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21,800 | Find a positive integer that is divisible by 20 and whose cube root is a number between 8.2 and 8.3. | 560 | 92.1875 |
21,801 | Painting the surface of a large metal ball requires 2.4 kilograms of paint. If this large metal ball is melted down to make 64 identical small metal balls, without considering any loss, the amount of paint needed to coat the surfaces of these small metal balls is \_\_\_\_\_\_ kilograms. | 9.6 | 6.25 |
21,802 | Given a 2x3 rectangle with six unit squares, the lower left corner at the origin, find the value of $c$ such that a slanted line extending from $(c,0)$ to $(4,4)$ divides the entire region into two regions of equal area. | \frac{5}{2} | 39.84375 |
21,803 | Jia and Yi are playing a guessing game with the following rules: It is known that there are five cards, each with the numbers $1-\left( \frac{1}{2} \right)^n$ ($n\in \mathbf{N}^*, 1\leqslant n\leqslant 5$) written on them. Now, Jia and Yi each randomly draw one card and then try to guess who has the larger number based on the number they drew. After looking at his number, Jia thinks for a moment and says: "I don't know who has the larger number"; after hearing Jia's judgment, Yi thinks for a moment and says: "I also don't know who has the larger number." Assuming that the reasoning made by Jia and Yi is correct, then the number Yi holds is $\boxed{\frac{7}{8}}$. | \frac{7}{8} | 58.59375 |
21,804 | When two fair 12-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that both the two-digit number $ab$ (where $a$ and $b$ are digits) and each of $a$ and $b$ individually are divisible by 4? | \frac{1}{16} | 65.625 |
21,805 | Calculate $180 \div \left( 12 + 9 \times 3 - 4 \right)$. | \frac{36}{7} | 32.8125 |
21,806 | A marathon is $26$ miles and $400$ yards. One mile equals $1760$ yards.
Mark has run fifteen marathons in his life. If the total distance Mark covered in these marathons is $m$ miles and $y$ yards, where $0 \leq y < 1760$, what is the value of $y$? | 720 | 0.78125 |
21,807 | Using the systematic sampling method to select 32 people for a questionnaire survey from 960 people, determine the number of people among the 32 whose numbers fall within the interval [200, 480]. | 10 | 19.53125 |
21,808 | Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards). | 89 | 98.4375 |
21,809 | The first few rows of a new sequence are given as follows:
- Row 1: $3$
- Row 2: $6, 6, 6, 6$
- Row 3: $9, 9, 9, 9, 9, 9$
- Row 4: $12, 12, 12, 12, 12, 12, 12, 12$
What is the value of the $40^{\mathrm{th}}$ number if this arrangement were continued? | 18 | 34.375 |
21,810 | Given a function $f(x)=\log _{a}\left(\sqrt {x^{2}+1}+x\right)+\dfrac{1}{a^{x}-1}+\dfrac{3}{2}$, where $a > 0$ and $a \neq 1$. If $f\left(\log _{3}b\right)=5$ for $b > 0$ and $b \neq 1$, find the value of $f\left(\log _{\frac{1}{3}}b\right)$. | -3 | 21.875 |
21,811 | Given vectors $\overrightarrow{a}=(\sin x,\cos x),\overrightarrow{b}=(2\sqrt{3}\cos x-\sin x,\cos x)$, and $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
$(1)$ Find the interval where the function $f(x)$ is monotonically decreasing.
$(2)$ If $f(x_0)=\frac{2\sqrt{3}}{3}$ and $x_0\in\left[\frac{\pi}{6},\frac{\pi}{2}\right]$, find the value of $\cos 2x_0$. | \frac{\sqrt{3}-3\sqrt{2}}{6} | 57.8125 |
21,812 | In an experimental field, the number of fruits grown on a single plant of a certain crop, denoted as $x$, follows a normal distribution $N(90, \sigma ^{2})$, and $P(x < 70) = 0.2$. Ten plants are randomly selected from the field, and the number of plants with fruit numbers in the range $[90, 110]$ is denoted as the random variable $X$, which follows a binomial distribution. The variance of $X$ is ______. | 2.1 | 63.28125 |
21,813 | Determine the largest value the expression $$ \sum_{1\le i<j\le 4} \left( x_i+x_j \right)\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-negative real numbers, and add up to $ 1. $ Find also the specific values of this numbers that make the above sum achieve the asked maximum. | 3/4 | 79.6875 |
21,814 | Given that the plane unit vectors $\overrightarrow{{e}_{1}}$ and $\overrightarrow{{e}_{2}}$ satisfy $|2\overrightarrow{{e}_{1}}-\overrightarrow{{e}_{2}}|\leqslant \sqrt{2}$. Let $\overrightarrow{a}=\overrightarrow{{e}_{1}}+\overrightarrow{{e}_{2}}$, $\overrightarrow{b}=3\overrightarrow{{e}_{1}}+\overrightarrow{{e}_{2}}$. If the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\theta$, then the minimum value of $\cos^{2}\theta$ is ____. | \frac{28}{29} | 42.96875 |
21,815 | Let \( x \in \mathbf{R} \). Find the minimum value of the algebraic expression \( (x+1)(x+2)(x+3)(x+4) + 2019 \). | 2018 | 90.625 |
21,816 | For how many integers \( n \) between 1 and 20 (inclusive) is \( \frac{n}{18} \) a repeating decimal? | 14 | 63.28125 |
21,817 | Given $6$ cards labeled $1$, $2$, $3$, $4$, $5$, and $6$ are drawn without replacement, calculate the probability that the product of the numbers of the $2$ cards is a multiple of $4$. | \frac{2}{5} | 37.5 |
21,818 | In a circle centered at $O$, points $A$ and $C$ lie on the circle, each of $\overline{BA}$ and $\overline{BC}$ are tangent to the circle. Triangle $ABC$ is isosceles with $AB = BC$ and $\angle ABC = 100^\circ$. The circle intersects $\overline{BO}$ at $D$. Determine $\frac{BD}{BO}$.
A) $\frac{1}{3}$
B) $\frac{1}{2}$
C) $\frac{2}{3}$
D) $\frac{3}{4}$ | \frac{1}{2} | 50.78125 |
21,819 | Determine the smallest constant $n$, such that for any positive real numbers $x$, $y$, and $z$,
\[\sqrt{\frac{x}{y + 2z}} + \sqrt{\frac{y}{2x + z}} + \sqrt{\frac{z}{x + 2y}} > n.\] | \sqrt{3} | 86.71875 |
21,820 | Let \(a_{1}, a_{2}, \cdots, a_{k}\) be a finite arithmetic sequence, such that \(a_{4} + a_{7} + a_{10} = 17\), and \(a_{4} + a_{5} + a_{6} + \cdots + a_{14} = 77\), and \(a_{k} = 13\). Calculate the value of \(k\). | 18 | 75 |
21,821 | In cube \( ABCD A_{1} B_{1} C_{1} D_{1} \), with an edge length of 6, points \( M \) and \( N \) are the midpoints of edges \( AB \) and \( B_{1} C_{1} \) respectively. Point \( K \) is located on edge \( DC \) such that \( D K = 2 K C \). Find:
a) The distance from point \( N \) to line \( AK \);
b) The distance between lines \( MN \) and \( AK \);
c) The distance from point \( A_{1} \) to the plane of triangle \( MNK \). | \frac{66}{\sqrt{173}} | 2.34375 |
21,822 | Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$ . Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$ . | 370 | 100 |
21,823 | In the diagram, $ABC$ is a straight line. What is the value of $y$?
[asy]
draw((-2,0)--(8,0),linewidth(0.7)); draw((8,0)--(5,-5.5)--(0,0),linewidth(0.7));
label("$A$",(-2,0),W); label("$B$",(0,0),N); label("$C$",(8,0),E); label("$D$",(5,-5.5),S);
label("$148^\circ$",(0,0),SW); label("$58^\circ$",(7,0),S);
label("$y^\circ$",(5,-4.5));
[/asy] | 90 | 29.6875 |
21,824 | Let \(x\) and \(y\) be positive real numbers such that
\[
\frac{1}{x + 1} + \frac{1}{y + 1} = \frac{1}{2}.
\]
Find the minimum value of \(x + 3y.\) | 4 + 4 \sqrt{3} | 68.75 |
21,825 | There are $168$ primes below $1000$ . Then sum of all primes below $1000$ is, | 76127 | 100 |
21,826 | Given that $f(\alpha)= \frac{\sin(\pi - \alpha)\cos(-\alpha)\cos(-\alpha + \frac{3\pi}{2})}{\cos(\frac{\pi}{2} - \alpha)\sin(-\pi - \alpha)}$.
(1) Find the value of $f(-\frac{41\pi}{6})$;
(2) If $\alpha$ is an angle in the third quadrant and $\cos(\alpha - \frac{3\pi}{2}) = \frac{1}{3}$, find the value of $f(\alpha)$. | \frac{2\sqrt{2}}{3} | 95.3125 |
21,827 | In the Cartesian coordinate system $xOy$, line $l_{1}$: $kx-y+2=0$ intersects with line $l_{2}$: $x+ky-2=0$ at point $P$. When the real number $k$ varies, the maximum distance from point $P$ to the line $x-y-4=0$ is \_\_\_\_\_\_. | 3\sqrt{2} | 67.1875 |
21,828 | In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l_{1}$ are $\left\{\begin{array}{l}{x=t}\\{y=kt}\end{array}\right.$ (where $t$ is the parameter), and the parametric equations of the line $l_{2}$ are $\left\{\begin{array}{l}{x=-km+2}\\{y=m}\end{array}\right.$ (where $m$ is the parameter). Let $P$ be the intersection point of the lines $l_{1}$ and $l_{2}$. As $k$ varies, the locus of point $P$ is curve $C_{1}$. <br/>$(Ⅰ)$ Find the equation of the locus of curve $C_{1}$; <br/>$(Ⅱ)$ Using the origin as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of line $C_{2}$ is $\rho \sin (\theta +\frac{π}{4})=3\sqrt{2}$. Point $Q$ is a moving point on curve $C_{1}$. Find the maximum distance from point $Q$ to line $C_{2}$. | 1+\frac{5\sqrt{2}}{2} | 7.03125 |
21,829 | A rectangle can be divided into \( n \) equal squares. The same rectangle can also be divided into \( n + 76 \) equal squares. Find all possible values of \( n \). | 324 | 17.1875 |
21,830 | Simplify \[\frac{1}{\dfrac{3}{\sqrt{5}+2} + \dfrac{4}{\sqrt{7}-2}}.\] | \frac{3}{9\sqrt{5} + 4\sqrt{7} - 10} | 70.3125 |
21,831 | In a certain group, the probability that each member uses mobile payment is $p$, and the payment methods of each member are independent of each other. Let $X$ be the number of members in the group of $10$ who use mobile payment, $D\left(X\right)=2.4$, $P\left(X=4\right) \lt P\left(X=6\right)$. Find the value of $p$. | 0.6 | 56.25 |
21,832 | The sequence \\(\{a_n\}\) consists of numbers \\(1\\) or \\(2\\), with the first term being \\(1\\). Between the \\(k\\)-th \\(1\\) and the \\(k+1\\)-th \\(1\\), there are \\(2k-1\\) \\(2\\)s, i.e., the sequence \\(\{a_n\}\) is \\(1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, \ldots\\). Let the sum of the first \\(n\\) terms of the sequence \\(\{a_n\}\) be \\(S_n\\), then \\(S_{20} =\\) , \\(S_{2017} =\\) . | 3989 | 7.8125 |
21,833 | A and B plan to meet between 8:00 and 9:00 in the morning, and they agreed that the person who arrives first will wait for the other for 10 minutes before leaving on their own. Calculate the probability that they successfully meet. | \dfrac{11}{36} | 82.03125 |
21,834 | Given a quadratic equation \( x^{2} + bx + c = 0 \) with roots 98 and 99, within the quadratic function \( y = x^{2} + bx + c \), if \( x \) takes on values 0, 1, 2, 3, ..., 100, how many of the values of \( y \) are divisible by 6? | 67 | 14.84375 |
21,835 | Fold a 10m long rope in half 5 times, then cut it in the middle with scissors. How many segments is the rope cut into? | 33 | 5.46875 |
21,836 | Given the sequence $\{a\_n\}$ satisfies $a\_1=1$, and for any $n∈N^∗$, $a_{n+1}=a\_n+n+1$, find the value of $$\frac {1}{a_{1}}+ \frac {1}{a_{2}}+…+ \frac {1}{a_{2017}}+ \frac {1}{a_{2016}}+ \frac {1}{a_{2019}}$$. | \frac{2019}{1010} | 32.8125 |
21,837 | The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. What is the smallest possible difference between the two 2-digit numbers? | 14 | 2.34375 |
21,838 | Let $a_n= \frac {1}{n}\sin \frac {n\pi}{25}$, and $S_n=a_1+a_2+\ldots+a_n$. Find the number of positive terms among $S_1, S_2, \ldots, S_{100}$. | 100 | 7.03125 |
21,839 | Given the parametric equations of line $l$ as $\begin{cases} x=t\cos α \\ y=1+t\sin α \end{cases}\left(t \text{ is a parameter, } \frac{π}{2}\leqslant α < π\right)$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of circle $C$ is $ρ =2\cos θ$.
(I) Discuss the number of common points between line $l$ and circle $C$;
(II) Draw a perpendicular line to line $l$ passing through the pole, with the foot of the perpendicular denoted as $P$, find the length of the chord formed by the intersection of the trajectory of point $P$ and circle $C$. | \frac{2\sqrt{5}}{5} | 5.46875 |
21,840 | Given Josie makes lemonade by using 150 grams of lemon juice, 200 grams of sugar, and 300 grams of honey, and there are 30 calories in 100 grams of lemon juice, 386 calories in 100 grams of sugar, and 304 calories in 100 grams of honey, determine the total number of calories in 250 grams of her lemonade. | 665 | 35.15625 |
21,841 | How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? | 90 | 92.96875 |
21,842 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given $\overrightarrow{m}=(\sin C,\sin B\cos A)$ and $\overrightarrow{n}=(b,2c)$ with $\overrightarrow{m}\cdot \overrightarrow{n}=0$.
(1) Find angle $A$;
(2) If $a=2 \sqrt {3}$ and $c=2$, find the area of $\triangle ABC$. | \sqrt {3} | 0 |
21,843 | Given that angle $A$ is an internal angle of a triangle and $\cos A= \frac{3}{5}$, find $\tan A=$ \_\_\_\_\_\_ and $\tan (A+ \frac{\pi}{4})=$ \_\_\_\_\_\_. | -7 | 99.21875 |
21,844 | Given a set of data pairs (3,y_{1}), (5,y_{2}), (7,y_{3}), (12,y_{4}), (13,y_{5}) corresponding to variables x and y, the linear regression equation obtained is \hat{y} = \frac{1}{2}x + 20. Calculate the value of \sum\limits_{i=1}^{5}y_{i}. | 120 | 35.9375 |
21,845 | Let the function \( g(x) \) take positive real numbers to real numbers such that
\[ xg(y) - yg(x) = g \left( \frac{x}{y} \right) + x - y \]
for all positive real numbers \( x \) and \( y \). Find all possible values of \( g(50) \). | -24.5 | 1.5625 |
21,846 | Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\mathcal{T}$. | 45 | 26.5625 |
21,847 | Triangle $PQR$ has side-lengths $PQ = 20, QR = 40,$ and $PR = 30.$ The line through the incenter of $\triangle PQR$ parallel to $\overline{QR}$ intersects $\overline{PQ}$ at $X$ and $\overline{PR}$ at $Y.$ What is the perimeter of $\triangle PXY?$ | 50 | 14.84375 |
21,848 | A parallelogram-shaped paper WXYZ with an area of 7.17 square centimeters is placed on another parallelogram-shaped paper EFGH, as shown in the diagram. The intersection points A, C, B, and D are formed, and AB // EF and CD // WX. What is the area of the paper EFGH in square centimeters? Explain the reasoning. | 7.17 | 25 |
21,849 | Complete the following questions:
$(1)$ Calculate: $(\sqrt{8}-\sqrt{\frac{1}{2}})\div \sqrt{2}$.
$(2)$ Calculate: $2\sqrt{3}\times (\sqrt{12}-3\sqrt{75}+\frac{1}{3}\sqrt{108})$.
$(3)$ Given $a=3+2\sqrt{2}$ and $b=3-2\sqrt{2}$, find the value of the algebraic expression $a^{2}-3ab+b^{2}$.
$(4)$ Solve the equation: $\left(2x-1\right)^{2}=x\left(3x+2\right)-7$.
$(5)$ Solve the equation: $2x^{2}-3x+\frac{1}{2}=0$.
$(6)$ Given that real numbers $a$ and $b$ are the roots of the equation $x^{2}-x-1=0$, find the value of $\frac{b}{a}+\frac{a}{b}$. | -3 | 64.84375 |
21,850 | In a large square of area 100 square units, points $P$, $Q$, $R$, and $S$ are the midpoints of the sides of the square. A line is drawn from each corner of the square to the midpoint of the opposite side, creating a new, smaller, central polygon. What is the area of this central polygon? | 25 | 59.375 |
21,851 | Find the minimum value of
\[(15 - x)(8 - x)(15 + x)(8 + x).\] | -6480.25 | 57.03125 |
21,852 | Given that the direction vector of line $l$ is $(4,2,m)$, the normal vector of plane $\alpha$ is $(2,1,-1)$, and $l \perp \alpha$, find the value of $m$. | -2 | 85.9375 |
21,853 | Either increasing the radius of a cylinder by 4 inches or the height by 10 inches results in the same volume. The original height of the cylinder is 5 inches. What is the original radius in inches? | 2 + 2\sqrt{3} | 95.3125 |
21,854 | In a circle, parallel chords of lengths 5, 12, and 13 determine central angles of $\theta$, $\phi$, and $\theta + \phi$ radians, respectively, where $\theta + \phi < \pi$. If $\sin \theta$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator? | 18 | 0.78125 |
21,855 | Let \( p, q, \) and \( r \) be the roots of the equation \( x^3 - 15x^2 + 25x - 10 = 0 \). Find the value of \( (1+p)(1+q)(1+r) \). | 51 | 94.53125 |
21,856 | Given that 800 students were surveyed, and their pasta and pizza preferences were lasagna (150 students), manicotti (120 students), ravioli (180 students), spaghetti (200 students), and pizza (150 students), calculate the ratio of the number of students who preferred spaghetti to the number of students who preferred pizza. | \frac{4}{3} | 88.28125 |
21,857 | The Lions beat the Eagles 3 out of the 4 times they played, then played N more times, and the Eagles ended up winning at least 98% of all the games played; find the minimum possible value for N. | 146 | 2.34375 |
21,858 | How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 42.1875 |
21,859 | In triangle $ABC$, $BC = 40$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ at $D$ and extend to meet an extension of $AB$ at $E$. Find the length of $DE$. | 20 | 32.03125 |
21,860 | Given two points $A(-2,0)$ and $B(0,2)$, and point $C$ is any point on the circle $x^{2}+y^{2}-2x=0$, find the minimum area of $\triangle ABC$. | 3 - \sqrt{2} | 63.28125 |
21,861 | How many natural numbers from 1 to 700, inclusive, contain the digit 7 at least once? | 133 | 0 |
21,862 | Given $\sqrt{20} \approx 4.472, \sqrt{2} \approx 1.414$, find $-\sqrt{0.2} \approx$____. | -0.4472 | 34.375 |
21,863 | A ray passing through the focus $F$ of the parabola $y^2 = 4x$ intersects the parabola at point $A$. Determine the equation of the line to which a circle with diameter $FA$ must be tangent. | y=0 | 6.25 |
21,864 | In the sequence $\{a_n\}$, $a_1=5$, and $a_{n+1}-a_n=3+4(n-1)$. Then, calculate the value of $a_{50}$. | 4856 | 94.53125 |
21,865 | If \(x\) and \(y\) are positive real numbers such that \(6x^2 + 12xy + 6y^2 = x^3 + 3x^2 y + 3xy^2\), find the value of \(x\). | \frac{24}{7} | 21.09375 |
21,866 | Given the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\ (a > b > 0)$, with $F\_{1}$ as the left focus, $A$ as the right vertex, and $B\_{1}$, $B\_{2}$ as the upper and lower vertices respectively. If the four points $F\_{1}$, $A$, $B\_{1}$, and $B\_{2}$ lie on the same circle, find the eccentricity of this ellipse. | \dfrac{\sqrt{5}-1}{2} | 40.625 |
21,867 | A line $l$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $90^\circ$ counterclockwise about the point $(15,10)$ to obtain line $k'$. Find the $x$-coordinate of the $x$-intercept of $k'$. | \frac{65}{3} | 76.5625 |
21,868 | In triangle $\triangle ABC$, $A=60^{\circ}$, $a=\sqrt{6}$, $b=2$.
$(1)$ Find $\angle B$;
$(2)$ Find the area of $\triangle ABC$. | \frac{3 + \sqrt{3}}{2} | 97.65625 |
21,869 | Given two circles $C\_1$: $x^{2}+y^{2}=1$ and $C\_2$: $(x-2)^{2}+(y-4)^{2}=1$, a moving point $P(a,b)$ passes through and forms tangent lines $PM$ and $PN$ to circles $C\_1$ and $C\_2$ respectively with $M$ and $N$ being the points of tangency. If $PM=PN$, find the minimum value of $\sqrt{a^{2}+b^{2}}+\sqrt{(a-5)^{2}+(b+1)^{2}}$. | \sqrt{34} | 1.5625 |
21,870 | Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?
The order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as selecting cards with numbers 16 and 5, is considered the same way. | 15 | 77.34375 |
21,871 | Let $ABCD$ be a parallelogram with vertices $A(0, 0)$, $B(0, 5)$, $C(x, 5)$, and $D(x, 0)$ where $x > 0$. The parallelogram is titled such that $AD$ and $BC$ make an angle of $30^\circ$ with the horizontal axis. If the area of the parallelogram is 35 square units, find the value of $x$. | 14 | 4.6875 |
21,872 | Given a sequence $\{a_{n}\}$ such that $a_{4}+a_{7}=2$, $a_{5}a_{6}=-8$. If $\{a_{n}\}$ is an arithmetic progression, then $a_{1}a_{10}=$____; if $\{a_{n}\}$ is a geometric progression, then $a_{1}+a_{10}=$____. | -7 | 34.375 |
21,873 | A bag contains four balls of identical shape and size, numbered $1$, $2$, $3$, $4$.
(I) A ball is randomly drawn from the bag, its number recorded as $a$, then another ball is randomly drawn from the remaining three, its number recorded as $b$. Find the probability that the quadratic equation $x^{2}+2ax+b^{2}=0$ has real roots.
(II) A ball is randomly drawn from the bag, its number recorded as $m$, then the ball is returned to the bag, and another ball is randomly drawn, its number recorded as $n$. If $(m,n)$ is used as the coordinates of point $P$, find the probability that point $P$ falls within the region $\begin{cases} x-y\geqslant 0 \\ x+y-5 < 0\end{cases}$. | \frac {1}{4} | 22.65625 |
21,874 | Given that the curves $y=x^2-1$ and $y=1+x^3$ have perpendicular tangents at $x=x_0$, find the value of $x_0$. | -\frac{1}{\sqrt[3]{6}} | 22.65625 |
21,875 | Given vectors $\overrightarrow{m}=(1,3\cos \alpha)$ and $\overrightarrow{n}=(1,4\tan \alpha)$, where $\alpha \in (-\frac{\pi}{2}, \;\;\frac{\pi}{2})$, and their dot product $\overrightarrow{m} \cdot \overrightarrow{n} = 5$.
(I) Find the magnitude of $|\overrightarrow{m}+ \overrightarrow{n}|$;
(II) Let the angle between vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ be $\beta$, find the value of $\tan(\alpha + \beta)$. | \frac{\sqrt{2}}{2} | 67.96875 |
21,876 | The line \( L \) crosses the \( x \)-axis at \((-8,0)\). The area of the shaded region is 16. Find the slope of the line \( L \). | \frac{1}{2} | 67.1875 |
21,877 | Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. Find the ratio of the speed of $A$ to the speed of $B$. | \frac{2}{3} | 7.8125 |
21,878 | Determine, with proof, the smallest positive integer \( n \) with the following property: For every choice of \( n \) integers, there exist at least two whose sum or difference is divisible by 2009. | 1006 | 49.21875 |
21,879 | A certain pharmaceutical company has developed a new drug to treat a certain disease, with a cure rate of $p$. The drug is now used to treat $10$ patients, and the number of patients cured is denoted as $X$.
$(1)$ If $X=8$, two patients are randomly selected from these $10$ people for drug interviews. Find the distribution of the number of patients cured, denoted as $Y$, among the selected patients.
$(2)$ Given that $p\in \left(0.75,0.85\right)$, let $A=\{k\left|\right.$ probability $P\left(X=k\right)$ is maximum$\}$, and $A$ contains only two elements. Find $E\left(X\right)$. | \frac{90}{11} | 5.46875 |
21,880 | If $α∈(0, \dfrac{π}{2})$, $\cos ( \dfrac{π}{4}-α)=2 \sqrt{2}\cos 2α$, then $\sin 2α=$____. | \dfrac{15}{16} | 21.875 |
21,881 | Marla has a large white cube that has an edge of 8 feet. She also has enough green paint to cover 200 square feet. Marla uses all the paint to create a white circle centered on each face, surrounded by a green border. What is the area of one of the white circles, in square feet?
A) $15.34$ sq ft
B) $30.67$ sq ft
C) $45.23$ sq ft
D) $60.89$ sq ft | 30.67 | 89.84375 |
21,882 | In a sequence, the first term is \(a_1 = 2010\) and the second term is \(a_2 = 2011\). The values of the other terms satisfy the relation:
\[ a_n + a_{n+1} + a_{n+2} = n \]
for all \(n \geq 1\). Determine \(a_{500}\). | 2177 | 29.6875 |
21,883 | Consider all polynomials of the form
\[x^7 + b_6 x^6 + b_5 x^5 + \dots + b_2 x^2 + b_1 x + b_0,\]
where \( b_i \in \{0,1\} \) for all \( 0 \le i \le 6 \). Find the number of such polynomials that have exactly two different integer roots, -1 and 0. | 15 | 0.78125 |
21,884 | Let $x$ be Walter's age in 1996. If Walter was one-third as old as his grandmother in 1996, then his grandmother's age in 1996 is $3x$. The sum of their birth years is $3864$, and the sum of their ages in 1996 plus $x$ and $3x$ is $1996+1997$. Express Walter's age at the end of 2001 in terms of $x$. | 37 | 89.84375 |
21,885 | An enterprise has four employees participating in a vocational skills assessment. Each employee can draw any one of the four available assessment projects to participate in. Calculate the probability that exactly one project is not selected. | \frac{9}{16} | 67.96875 |
21,886 | There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls? | 102 | 17.96875 |
21,887 | A person bought a bond for 1000 yuan with a maturity of one year. After the bond matured, he spent 440 yuan and then used the remaining money to buy the same type of bond again for another year. After the bond matured the second time, he received 624 yuan. Calculate the annual interest rate of this bond. | 4\% | 37.5 |
21,888 | 50 businessmen - Japanese, Koreans, and Chinese - are sitting at a round table. It is known that between any two nearest Japanese, there are exactly as many Chinese as there are Koreans at the table. How many Chinese can be at the table? | 32 | 0 |
21,889 | A batch of fragile goods totaling $10$ items is transported to a certain place by two trucks, A and B. Truck A carries $2$ first-class items and $2$ second-class items, while truck B carries $4$ first-class items and $2$ second-class items. Upon arrival at the destination, it was found that trucks A and B each broke one item. If one item is randomly selected from the remaining $8$ items, the probability of it being a first-class item is ______. (Express the result in simplest form) | \frac{29}{48} | 2.34375 |
21,890 | Given unit vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ in the same plane, if the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, find the maximum value of $(\overrightarrow{a} - \overrightarrow{b}) \cdot (\overrightarrow{a} - 2\overrightarrow{c})$. | \frac{5}{2} | 35.15625 |
21,891 | The distance from the origin to the line $3x+2y-13=0$ is $\sqrt {13}$. | \sqrt{13} | 39.0625 |
21,892 | In $\triangle ABC$, with $D$ on $AC$ and $F$ on $BC$, given $AB \perp AC$, $AF \perp BC$, and $BD = DF = FC = 1$. If also $D$ is the midpoint of $AC$, find the length of $AC$.
A) 1
B) $\sqrt{2}$
C) $\sqrt{3}$
D) 2
E) $\sqrt[3]{4}$ | \sqrt{2} | 13.28125 |
21,893 | In triangle $XYZ$, $\angle X = 90^\circ$ and $\sin Y = \frac{3}{5}$. Find $\cos Z$. | \frac{3}{5} | 85.15625 |
21,894 | Given the function $f(x)=2ax- \frac {3}{2}x^{2}-3\ln x$, where $a\in\mathbb{R}$ is a constant,
$(1)$ If $f(x)$ is a decreasing function on $x\in[1,+\infty)$, find the range of the real number $a$;
$(2)$ If $x=3$ is an extremum point of $f(x)$, find the maximum value of $f(x)$ on $x\in[1,a]$. | \frac {33}{2}-3\ln 3 | 9.375 |
21,895 | The sum of the series $\frac{3}{4} + \frac{5}{8} + \frac{9}{16} + \frac{17}{32} + \frac{33}{64} + \frac{65}{128} - 3.5$. | \frac{-1}{128} | 0 |
21,896 | The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal asymptote below 1 but above 0.5. | -24 | 8.59375 |
21,897 | A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads. | \frac{1}{2} | 13.28125 |
21,898 | Given that the hotel has 80 suites, the daily rent is 160 yuan, and for every 20 yuan increase in rent, 3 guests are lost, determine the optimal daily rent to set in order to maximize profits, considering daily service and maintenance costs of 40 yuan for each occupied room. | 360 | 22.65625 |
21,899 | What is the $100^{\mathrm{th}}$ odd positive integer, and what even integer directly follows it? | 200 | 42.96875 |
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