)+ \sin C= \sqrt {2}\sin A$.
(I) Find the value of angle $B$;
(II) If $b=2$, find the maximum value of $a^{2}+c^{2}$, and find the values of angles $A$ and $C$ when the maximum value is obtained. | 8+4 \sqrt {2} | 0 |
21,075 | Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$ , $v_i$ is connected to $v_j$ if and only if $i$ divides $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color. | 10 | 34.375 |
21,076 | Given that the area of $\triangle ABC$ is $2 \sqrt {3}$, $BC=2$, $C=120^{\circ}$, find the length of side $AB$. | 2 \sqrt {7} | 0 |
21,077 | What is the sum of all the odd integers between $200$ and $600$? | 80000 | 61.71875 |
21,078 | Given $\overrightarrow{a}=(1+\cos \omega x,-1)$, $\overrightarrow{b}=( \sqrt {3},\sin \omega x)$ ($\omega > 0$), and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, with the smallest positive period of $f(x)$ being $2\pi$.
(1) Find the expression of the function $f(x)$.
(2) Let $\theta\in(0, \frac {\pi}{2})$, and $f(\theta)= \sqrt {3}+ \frac {6}{5}$, find the value of $\cos \theta$. | \frac {3 \sqrt {3}+4}{10} | 0 |
21,079 | Quadrilateral $ABCD$ has right angles at $B$ and $D$. The length of the diagonal $AC$ is $5$. If two sides of $ABCD$ have integer lengths and one of these lengths is an odd integer, determine the area of $ABCD$. | 12 | 50 |
21,080 | James used a calculator to find the product $0.005 \times 3.24$. He forgot to enter the decimal points, and the calculator showed $1620$. If James had entered the decimal points correctly, what would the answer have been?
A) $0.00162$
B) $0.0162$
C) $0.162$
D) $0.01620$
E) $0.1620$ | 0.0162 | 58.59375 |
21,081 | Given that $60$ people are wearing sunglasses, $40$ people are wearing caps, and a randomly selected person wearing a cap has a $\frac{2}{5}$ probability of also wearing sunglasses, find the probability that a person wearing sunglasses is also wearing a cap. | \frac{4}{15} | 89.84375 |
21,082 | Given the function $f(x) = 4\cos(\omega x - \frac{\pi}{6})\sin \omega x - \cos(2\omega x + \pi)$, where $\omega > 0$.
(I) Find the range of the function $y = f(x)$.
(II) If $f(x)$ is an increasing function on the interval $[-\frac{3\pi}{2}, \frac{\pi}{2}]$, find the maximum value of $\omega$. | \frac{1}{6} | 21.875 |
21,083 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. It is known that $c^{2}=a^{2}+b^{2}-4bc\cos C$, and $A-C= \frac {\pi}{2}$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) Find the value of $\cos \left(B+ \frac {\pi}{3}\right)$. | \frac {4-3 \sqrt {3}}{10} | 0 |
21,084 | Evaluate the expression \[(5^{1001} + 6^{1002})^2 - (5^{1001} - 6^{1002})^2\] and express it in the form \(k \cdot 30^{1001}\) for some integer \(k\). | 24 | 94.53125 |
21,085 | Let $ ABC$ be an isosceles triangle with $ \left|AB\right| \equal{} \left|AC\right| \equal{} 10$ and $ \left|BC\right| \equal{} 12$ . $ P$ and $ R$ are points on $ \left[BC\right]$ such that $ \left|BP\right| \equal{} \left|RC\right| \equal{} 3$ . $ S$ and $ T$ are midpoints of $ \left[AB\right]$ and $ \left[AC\right]$ , respectively. If $ M$ and $ N$ are the foot of perpendiculars from $ S$ and $ R$ to $ PT$ , then find $ \left|MN\right|$ . | $ \frac {10\sqrt {13} }{13} $ | 0 |
21,086 | If the letters in the English word "good" are written in the wrong order, then the total number of possible mistakes is _____ | 11 | 50.78125 |
21,087 | A 12-hour digital clock has a glitch such that whenever it is supposed to display a 5, it mistakenly displays a 7. For example, when it is 5:15 PM the clock incorrectly shows 7:77 PM. What fraction of the day will the clock show the correct time? | \frac{33}{40} | 6.25 |
21,088 | A nine-digit integer is formed by repeating a three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form? | 1001001 | 45.3125 |
21,089 | Given $\tan \alpha = 3$, evaluate the expression $2\sin ^{2}\alpha + 4\sin \alpha \cos \alpha - 9\cos ^{2}\alpha$. | \dfrac{21}{10} | 91.40625 |
21,090 | Given the geometric sequence $\{a_{n}\}$, $a_{2}$ and $a_{18}$ are the two roots of the equation $x^{2}+15x+16=0$, find the value of $a_{3}a_{10}a_{17}$. | -64 | 10.15625 |
21,091 | Team A and Team B each have $n$ members. It is known that each member of Team A shakes hands with each member of Team B exactly once (members within the same team do not shake hands). From these $n^2$ handshakes, two are randomly selected. Let event A be the event that exactly 3 members are involved in these two handshakes. If the probability of event A happening, $P$, is less than $\frac{1}{10}$, then the minimum value of $n$ is \_\_\_\_\_\_. | 20 | 28.125 |
21,092 | What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 1980 = 0$ has integral solutions? | 290 | 1.5625 |
21,093 | Suppose
\[\frac{1}{x^3-7x^2+11x+15} = \frac{A}{x-5} + \frac{B}{x+3} + \frac{C}{(x+3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | \frac{1}{64} | 85.15625 |
21,094 | If 2035 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 50 | 0 |
21,095 | Let $a$ and $b$ be randomly selected numbers from the set $\{1,2,3,4\}$. Given the line $l:y=ax+b$ and the circle $O:x^{2}+y^{2}=1$, the probability that the line $l$ intersects the circle $O$ is ______. The expected number of intersection points between the line $l$ and the circle $O$ is ______. | \frac{5}{4} | 43.75 |
21,096 | In square \(ABCD\), \(P\) is the midpoint of \(DC\) and \(Q\) is the midpoint of \(AD\). If the area of the quadrilateral \(QBCP\) is 15, what is the area of square \(ABCD\)? | 24 | 69.53125 |
21,097 | A point $P$ is chosen at random from the interior of a square $ABCD$. What is the probability that the triangle $ABP$ has a greater area than each of the triangles $BCP$, $CDP$, and $DAP$? | \frac{1}{4} | 47.65625 |
21,098 | The value of \(2 \frac{1}{10} + 3 \frac{11}{100} + 4 \frac{111}{1000}\) is | 9.321 | 16.40625 |
21,099 | In $\triangle ABC$, it is known that $\cos C+(\cos A- \sqrt {3}\sin A)\cos B=0$.
(1) Find the measure of angle $B$;
(2) If $\sin (A- \frac {π}{3})= \frac {3}{5}$, find $\sin 2C$. | \frac {24+7 \sqrt {3}}{50} | 0 |
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