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23,600 | Given plane vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(1,-4)$.
$(1)$ If $4\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $k\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$.
$(2)$ If $\theta$ is the angle between $4\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}+\overrightarrow{b}$, find the value of $\tan \theta$. | -\frac{3}{4} | 89.84375 |
23,601 | What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)
Alexandru Mihalcu | 16 | 28.125 |
23,602 | Given the function $f(x)=-\cos^2 x + \sqrt{3}\sin x\sin\left(x + \frac{\pi}{2}\right)$, find the sum of the minimum and maximum values of $f(x)$ when $x \in \left[0, \frac{\pi}{2}\right]$. | -\frac{1}{2} | 79.6875 |
23,603 | Given circle $C$: $(x-1)^{2}+(y-2)^{2}=25$, and line $l$: $(2m+1)x+(m+1)y-7m-4=0$, if the length of the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is _____. | -\frac{3}{4} | 0.78125 |
23,604 | In $\triangle ABC$, the three internal angles are $A$, $B$, and $C$. If $\dfrac{\sqrt{3}\cos A + \sin A}{\sqrt{3}\sin A - \cos A} = \tan(-\dfrac{7}{12}\pi)$, find the maximum value of $2\cos B + \sin 2C$. | \dfrac{3}{2} | 27.34375 |
23,605 | Given that tetrahedron $ABCD$ is inscribed in sphere $O$, and $AD$ is the diameter of sphere $O$. If triangles $\triangle ABC$ and $\triangle BCD$ are equilateral triangles with side length 1, what is the volume of tetrahedron $ABCD$? | $\frac{\sqrt{2}}{12}$ | 0 |
23,606 | If α is in the interval (0, π) and $\frac{1}{2}\cos2α = \sin\left(\frac{π}{4} + α\right)$, then find the value of $\sin2α$. | -1 | 11.71875 |
23,607 | Let \( n \geq 2 \) be a fixed integer. Find the least constant \( C \) such that the inequality
\[ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} \]
holds for every \( x_{1}, \ldots, x_{n} \geq 0 \) (the sum on the left consists of \(\binom{n}{2}\) summands). For this constant \( \bar{C} \), characterize the instances of equality. | \frac{1}{8} | 59.375 |
23,608 | Given $\triangle ABC$, where $AB=6$, $BC=8$, $AC=10$, and $D$ is on $\overline{AC}$ with $BD=6$, find the ratio of $AD:DC$. | \frac{18}{7} | 13.28125 |
23,609 | Given the hyperbola $mx^{2}+y^{2}=1$ and one of its asymptotes has a slope of $2$, find the value of $m$. | -4 | 85.15625 |
23,610 | Find the minimum area of the part bounded by the parabola $ y\equal{}a^3x^2\minus{}a^4x\ (a>0)$ and the line $ y\equal{}x$ . | \frac{4}{3} | 75.78125 |
23,611 | The line passing through the points (3,9) and (-1,1) has an x-intercept of ( ). | -\frac{3}{2} | 53.90625 |
23,612 | In the polar coordinate system $Ox$, the polar equation of curve $C_{1}$ is $ρ=\frac{2\sqrt{2}}{sin(θ+\frac{π}{4})}$, with the pole $O$ as the origin and the polar axis $Ox$ as the x-axis. A Cartesian coordinate system $xOy$ is established with the same unit length. It is known that the general equation of curve $C_{2}$ is $\left(x-2\right)^{2}+\left(y-1\right)^{2}=9$.<br/>$(1)$ Write down the Cartesian equation of curve $C_{1}$ and the polar equation of curve $C_{2}$;<br/>$(2)$ Let point $M\left(2,2\right)$, and curves $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Find the value of $\overrightarrow{MA}•\overrightarrow{MB}$. | -8 | 92.96875 |
23,613 | How many nonnegative integers can be represented in the form \[
a_7 \cdot 4^7 + a_6 \cdot 4^6 + a_5 \cdot 4^5 + a_4 \cdot 4^4 + a_3 \cdot 4^3 + a_2 \cdot 4^2 + a_1 \cdot 4^1 + a_0 \cdot 4^0,
\]
where $a_i \in \{0, 1, 2\}$ for $0 \leq i \leq 7$? | 6561 | 96.875 |
23,614 | (In the 15th Jiangsu Grade 7 First Trial) On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The building numbered $k$ has exactly $k$ (where $k=1, 2, 3, 4, 5$) workers from Factory A. The distance between two adjacent buildings is 50 meters. Factory A plans to build a station on this street. To minimize the total walking distance for all workers from these 5 buildings to the station, the station should be built at a distance of ______ meters from Building 1. | 150 | 27.34375 |
23,615 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a\sin 2B= \sqrt{3}b\sin A$.
1. Find $B$;
2. If $\cos A= \dfrac{1}{3}$, find the value of $\sin C$. | \dfrac{2\sqrt{6}+1}{6} | 75.78125 |
23,616 | Let $(2-x)^{6}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{6}x^{6}$, then the value of $|a_{1}|+|a_{2}|+\ldots+|a_{6}|$ is \_\_\_\_\_\_. | 665 | 79.6875 |
23,617 | How many 11 step paths are there from point A to point D, which pass through points B and C in that order? Assume the grid layout permits only right and down steps, where B is 2 right and 2 down steps from A, and C is 1 right and 3 down steps from B, and finally, D is 3 right and 1 down step from C. | 96 | 98.4375 |
23,618 | Three lines are drawn parallel to each of the three sides of $\triangle ABC$ so that the three lines intersect in the interior of $ABC$ . The resulting three smaller triangles have areas $1$ , $4$ , and $9$ . Find the area of $\triangle ABC$ .
[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); }
pair B = (0,0), C = (1,0), A = (0.3, 0.8), D = relpt(relpt(A,B,3,3),relpt(A,C,3,3),1,2);
draw(A--B--C--cycle);
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);
filldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7));
filldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7));
filldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy] | 36 | 52.34375 |
23,619 | In an equilateral triangle $ABC$ with side length $6$, point $D$ is the midpoint of $BC$. Calculate $\tan{\angle BAD}$. | \frac{1}{\sqrt{3}} | 0.78125 |
23,620 | For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$ . | 125 | 81.25 |
23,621 | Amy, Beth, and Claire each have some sweets. Amy gives one third of her sweets to Beth. Beth gives one third of all the sweets she now has to Claire. Then Claire gives one third of all the sweets she now has to Amy. All the girls end up having the same number of sweets.
Claire begins with 40 sweets. How many sweets does Beth have originally? | 50 | 26.5625 |
23,622 | (1) Given $\tan(\alpha+\beta)= \frac{2}{5}$ and $\tan\left(\beta- \frac{\pi}{4}\right)= \frac{1}{4}$, find the value of $\frac{\cos\alpha+\sin\alpha}{\cos\alpha-\sin\alpha}$;
(2) Given $\alpha$ and $\beta$ are acute angles, and $\cos(\alpha+\beta)= \frac{\sqrt{5}}{5}$, $\sin(\alpha-\beta)= \frac{\sqrt{10}}{10}$, find $2\beta$. | \frac{\pi}{4} | 75 |
23,623 | Given a regular triangular prism $ABC-A_1B_1C_1$ with side edges equal to the edges of the base, the sine of the angle formed by $AB_1$ and the lateral face $ACCA_1$ is equal to _______. | \frac{\sqrt{6}}{4} | 55.46875 |
23,624 | Juan rolls a fair twelve-sided die marked with the numbers 1 through 12. Then Amal rolls a fair ten-sided die. What is the probability that the product of the two rolls is a multiple of 3? | \frac{8}{15} | 83.59375 |
23,625 | Select 4 shoes from 5 pairs of different-sized shoes, find the number of possibilities where at least 2 of the 4 shoes can be paired together. | 130 | 15.625 |
23,626 | Calculate the following expressions:
1. $(-51) + (-37)$;
2. $(+2) + (-11)$;
3. $(-12) + (+12)$;
4. $8 - 14$;
5. $15 - (-8)$;
6. $(-3.4) + 4.3$;
7. $|-2\frac{1}{4}| + (-\frac{1}{2})$;
8. $-4 \times 1\frac{1}{2}$;
9. $-3 \times (-6)$. | 18 | 99.21875 |
23,627 | Given the parametric equations of curve $C$ are
$$
\begin{cases}
x=3+ \sqrt {5}\cos \alpha \\
y=1+ \sqrt {5}\sin \alpha
\end{cases}
(\alpha \text{ is the parameter}),
$$
with the origin of the Cartesian coordinate system as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system.
$(1)$ Find the polar equation of curve $C$;
$(2)$ If the polar equation of a line is $\sin \theta-\cos \theta= \frac {1}{\rho }$, find the length of the chord cut from curve $C$ by the line. | \sqrt {2} | 0 |
23,628 | What is the base five product of the numbers $132_{5}$ and $12_{5}$? | 2114_5 | 0 |
23,629 | Given the parabola $C: y^2 = 8x$ with focus $F$ and directrix $l$. Let $P$ be a point on $l$, and $Q$ be a point where the line $PF$ intersects $C$. If $\overrightarrow{FQ} = -4\overrightarrow{FP}$, calculate $|QF|$. | 20 | 43.75 |
23,630 | A circle with radius 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Determine the length of the longer side of the rectangle, expressed in centimeters and in terms of $\pi$. | 9\pi | 78.90625 |
23,631 | 8 people are sitting around a circular table for a meeting, including one leader, one deputy leader, and one recorder. If the recorder is sitting between the leader and the deputy leader, how many different seating arrangements are possible (seating arrangements that can be made identical through rotation are considered the same). | 240 | 71.09375 |
23,632 | The graph of the function $f(x)$ is symmetric about the $y$-axis, and for any $x \in \mathbb{R}$, it holds that $f(x+3)=-f(x)$. If $f(x)=(\frac{1}{2})^{x}$ when $x \in \left( \frac{3}{2}, \frac{5}{2} \right)$, then find $f(2017)$. | -\frac{1}{4} | 36.71875 |
23,633 | There are 5 students who signed up to participate in a two-day volunteer activity at the summer district science museum. Each day, two students are needed for the activity. The probability that exactly 1 person will participate in the volunteer activity for two consecutive days is ______. | \frac{3}{5} | 25 |
23,634 | In triangle $\triangle ABC$, $AC=2$, $D$ is the midpoint of $AB$, $CD=\frac{1}{2}BC=\sqrt{7}$, $P$ is a point on $CD$, and $\overrightarrow{AP}=m\overrightarrow{AC}+\frac{1}{3}\overrightarrow{AB}$. Find $|\overrightarrow{AP}|$. | \frac{2\sqrt{13}}{3} | 14.84375 |
23,635 | Let $F$ be the focus of the parabola $C: y^2=4x$, point $A$ lies on $C$, and point $B(3,0)$. If $|AF|=|BF|$, then calculate the distance of point $A$ from point $B$. | 2\sqrt{2} | 85.9375 |
23,636 | Given that angle $\alpha$ is in the fourth quadrant, and the x-coordinate of the point where its terminal side intersects the unit circle is $\frac{1}{3}$.
(1) Find the value of $\tan \alpha$;
(2) Find the value of $\frac{\sin^{2}\alpha - \sqrt{2}\sin \alpha\cos \alpha}{1 + \cos^{2}\alpha}$. | \frac{6}{5} | 87.5 |
23,637 | Given that all terms are positive in the geometric sequence $\{a_n\}$, and the sum of the first $n$ terms is $S_n$, if $S_1 + 2S_5 = 3S_3$, then the common ratio of $\{a_n\}$ equals \_\_\_\_\_\_. | \frac{\sqrt{2}}{2} | 66.40625 |
23,638 | It is known that $x^5 = a_0 + a_1 (1+x) + a_2 (1+x)^2 + a_3 (1+x)^3 + a_4 (1+x)^4 + a_5 (1+x)^5$, find the value of $a_0 + a_2 + a_4$. | -16 | 43.75 |
23,639 | Triangle $DEF$ is isosceles with angle $E$ congruent to angle $F$. The measure of angle $F$ is three times the measure of angle $D$. What is the number of degrees in the measure of angle $E$? | \frac{540}{7} | 71.09375 |
23,640 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $a\cos B - b\cos A = \frac{1}{2}c$. When $\tan(A-B)$ takes its maximum value, the value of angle $B$ is \_\_\_\_\_\_. | \frac{\pi}{6} | 42.96875 |
23,641 | The volume of the parallelepiped generated by the vectors $\begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix}$, $\begin{pmatrix} 2 \\ m \\ 3 \end{pmatrix}$, and $\begin{pmatrix} 2 \\ 3 \\ m \end{pmatrix}$ is 20. Find $m$, where $m > 0$. | 3 + \frac{2\sqrt{15}}{3} | 48.4375 |
23,642 | Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$ , $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers. | 6210001000 | 67.96875 |
23,643 | Find the smallest composite number that has no prime factors less than 15. | 323 | 11.71875 |
23,644 | Two players, A and B, take turns shooting baskets. The probability of A making a basket on each shot is $\frac{1}{2}$, while the probability of B making a basket is $\frac{1}{3}$. The rules are as follows: A goes first, and if A makes a basket, A continues to shoot; otherwise, B shoots. If B makes a basket, B continues to shoot; otherwise, A shoots. They continue to shoot according to these rules. What is the probability that the fifth shot is taken by player A? | \frac{247}{432} | 0.78125 |
23,645 | Given that all vertices of the tetrahedron S-ABC are on the surface of sphere O, SC is the diameter of sphere O, and if plane SCA is perpendicular to plane SCB, with SA = AC and SB = BC, and the volume of tetrahedron S-ABC is 9, find the surface area of sphere O. | 36\pi | 39.0625 |
23,646 | The Fahrenheit temperature ( $F$ ) is related to the Celsius temperature ( $C$ ) by $F = \tfrac{9}{5} \cdot C + 32$ . What is the temperature in Fahrenheit degrees that is one-fifth as large if measured in Celsius degrees? | -4 | 96.09375 |
23,647 | On a Cartesian plane, consider the points \(A(-8, 2), B(-4, -2)\) and \(X(1, y)\) and \(Y(10, 3)\). If segment \(AB\) is parallel to segment \(XY\), find the value of \(y\). | 12 | 98.4375 |
23,648 | If $f(1) = 3$, $f(2)= 12$, and $f(x) = ax^2 + bx + c$, what is the value of $f(3)$? | 21 | 6.25 |
23,649 | Given $$\frac{\pi}{2} < \alpha < \pi$$ and $$0 < \beta < \frac{\pi}{2}$$, with $\tan\alpha = -\frac{3}{4}$ and $\cos(\beta - \alpha) = \frac{5}{13}$, find the value of $\sin\beta$. | \frac{63}{65} | 27.34375 |
23,650 | Calculate $3 \cdot 7^{-1} + 6 \cdot 13^{-1} \pmod{60}$.
Express your answer as an integer from $0$ to $59$, inclusive. | 51 | 28.125 |
23,651 |
Karl the old shoemaker made a pair of boots and sent his son Hans to the market to sell them for 25 talers. At the market, two people, one missing his left leg and the other missing his right leg, approached Hans and asked to buy one boot each. Hans agreed and sold each boot for 12.5 talers.
When Hans came home and told his father everything, Karl decided that he should have sold the boots cheaper to the disabled men, for 10 talers each. He gave Hans 5 talers and instructed him to return 2.5 talers to each person.
While Hans was looking for the individuals in the market, he saw sweets for sale, couldn't resist, and spent 3 talers on candies. He then found the men and gave them the remaining money – 1 taler each. On his way back home, Hans realized how bad his actions were. He confessed everything to his father and asked for forgiveness. The shoemaker was very angry and punished his son by locking him in a dark closet.
While sitting in the closet, Hans thought deeply. Since he returned 1 taler to each man, they effectively paid 11.5 talers for each boot: $12.5 - 1 = 11.5$. Therefore, the boots cost 23 talers: $2 \cdot 11.5 = 23$. And Hans had spent 3 talers on candies, resulting in a total of 26 talers: $23 + 3 = 26$. But there were initially only 25 talers! Where did the extra taler come from? | 25 | 28.90625 |
23,652 | Xiaoming's family raises chickens and pigs in a ratio of 26:5, and sheep to horses in a ratio of 25:9, while the ratio of pigs to horses is 10:3. Find the ratio of chickens, pigs, horses, and sheep. | 156:30:9:25 | 21.875 |
23,653 | The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours? | 30 | 89.84375 |
23,654 | Two numbers need to be inserted between $4$ and $16$ such that the first three numbers are in arithmetic progression and the last three numbers are in geometric progression. What is the sum of those two numbers?
A) $6\sqrt{3} + 8$
B) $10\sqrt{3} + 6$
C) $6 + 10\sqrt{3}$
D) $16\sqrt{3}$ | 6\sqrt{3} + 8 | 50 |
23,655 | Given vectors $\overset{ .}{a}=(\sin x, \frac{1}{2})$, $\overset{ .}{b}=( \sqrt {3}\cos x+\sin x,-1)$, and the function $f(x)= \overset{ .}{a}\cdot \overset{ .}{b}$:
(1) Find the smallest positive period of the function $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ on the interval $[\frac{\pi}{4}, \frac{\pi}{2}]$. | \frac{1}{2} | 83.59375 |
23,656 | The diameter of the semicircle $AB=4$, with $O$ as the center, and $C$ is any point on the semicircle different from $A$ and $B$. Find the minimum value of $(\vec{PA}+ \vec{PB})\cdot \vec{PC}$. | -2 | 5.46875 |
23,657 | Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | \dfrac{1}{52} | 51.5625 |
23,658 | Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$ | 550 | 0.78125 |
23,659 | There is a moving point \( P \) on the \( x \)-axis. Given the fixed points \( A(0,2) \) and \( B(0,4) \), what is the maximum value of \( \sin \angle APB \) when \( P \) moves along the entire \( x \)-axis? | \frac{1}{3} | 27.34375 |
23,660 | Quadrilateral \(ABCD\) is inscribed in a circle with diameter \(AD\) having a length of 4. If the lengths of \(AB\) and \(BC\) are each 1, calculate the length of \(CD\). | \frac{7}{2} | 0 |
23,661 | Given that the distance from point $P(x,y)$ to $A(0,4)$ and $B(-2,0)$ is equal, the minimum value of ${2}^{x}+{4}^{y}$ is. | 4\sqrt{2} | 90.625 |
23,662 | The number 2017 has 7 ones and 4 zeros in binary representation. When will the nearest year come where the binary representation of the year has no more ones than zeros? (Enter the year.) | 2048 | 65.625 |
23,663 | Given that $S = 6 \times 10,000 + 5 \times 1000 + 4 \times 10 + 3 \times 1$, calculate the value of $S$. | 65043 | 73.4375 |
23,664 | Pascal's Triangle starting with row 1 has the sum of elements in row $n$ given by $2^{n-1}$. What is the sum of the interior numbers of the ninth row, considering interior numbers are all except the first and last numbers in the row? | 254 | 88.28125 |
23,665 | In a household, when someone is at home, the probability of the phone being answered at the 1st ring is 0.1, at the 2nd ring is 0.3, at the 3rd ring is 0.4, and at the 4th ring is 0.1. What is the probability that the phone is not answered within the first 4 rings? | 0.1 | 50.78125 |
23,666 | A point is chosen randomly from within a circular region with radius $r$. A related concentric circle with radius $\sqrt{r}$ contains points that are closer to the center than to the boundary. Calculate the probability that a randomly chosen point lies closer to the center than to the boundary. | \frac{1}{4} | 8.59375 |
23,667 | In a sequence of positive integers that starts with 1, certain numbers are sequentially colored red according to the following rules. First, 1 is colored red. Then, the next 2 even numbers, 2 and 4, are colored red. After 4, the next three consecutive odd numbers, 5, 7, and 9, are colored red. Following 9, the next four consecutive even numbers, 10, 12, 14, and 16, are colored red. Afterward, the next five consecutive odd numbers, 17, 19, 21, 23, and 25, are colored red. This pattern continues indefinitely. Thus, the red-colored subsequence obtained is 1, 2, 4, 5, 7, 9, 12, 14, 16, 17, etc. What is the 2003rd number in this red-colored subsequence? | 3943 | 3.90625 |
23,668 | The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Determine the last digit to appear in the units position of a number in the Fibonacci sequence when considered modulo 12. | 11 | 12.5 |
23,669 | A parallelogram $ABCD$ is inscribed in an ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$. The slope of line $AB$ is $k_1=1$. Calculate the slope of line $AD$. | -\frac{1}{2} | 7.03125 |
23,670 | Xiao Li was doing a subtraction problem and mistook the tens digit 7 for a 9 and the ones digit 3 for an 8, resulting in a difference of 76. The correct difference is ______. | 51 | 17.96875 |
23,671 | Calculate:<br/>$(1)-1^{2023}-\sqrt{2\frac{1}{4}}+\sqrt[3]{-1}+\frac{1}{2}$;<br/>$(2)2\sqrt{3}+|1-\sqrt{3}|-\left(-1\right)^{2022}+2$. | 3\sqrt{3} | 81.25 |
23,672 | What integer $n$ satisfies $0 \leq n < 201$ and $$200n \equiv 144 \pmod {101}~?$$ | 29 | 3.90625 |
23,673 | The fifth grade has 120 teachers and students going to visit the Natural History Museum. A transportation company offers two types of vehicles to choose from:
(1) A bus with a capacity of 40 people, with a ticket price of 5 yuan per person. If the bus is full, the ticket price can be discounted by 20%.
(2) A minivan with a capacity of 10 people, with a ticket price of 6 yuan per person. If the minivan is full, the ticket price can be discounted to 75% of the original price.
Please design the most cost-effective rental plan for the fifth-grade teachers and students based on the information above, and calculate the total rental cost. | 480 | 50.78125 |
23,674 | Given that the chord common to circle C: x²+(y-4)²=18 and circle D: (x-1)²+(y-1)²=R² has a length of $6\sqrt {2}$, find the radius of circle D. | 2\sqrt {7} | 0 |
23,675 | Simplify the expression $(-\frac{1}{343})^{-2/3}$. | 49 | 92.96875 |
23,676 | Calculate the number of four-digit numbers without repeating digits that can be formed by taking any two odd numbers and two even numbers from the six digits 0, 1, 2, 3, 4, 5. | 180 | 34.375 |
23,677 | For an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, it is known that (a_2 - 1)^3 + 2014(a_2 - 1) = sin \frac{2011\pi}{3} and (a_{2013} - 1)^3 + 2014(a_{2013} - 1) = cos \frac{2011\pi}{6}. Determine the value of S_{2014}. | 2014 | 82.03125 |
23,678 | When simplified, $\log_{16}{32} \div \log_{16}{\frac{1}{2}}$, calculate the result. | -5 | 89.84375 |
23,679 | Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.
[asy]
defaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
draw(unitcircle,dg);
for(int i = 0; i < 12; ++i) {
draw(dir(30*i+theta)--dir(30*(i+1)+theta), db);
dot(dir(30*i+theta),Fill(rgb(0.8,0,0)));
} dot(dir(theta),Fill(dr)); dot((0,0),Fill(dr));
[/asy] | 300 | 96.875 |
23,680 | Given a sequence $\{a_n\}$ that satisfies $a_{n+1} = a_n - a_{n-1}$ ($n \in N^*, n \geqslant 2$), with $a_1 = 2018$ and $a_2 = 2017$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. The value of $S_{100}$ is ______. | 2016 | 65.625 |
23,681 | In right triangle $DEF$, $DE=15$, $DF=9$ and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? | 7.5 | 22.65625 |
23,682 | To examine the effectiveness of a certain influenza vaccine, a laboratory randomly selected 100 healthy mice for an experiment and obtained the following contingency table:
| | Infection | Not Infected |
|----------|-----------|--------------|
| Injected | 10 | 40 |
| Not Injected | 20 | 30 |
Given $P(K^{2}\geqslant k_{0})$:
| $k_{0}$ | 3.841 | 5.024 | 6.635 |
|---------|-------|-------|-------|
| $P(K^{2}\geqslant k_{0})$ | 0.05 | 0.025 | 0.010 |
Under the condition that the probability of making an error does not exceed ______, it can be considered that "vaccination" is related to "influenza infection." Reference formula: ${K}^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$. | 0.05 | 50 |
23,683 | Four fair coins are tossed once. For every head that appears, two six-sided dice are rolled. What is the probability that the sum of all dice rolled is exactly ten?
A) $\frac{1} {48}$
B) $\frac{1} {20}$
C) $\frac{1} {16}$
D) $\frac{1} {30}$ | \frac{1} {20} | 24.21875 |
23,684 | In a box, there are 8 identical balls, including 3 red balls, 4 white balls, and 1 black ball.
$(1)$ If two balls are drawn consecutively without replacement from the box, one at a time, find the probability of drawing a red ball on the first draw and then drawing a red ball on the second draw.
$(2)$ If three balls are drawn with replacement from the box, find the probability distribution and the expected value of the number of white balls drawn. | \frac{3}{2} | 90.625 |
23,685 | The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{2} | 40.625 |
23,686 | Given the function $f(x) = x^3 - 3x^2 - 9x + 1$,
(1) Determine the monotonicity of the function on the interval $[-4, 4]$.
(2) Calculate the function's local maximum and minimum values as well as the absolute maximum and minimum values on the interval $[-4, 4]$. | -75 | 16.40625 |
23,687 | Some mice live in three neighboring houses. Last night, every mouse left its house and moved to one of the other two houses, always taking the shortest route. The numbers in the diagram show the number of mice per house, yesterday and today. How many mice used the path at the bottom of the diagram?
A 9
B 11
C 12
D 16
E 19 | 11 | 32.03125 |
23,688 | If there exists a real number $x$ such that the inequality $\left(e^{x}-a\right)^{2}+x^{2}-2ax+a^{2}\leqslant \dfrac{1}{2}$ holds with respect to $x$, determine the range of real number $a$. | \left\{\dfrac{1}{2}\right\} | 3.125 |
23,689 | A store sells a type of notebook. The retail price for each notebook is 0.30 yuan, a dozen (12 notebooks) is priced at 3.00 yuan, and for purchases of more than 10 dozen, each dozen can be paid for at 2.70 yuan.
(1) There are 57 students in the ninth grade class 1, and each student needs one notebook of this type. What is the minimum amount the class has to pay if they buy these notebooks collectively?
(2) There are 227 students in the ninth grade, and each student needs one notebook of this type. What is the minimum amount the grade has to pay if they buy these notebooks collectively? | 51.30 | 25.78125 |
23,690 | Marie has 75 raspberry lollipops, 132 mint lollipops, 9 blueberry lollipops, and 315 coconut lollipops. She decides to distribute these lollipops equally among her 13 friends, distributing as many as possible. How many lollipops does Marie end up keeping for herself? | 11 | 20.3125 |
23,691 | Given that $\underbrace{9999\cdots 99}_{80\text{ nines}}$ is multiplied by $\underbrace{7777\cdots 77}_{80\text{ sevens}}$, calculate the sum of the digits in the resulting product. | 720 | 3.90625 |
23,692 | Given the regression equation $\hat{y}$=4.4x+838.19, estimate the ratio of the growth rate between x and y, denoted as \_\_\_\_\_\_. | \frac{5}{22} | 0 |
23,693 | Alexis imagines a $2008\times 2008$ grid of integers arranged sequentially in the following way:
\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&20082009,&2010,&2011,&\ldots,&40264017,&4018,&4019,&\ldots,&6024\vdots&&&&\vdots2008^2-2008+1,&2008^2-2008+2,&2008^2-2008+3,&\ldots,&2008^2\end{array}\]
She picks one number from each row so that no two numbers she picks are in the same column. She them proceeds to add them together and finds that $S$ is the sum. Next, she picks $2008$ of the numbers that are distinct from the $2008$ she picked the first time. Again she picks exactly one number from each row and column, and again the sum of all $2008$ numbers is $S$ . Find the remainder when $S$ is divided by $2008$ . | 1004 | 82.03125 |
23,694 | Given a function $f(x)$ $(x \in \mathbb{R})$ that satisfies the equation $f(-x) = 8 - f(4 + x)$, and another function $g(x) = \frac{4x + 3}{x - 2}$. If the graph of $f(x)$ has 168 intersection points with the graph of $g(x)$, denoted as $P_i(x_i, y_i)$ $(i = 1,2, \dots, 168)$, calculate the value of $(x_{1} + y_{1}) + (x_{2} + y_{2}) + \dots + (x_{168} + y_{168})$. | 1008 | 64.0625 |
23,695 | Given triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $9\sin ^{2}B=4\sin ^{2}A$ and $\cos C=\frac{1}{4}$, calculate $\frac{c}{a}$. | \frac{\sqrt{10}}{3} | 72.65625 |
23,696 | Given that $\cos (α+ \dfrac {π}{6})= \dfrac {1}{3}$, where $α∈(0\,,\,\, \dfrac {π}{2})$, find $\sin α$ and $\sin (2α+ \dfrac {5π}{6})$. | -\dfrac {7}{9} | 7.03125 |
23,697 | A car's clock is running at a constant speed but is inaccurate. One day, when the driver begins shopping, he notices both the car clock and his wristwatch (which is accurate) show 12:00 noon. After shopping, the wristwatch reads 12:30, and the car clock reads 12:35. Later that day, he loses his wristwatch and looks at the car clock, which shows 7:00. What is the actual time? | 6:00 | 83.59375 |
23,698 | If $f(x)$ is a monic quartic polynomial such that $f(-2)=-4$, $f(1)=-1$, $f(3)=-9$, and $f(5)=-25$, find $f(0)$. | -30 | 0 |
23,699 | Let \(P_1\) be a regular \(r\)-sided polygon and \(P_2\) be a regular \(s\)-sided polygon with \(r \geq s \geq 3\), such that each interior angle of \(P_1\) is \(\frac{61}{60}\) as large as each interior angle of \(P_2\). What is the largest possible value of \(s\)? | 121 | 14.84375 |
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