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23,700 | $908 \times 501 - [731 \times 1389 - (547 \times 236 + 842 \times 731 - 495 \times 361)] =$ | 5448 | 0.78125 |
23,701 | Given that $\cos (\alpha - \frac{\pi }{3}) - \cos \alpha = \frac{1}{3}$, find the value of $\sin (\alpha - \frac{\pi }{6})$. | \frac{1}{3} | 85.15625 |
23,702 | Given the function $f(x)=2 \sqrt {3}\sin x\cos x-\cos 2x$, where $x\in R$.
(1) Find the interval where the function $f(x)$ is monotonically increasing.
(2) In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(A)=2$, $C= \frac {\pi}{4}$, and $c=2$, find the value of the area $S_{\triangle ABC}$. | \frac {3+ \sqrt {3}}{2} | 0 |
23,703 | A) For a sample of size $n$ taken from a normal population with a known standard deviation $\sigma$, the sample mean $\bar{x}$ is found. At a significance level $\alpha$, it is required to find the power function of the test of the null hypothesis $H_{0}: a=a_{0}$ regarding the population mean $a$ with the hypothetical value $a_{0}$, under the competing hypothesis $H_{1}: a=a_{1} \neq a_{0}$.
B) For a sample of size $n=16$ taken from a normal population with a known standard deviation $\sigma=5$, at a significance level of 0.05, the null hypothesis $H_{0}: a=a_{0}=20$ regarding the population mean $a$ with the hypothetical value $a_{0}=20$ is tested against the competing hypothesis $H_{1}: a \neq 20$. Calculate the power of the two-sided test for the hypothesized value of the population mean $a_{1}=24$. | 0.8925 | 17.1875 |
23,704 | For a French class, I need to master a list of 600 vocabulary words for an upcoming test. The score on the test is based on the percentage of words I recall correctly. In this class, I have noticed that even when guessing the words I haven't studied, I have about a 10% chance of getting them right due to my prior knowledge. What is the minimum number of words I need to learn in order to guarantee at least a 90% score on this test? | 534 | 90.625 |
23,705 | Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$ . Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$ . Find $100p+q$ .
*Proposed by Michael Ren* | 301 | 7.03125 |
23,706 | An infinite geometric series has a first term of $540$ and a sum of $4500$. What is its common ratio, and what is the second term of the series? | 475.2 | 40.625 |
23,707 | Consider three coins where two are fair and a third coin lands on heads with a probability of $\frac{3}{5}$. Alice flips the three coins, and then Bob flips the same three coins. Let $\frac{p}{q}$ be the probability that Alice and Bob get the same number of heads, where $p$ and $q$ are coprime integers. Find $p + q$. | 263 | 25.78125 |
23,708 | The polynomial $(x+y)^{10}$ is expanded in decreasing powers of $x$. The second and third terms have equal values when evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum of $p$ plus twice $q$ equals one. Determine the value of $p$. | \frac{9}{13} | 78.90625 |
23,709 | What is the product of all real numbers that are tripled when added to their reciprocals? | -\frac{1}{2} | 85.9375 |
23,710 | The sum of the first $n$ terms of a geometric sequence $\{a_n\}$, where each term is positive, is $S_n$. Given that $S_n = 2$ and $S_{3n} = 14$, calculate $S_{4n}$. | 30 | 86.71875 |
23,711 | Let $X \sim B(4, p)$, and $P(X=2)=\frac{8}{27}$, find the probability of success in one trial. | \frac{2}{3} | 15.625 |
23,712 | When plotted in the standard rectangular coordinate system, trapezoid $PQRS$ has vertices $P(2, -4)$, $Q(2, 3)$, $R(7, 10)$, and $S(7, 2)$. What is the area of trapezoid $PQRS$? | 37.5 | 95.3125 |
23,713 | The remainder when \( 104^{2006} \) is divided by 29 is ( ) | 28 | 28.90625 |
23,714 | **How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 3, 6, \ldots, 45\}$, where the numbers form an arithmetic sequence? Already given that the common difference between successive elements is 3.** | 14 | 42.96875 |
23,715 | A modified octahedron consists of two pyramids, each with a pentagonal base, glued together along their pentagonal bases, forming a polyhedron with twelve faces. An ant starts at the top vertex and walks randomly to one of the five adjacent vertices in the middle ring. From this vertex, the ant walks again to another randomly chosen adjacent vertex among five possibilities. What is the probability that the second vertex the ant reaches is the bottom vertex of the polyhedron? Express your answer as a common fraction. | \frac{1}{5} | 45.3125 |
23,716 | Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | \frac{33}{105} | 0 |
23,717 | In the diagram, $CP$ and $CQ$ trisect $\angle ACB$. $CM$ bisects $\angle PCQ$. Find the ratio of the measure of $\angle MCQ$ to the measure of $\angle ACQ$. | \frac{1}{4} | 53.90625 |
23,718 | Given that \(AD\), \(BE\), and \(CF\) are the altitudes of the acute triangle \(\triangle ABC\). If \(AB = 26\) and \(\frac{EF}{BC} = \frac{5}{13}\), what is the length of \(BE\)? | 24 | 49.21875 |
23,719 | Given $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \ldots + a_{11}(x+2)^{11}$, find the value of $a_0 + a_1 + a_2 + \ldots + a_{11}$. | -2 | 89.84375 |
23,720 | In triangle $ABC$ , $AB=13$ , $BC=14$ and $CA=15$ . Segment $BC$ is split into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from $A$ . Find the smallest possible value of $n$ .
*Proposed by Evan Chen* | 27 | 6.25 |
23,721 | What is the value of $x + y$ if the sequence $3, ~9, ~x, ~y, ~30$ is an arithmetic sequence? | 36 | 80.46875 |
23,722 | What is the sum of all odd integers between $400$ and $600$? | 50000 | 50.78125 |
23,723 | Given $cos({\frac{π}{4}+α})=\frac{{\sqrt{2}}}{3}$, then $\frac{{sin2α}}{{1-sinα+cosα}}=$____. | \frac{1}{3} | 71.09375 |
23,724 | In the rectangular coordinate system $xOy$, curve $C_1$ passes through point $P(a, 1)$ with parametric equations $$\begin{cases} x=a+ \frac { \sqrt {2}}{2}t \\ y=1+ \frac { \sqrt {2}}{2}t\end{cases}$$ where $t$ is a parameter and $a \in \mathbb{R}$. In the polar coordinate system with the pole at $O$ and the non-negative half of the $x$-axis as the polar axis, the polar equation of curve $C_2$ is $\rho\cos^2\theta + 4\cos\theta - \rho = 0$.
(I) Find the Cartesian equation of the curve $C_1$ and the polar equation of the curve $C_2$;
(II) Given that the curves $C_1$ and $C_2$ intersect at points $A$ and $B$, with $|PA| = 2|PB|$, find the value of the real number $a$. | \frac{1}{36} | 57.03125 |
23,725 | Find the real solution \( x, y, z \) to the equations \( x + y + z = 5 \) and \( xy + yz + zx = 3 \) such that \( z \) is the largest possible value. | \frac{13}{3} | 27.34375 |
23,726 | Given the function \( f(x)=\cos x + m \left(x+\frac{\pi}{2}\right) \sin x \) where \( m \leqslant 1 \):
(1) Discuss the number of zeros of \( f(x) \) in the interval \( (-\pi, 0) \).
(2) If there exists \( t > 0 \) such that \( |f(x)| < -2x - \pi \) holds for \( x \in \left(-\frac{\pi}{2} - t, -\frac{\pi}{2}\right) \), determine the minimum value of \( m \) that satisfies this condition.
| m = -1 | 11.71875 |
23,727 | Given that $F\_1$ and $F\_2$ are the foci of a hyperbola, a line passing through $F\_2$ perpendicular to the real axis intersects the hyperbola at points $A$ and $B$. If $BF\_1$ intersects the $y$-axis at point $C$, and $AC$ is perpendicular to $BF\_1$, determine the eccentricity of the hyperbola. | \sqrt{3} | 63.28125 |
23,728 | A nine-joint bamboo tube has rice capacities of 4.5 *Sheng* in the lower three joints and 3.8 *Sheng* in the upper four joints. Find the capacity of the middle two joints. | 2.5 | 2.34375 |
23,729 | Given $f(x)=m\sin (πx+α)+n\cos (πx+β)+8$, where $m$, $n$, $α$, $β$ are all real numbers. If $f(2000)=-2000$, find $f(2015)$ = \_\_\_\_\_\_. | 2016 | 89.84375 |
23,730 | A sign painter paints numbers for each of 150 houses, numbered consecutively from 1 to 150. How many times does the digit 9 appear in total on all the house numbers? | 25 | 67.96875 |
23,731 | How many ways are there to choose 4 cards from a standard deck of 52 cards, where two cards come from one suit and the other two each come from different suits? | 158184 | 0.78125 |
23,732 | With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team.
There are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays. The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together. Wendy notices that there are even numbers of groups she could select whether she chooses $4$ or $5$ students at a time to tutor geometry each week:
\begin{align*}\dbinom{16}4&=1820,\dbinom{16}5&=4368.\end{align*}
Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even. This gives her an idea for a problem for the $2008$ Jupiter Falls High School Math Meet team test:
\[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\]
Wendy works the solution out correctly. What is her answer? | 1881 | 59.375 |
23,733 | A residential building has a construction cost of 250 yuan per square meter. Considering a useful life of 50 years and an annual interest rate of 5%, what monthly rent per square meter is required to recoup the entire investment? | 1.14 | 17.1875 |
23,734 | Add 22 and 62. | 84 | 91.40625 |
23,735 | What is the volume of the pyramid whose net is shown, if the base is a square with a side length of $1$? | \frac{\sqrt{3}}{6} | 10.15625 |
23,736 | Simplify first, then evaluate: $(1-\frac{a}{a+1})\div \frac{{a}^{2}-1}{{a}^{2}+2a+1}$, where $a=\sqrt{2}+1$. | \frac{\sqrt{2}}{2} | 85.9375 |
23,737 | Compute the product
\[
\prod_{n = 1}^{15} \frac{n^2 + 5n + 6}{n+2}.
\] | \frac{18!}{6} | 0 |
23,738 | For a real number \( x \), \([x]\) denotes the greatest integer less than or equal to \( x \). Given a sequence of positive numbers \( \{a_n\} \) such that \( a_1 = 1 \) and \( S_n = \frac{1}{2} \left( a_n + \frac{1}{a_n} \right) \), where \( S_n \) is the sum of the first \( n \) terms of the sequence \( \{a_n\} \), then \(\left[ \frac{1}{S_1} + \frac{1}{S_2} + \cdots + \frac{1}{S_{100}} \right] = \, \). | 18 | 66.40625 |
23,739 | The sum of 81 consecutive integers is $9^5$. What is their median? | 729 | 66.40625 |
23,740 | Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$ . If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$ , then what is the product of the lengths of the diagonals of $ABCD$ ? | 96 | 42.96875 |
23,741 | Given that $\sin \alpha = \frac{2\sqrt{2}}{3}$, $\cos(\alpha + \beta) = -\frac{1}{3}$, and both $\alpha$ and $\beta$ are within the interval $(0, \frac{\pi}{2})$, find the value of $\sin(\alpha - \beta)$. | \frac{10\sqrt{2}}{27} | 46.09375 |
23,742 | Simplify $21 \cdot \frac{8}{15} \cdot \frac{1}{14}$. | \frac{4}{5} | 71.875 |
23,743 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. If the area of $\triangle ABC$ equals $8$, $a=5$, and $\tan B= -\frac{4}{3}$, then find the value of $\frac{a+b+c}{\sin A+\sin B+\sin C}$. | \frac{5 \sqrt{65}}{4} | 27.34375 |
23,744 | Mountain bikes are a popular means of transportation for high school students, with a huge market and fierce competition. The total sales of type $A$ bikes operated by a certain brand dealer last year was $50,000. This year, the selling price per bike is $400 less than last year. If the quantity sold remains the same, the total sales will decrease by $20\%$.
Given the table below for the purchase and selling prices of type $A$ and type $B$ bikes this year:
| | Type $A$ | Type $B$ |
|----------|---------------|---------------|
| Purchase Price | $1100$ yuan per bike | $1400$ yuan per bike |
| Selling Price | $x$ yuan per bike | $2000$ yuan per bike |
$(1)$ Let the selling price per type $A$ bike this year be $x$ yuan. Find the value of $x$.
$(2)$ The brand dealer plans to purchase a new batch of $A$ bikes and new type $B$ bikes, totaling $60$ bikes. The quantity of type $B$ bikes purchased should not exceed twice the quantity of type $A$ bikes. How should the quantities of the two types of bikes be arranged to maximize profit after selling this batch of bikes? | 40 | 3.125 |
23,745 | In a bag, there are a total of 12 balls that are identical in shape and mass, including red, black, yellow, and green balls. When drawing a ball at random, the probability of getting a red ball is $\dfrac{1}{3}$, the probability of getting a black or yellow ball is $\dfrac{5}{12}$, and the probability of getting a yellow or green ball is also $\dfrac{5}{12}$.
(1) Please calculate the probabilities of getting a black ball, a yellow ball, and a green ball, respectively;
(2) When drawing a ball at random, calculate the probability of not getting a "red or green ball". | \dfrac{5}{12} | 75.78125 |
23,746 | Deepen and Expand: Suppose set $A$ contains 4 elements, and set $B$ contains 3 elements. Now, establish a mapping $f: A \rightarrow B$, and make every element in $B$ have a pre-image in $A$. Then, the number of such mappings is ____. | 36 | 91.40625 |
23,747 | Suppose two arithmetic sequences $\{a_n\}$ and $\{b_n\}$ have the sum of their first $n$ terms as $S_n$ and $T_n$, respectively. Given that $\frac{S_n}{T_n} = \frac{7n}{n+3}$, find the value of $\frac{a_5}{b_5}$. | \frac{21}{4} | 65.625 |
23,748 | Given that the sequence {a<sub>n</sub>} is a decreasing geometric sequence and satisfies the conditions $$a_{2}a_{7}= \frac {1}{2}$$ and $$a_{3}+a_{6}= \frac {9}{4}$$, find the maximum value of a<sub>1</sub>a<sub>2</sub>a<sub>3</sub>…a<sub>2n</sub>. | 64 | 31.25 |
23,749 | Given that $a+b+c=0$, calculate the value of $\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}+\frac{|ab|}{ab}+\frac{|ac|}{ac}+\frac{|bc|}{bc}+\frac{|abc|}{abc}$. | -1 | 56.25 |
23,750 | Determine how many integer palindromes are between 200 and 700. | 50 | 75 |
23,751 | A student, Theo, needs to earn a total of 30 homework points. For the first six homework points, he has to do one assignment each; for the next six points, he needs to do two assignments each; and so on, such that for every subsequent set of six points, the number of assignments he needs to complete doubles the previous set. Calculate the minimum number of homework assignments necessary for Theo to earn all 30 points. | 186 | 6.25 |
23,752 | Employees from department X are 30, while the employees from department Y are 20. Since employees from the same department do not interact, the number of employees from department X that will shake hands with the employees from department Y equals 30, and the number of employees from department Y that will shake hands with the employees from department X also equals 20. Find the total number of handshakes that occur between employees of different departments. | 600 | 99.21875 |
23,753 | Given the function \( y = y_1 + y_2 \), where \( y_1 \) is directly proportional to \( x^2 \) and \( y_2 \) is inversely proportional to \( x^2 \). When \( x = 1 \), \( y = 5 \); when \( x = \sqrt{3} \), \( y = 7 \). Find the value of \( x \) when \( y \) is minimized. | \sqrt[4]{\frac{3}{2}} | 28.90625 |
23,754 | Given a pedestrian signal light that alternates between red and green, with the red light lasting for 50 seconds, calculate the probability that a student needs to wait at least 20 seconds for the green light to appear. | \dfrac{3}{5} | 72.65625 |
23,755 | How many three-digit whole numbers contain at least one 6 or at least one 8? | 452 | 96.875 |
23,756 | In an arithmetic sequence $\{a_n\}$, the sum $a_1 + a_2 + \ldots + a_5 = 30$, and the sum $a_6 + a_7 + \ldots + a_{10} = 80$. Calculate the sum $a_{11} + a_{12} + \ldots + a_{15}$. | 130 | 66.40625 |
23,757 | A bag contains 3 red balls, 2 black balls, and 1 white ball. All 6 balls are identical in every aspect except for color and are well mixed. Balls are randomly drawn from the bag.
(1) With replacement, find the probability of drawing exactly 1 red ball in 2 consecutive draws;
(2) Without replacement, find the probability of drawing exactly 1 red ball in 2 consecutive draws. | \frac{3}{5} | 96.09375 |
23,758 | In the geometric sequence $\{a_{n}\}$, $a_{20}$ and $a_{60}$ are the two roots of the equation $(x^{2}-10x+16=0)$. Find the value of $\frac{{{a}\_{30}}\cdot {{a}\_{40}}\cdot {{a}\_{50}}}{2}$. | 32 | 58.59375 |
23,759 | In isosceles $\triangle ABC$, $|AB|=|AC|$, vertex $A$ is the intersection point of line $l: x-y+1=0$ with the y-axis, and $l$ bisects $\angle A$. If $B(1,3)$, find:
(I) The equation of line $BC$;
(II) The area of $\triangle ABC$. | \frac {3}{2} | 39.84375 |
23,760 | A certain product in a shopping mall sells an average of 70 items per day, with a profit of $50 per item. In order to reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that for each item, for every $1 decrease in price, the mall can sell an additional 2 items per day. Let $x$ represent the price reduction per item. Based on this rule, please answer:<br/>$(1)$ The daily sales volume of the mall increases by ______ items, and the profit per item is ______ dollars. (Express using algebraic expressions involving $x$)<br/>$(2)$ With the above conditions unchanged, how much should the price of each item be reduced so that the mall's daily profit reaches $3572$ dollars. | 12 | 9.375 |
23,761 | Consider a kite-shaped field with the following measurements and angles: sides AB = 120 m, BC = CD = 80 m, DA = 120 m. The angle between sides AB and BC is 120°. The angle between sides CD and DA is also 120°. The wheat harvested from any location in the field is brought to the nearest point on the field's perimeter. What fraction of the crop is brought to the longest side, which in this case is side BC? | \frac{1}{2} | 46.09375 |
23,762 | There is a tram with a starting station A and an ending station B. A tram departs from station A every 5 minutes towards station B, completing the journey in 15 minutes. A person starts cycling along the tram route from station B towards station A just as a tram arrives at station B. On his way, he encounters 10 trams coming towards him before reaching station A. At this moment, another tram is just departing from station A. How many minutes did it take for him to travel from station B to station A? | 50 | 24.21875 |
23,763 | Given that the random variable $X$ follows a normal distribution $N(0,\sigma^{2})$, if $P(X > 2) = 0.023$, determine the probability $P(-2 \leqslant X \leqslant 2)$. | 0.954 | 88.28125 |
23,764 | When two standard dice and one 8-sided die (with faces showing numbers from 1 to 8) are tossed, the numbers \(a, b, c\) are obtained respectively where \(a\) and \(b\) are from the standard dice and \(c\) is from the 8-sided die. Find the probability that \((a-1)(b-1)(c-1) \neq 0\). | \frac{175}{288} | 98.4375 |
23,765 | Given the function $f(x) = |2x+a| + |2x-2b| + 3$
(Ⅰ) If $a=1$, $b=1$, find the solution set of the inequality $f(x) > 8$;
(Ⅱ) When $a>0$, $b>0$, if the minimum value of $f(x)$ is $5$, find the minimum value of $\frac{1}{a} + \frac{1}{b}$. | \frac{3+2\sqrt{2}}{2} | 16.40625 |
23,766 | If the sines of the internal angles of $\triangle ABC$ form an arithmetic sequence, what is the minimum value of $\cos C$? | \frac{1}{2} | 7.8125 |
23,767 | Find the maximum value of
\[\frac{2x + 3y + 4}{\sqrt{x^2 + y^2 + 4}}\]
over all real numbers $x$ and $y$. | \sqrt{29} | 43.75 |
23,768 | If
\[
(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 89^\circ) = 2^m,
\]
then find $m.$ | 45 | 59.375 |
23,769 | Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to points $P_{2}$ on $CD$, $P_{3}$ on $DA$, and $P_{4}$ on $AB$, with the reflection angle equal to the incidence angle. If $P_{4}$ coincides with $P_{0}$, then find $\tan \theta$. | $\frac{1}{2}$ | 0 |
23,770 | Given the parabola $y^{2}=4x$, its focus intersects the parabola at two points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$. If $x_{1}+x_{2}=10$, find the length of the chord $AB$. | 12 | 34.375 |
23,771 | What is the sum of all the even integers between $200$ and $400$? | 30100 | 0 |
23,772 | Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle. | 16 | 0 |
23,773 | What is the smallest number of rectangles, each measuring $2 \mathrm{~cm}$ by $3 \mathrm{~cm}$, which are needed to fit together without overlap to form a rectangle whose sides are in the ratio 5:4? | 30 | 71.09375 |
23,774 | The output of a factory last year is denoted as $1$. If it is planned that the output of each of the next five years will increase by $10\%$ compared to the previous year, then the total output of this factory for the five years starting from this year will be approximately \_\_\_\_\_\_\_\_. (Keep one decimal place, take $1.1^{5} \approx 1.6$) | 6.6 | 2.34375 |
23,775 | Given $x = \frac{2}{3}$ and $y = \frac{5}{2}$, find the value of $\frac{1}{3}x^8y^9$. | \frac{5^9}{2 \cdot 3^9} | 0 |
23,776 | In quadrilateral ABCD, m∠B = m∠C = 120°, AB = 4, BC = 6, and CD = 7. Diagonal BD = 8. Calculate the area of ABCD. | 16.5\sqrt{3} | 0 |
23,777 | What are the last two digits in the sum of the factorials of the first 15 positive integers? | 13 | 27.34375 |
23,778 | Among the following propositions, the true one is numbered \_\_\_\_\_\_.
(1) The negation of the proposition "For all $x>0$, $x^2-x\leq0$" is "There exists an $x>0$ such that $x^2-x>0$."
(2) If $A>B$, then $\sin A > \sin B$.
(3) Given a sequence $\{a_n\}$, "The sequence $a_n, a_{n+1}, a_{n+2}$ forms a geometric sequence" is a necessary and sufficient condition for $a_{n+1}^2=a_{n}a_{n+2}$.
(4) Given the function $f(x)=\lg x+ \frac{1}{\lg x}$, then the minimum value of $f(x)$ is 2. | (1) | 0 |
23,779 | Given that points P1 and P2 are two adjacent centers of symmetry for the curve $y= \sqrt {2}\sin ωx-\cos ωx$ $(x\in\mathbb{R})$, if the tangents to the curve at points P1 and P2 are perpendicular to each other, determine the value of ω. | \frac{\sqrt{3}}{3} | 59.375 |
23,780 | In a circle with center $O$, the measure of $\angle RIP$ is $45^\circ$ and $OR=15$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\pi$. | 7.5\pi | 0.78125 |
23,781 | Given that the equation about $x$, $x^{2}-2a\ln x-2ax=0$ has a unique solution, find the value of the real number $a$. | \frac{1}{2} | 21.09375 |
23,782 | Matrices $A$ , $B$ are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0 1 & 2 & 0 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 2 & 4 & 0 0 & 0 & 12\end{pmatrix}\]
Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$ . | \frac{\pi}{3} | 32.03125 |
23,783 | Given that $({x-1})^4({x+2})^5=a_0+a_1x+a_2x^2+⋯+a_9x^9$, find the value of $a_{2}+a_{4}+a_{6}+a_{8}$. | -24 | 6.25 |
23,784 | The magnitude of the vector $\overset{→}{a} +2 \overset{→}{b}$, where $\overset{→}{a} =(2,0)$, $\overset{→}{b}$ is a unit vector with a magnitude of 1 and the angle between the two vectors is $60^{\circ}$. | 2\sqrt{3} | 90.625 |
23,785 | Given $m, n \in \mathbb{R}$, if the line $(m+1)x + (n+1)y - 2 = 0$ is tangent to the circle $x^2 + y^2 = 1$, find the maximum value of $m - n$. | 2\sqrt{2} | 94.53125 |
23,786 | I have two 10-sided dice where each die has 3 gold sides, 4 silver sides, 2 diamond sides, and 1 rainbow side. If I roll both dice, what is the probability that they come up showing the same color or pattern? | \frac{3}{10} | 61.71875 |
23,787 | Given the function $f(x)=\cos^{4}x-2\sin x\cos x-\sin^{4}x$.
(1) Find the smallest positive period of the function $f(x)$;
(2) When $x\in\left[0,\frac{\pi}{2}\right]$, find the minimum value of $f(x)$ and the set of $x$ values where the minimum value is obtained. | \left\{\frac{3\pi}{8}\right\} | 40.625 |
23,788 | In triangle $ABC$, point $D$ is on side $BC$ such that $BD:DC = 1:2$. A line through $A$ and $D$ intersects $BC$ at $E$. If the area of triangle $ABE$ is $30$, find the total area of triangle $ABC$. | 90 | 53.90625 |
23,789 | Abby, Bernardo, Carl, and Debra play a revised game where each starts with five coins and there are five rounds. In each round, five balls are placed in an urn—two green, two red, and one blue. Each player draws a ball at random without replacement. If a player draws a green ball, they give one coin to a player who draws a red ball. If anyone draws a blue ball, no transaction occurs for them. What is the probability that at the end of the fifth round, each of the players has five coins?
**A)** $\frac{1}{120}$
**B)** $\frac{64}{15625}$
**C)** $\frac{32}{3125}$
**D)** $\frac{1}{625}$
**E)** $\frac{4}{125}$ | \frac{64}{15625} | 32.8125 |
23,790 | Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth $4$ and tails is worth $2.$ What is the probability Katie’s score is strictly greater than Allie’s? | 25/72 | 2.34375 |
23,791 | A Saxon silver penny, from the reign of Ethelbert II in the eighth century, was sold in 2014 for £78000. A design on the coin depicts a circle surrounded by four equal arcs, each a quarter of a circle. The width of the design is 2 cm. What is the radius of the small circle, in centimetres?
A) \(\frac{1}{2}\)
B) \(2 - \sqrt{2}\)
C) \(\frac{1}{2} \sqrt{2}\)
D) \(5 - 3\sqrt{2}\)
E) \(2\sqrt{2} - 2\) | 2 - \sqrt{2} | 22.65625 |
23,792 | Bob's password consists of a positive single-digit number followed by a letter and another positive single-digit number. What is the probability that Bob's password consists of an even single-digit number followed by a vowel (from A, E, I, O, U) and a number greater than 5? | \frac{40}{1053} | 42.96875 |
23,793 | Calculate:<br/>$(1)\frac{\sqrt{20}+\sqrt{5}}{\sqrt{5}}-2$;<br/>$(2)\sqrt[3]{-8}+5\sqrt{\frac{1}{10}}-\sqrt{10}+\sqrt{4}$;<br/>$(3)(\sqrt{3}-\sqrt{2})^2•(5+2\sqrt{6})$;<br/>$(4)(π-3.14)^0+\frac{1}{2+\sqrt{3}}+(-\frac{1}{3})^{-1}+|1-\sqrt{3}|$. | -1 | 35.15625 |
23,794 | Calculate the amount of personal income tax (НДФЛ) paid by Sergey over the past year if he is a resident of the Russian Federation and had a stable income of 30,000 rubles per month and a one-time vacation bonus of 20,000 rubles during this period. Last year, Sergey sold his car, which he inherited two years ago, for 250,000 rubles and bought a plot of land for building a house for 300,000 rubles. Sergey applied all applicable tax deductions. (Provide the answer without spaces and units of measurement.) | 10400 | 21.09375 |
23,795 | The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangement for the 100th stage? | 400 | 48.4375 |
23,796 | A rectangular prism measuring 20 cm by 14 cm by 12 cm has a small cube of 4 cm on each side removed from each corner. What percent of the original volume is removed? | 15.24\% | 23.4375 |
23,797 | Calculate the integral $\int_{2}^{7}(x-3)^{2} d x$.
a) Using the substitution $z=x-3$.
b) Using the substitution $z=(x-3)^{2}$. | \frac{65}{3} | 32.03125 |
23,798 | There are 6 boxes, each containing a key that cannot be used interchangeably. If one key is placed in each box and all the boxes are locked, and it is required that after breaking open one box, the remaining 5 boxes can still be opened with the keys, then the number of ways to place the keys is ______. | 120 | 60.9375 |
23,799 | Two friends, Alice and Bob, start cycling towards a park 80 miles away. Alice cycles 3 miles per hour slower than Bob. Upon reaching the park, Bob immediately turns around and starts cycling back, meeting Alice 15 miles away from the park. Find the speed of Alice. | 6.5 | 37.5 |
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