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23,400 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $(2a-c)\cos B=b\cos C$.
(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) If $a=2$ and $c=3$, find the value of $\sin C$. | \frac {3 \sqrt {21}}{14} | 0 |
23,401 | Given the sample 7, 8, 9, x, y has an average of 8, and xy=60, then the standard deviation of this sample is \_\_\_\_\_\_. | \sqrt{2} | 96.09375 |
23,402 | Two boxes of candies have a total of 176 pieces. If 16 pieces are taken out from the second box and put into the first box, the number of pieces in the first box is 31 more than m times the number of pieces in the second box (m is an integer greater than 1). Then, determine the number of pieces originally in the first box. | 131 | 2.34375 |
23,403 | In triangle $XYZ$, $XY = 5$, $YZ = 12$, $XZ = 13$, and $YM$ is the angle bisector from vertex $Y$. If $YM = l \sqrt{2}$, find $l$. | \frac{60}{17} | 49.21875 |
23,404 | What is the least positive integer that is divisible by the first three prime numbers greater than 5? | 1001 | 100 |
23,405 | (12 points) A workshop has a total of 12 workers and needs to equip two types of machines. Each type A machine requires 2 people to operate, consumes 30 kilowatt-hours of electricity per day, and can produce products worth 40,000 yuan; each type B machine requires 3 people to operate, consumes 20 kilowatt-hours of electricity per day, and can produce products worth 30,000 yuan. Now, the daily electricity supply to the workshop is no more than 130 kilowatt-hours. How should the workshop equip these two types of machines to maximize the daily output value? What is the maximum output value in ten thousand yuan? | 18 | 0.78125 |
23,406 | Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM ,each mango costs $4$ RM , and each papaya costs $5$ RM . He bought at least one of each fruit, and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought? | 15 | 94.53125 |
23,407 | Let $M = 36 \cdot 36 \cdot 77 \cdot 330$. Find the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. | 1 : 62 | 0 |
23,408 | Given the function $f(x)=|2x-a|+|x+ \frac {2}{a}|$
$(1)$ When $a=2$, solve the inequality $f(x)\geqslant 1$;
$(2)$ Find the minimum value of the function $g(x)=f(x)+f(-x)$. | 4 \sqrt {2} | 0 |
23,409 | Consider a square where each side measures 1 unit. At each vertex of the square, a quarter circle is drawn outward such that each side of the square serves as the radius for two adjoining quarter circles. Calculate the total perimeter formed by these quarter circles. | 2\pi | 85.9375 |
23,410 | A fair six-sided die has faces numbered $1, 2, 3, 4, 5, 6$. The die is rolled four times, and the results are $a, b, c, d$. What is the probability that one of the numbers in the set $\{a, a+b, a+b+c, a+b+c+d\}$ is equal to 4? | $\frac{343}{1296}$ | 0 |
23,411 | How many integers between $\frac{23}{3}$ and $\frac{65}{2}$ are multiples of $5$ or $3$? | 11 | 10.9375 |
23,412 | What is the sum of all the odd divisors of $360$? | 78 | 78.90625 |
23,413 | In a polar coordinate system, the polar equation of curve C is $\rho=2\cos\theta+2\sin\theta$. Establish a Cartesian coordinate system with the pole as the origin and the positive x-axis as the polar axis. The parametric equation of line l is $\begin{cases} x=1+t \\ y= \sqrt{3}t \end{cases}$ (t is the parameter). Find the length of the chord that curve C cuts off on line l. | \sqrt{7} | 83.59375 |
23,414 | At the end of $1997$, Jason was one-third as old as his grandmother. The sum of the years in which they were born was $3852$. How old will Jason be at the end of $2004$?
A) $41.5$
B) $42.5$
C) $43.5$
D) $44.5$ | 42.5 | 75.78125 |
23,415 | A certain unit is planning to build a rectangular room that is backed against a wall with a ground surface area of 12 square meters. Due to geographical constraints, the side length x of the room cannot exceed 5 meters. The cost of constructing the front face of the room is 400 yuan per square meter, the cost for the sides is 150 yuan per square meter, and the total cost for the roof and ground is 5800 yuan. If the wall height is 3 meters, and the cost of the back face is disregarded, at what side length will the total construction cost be minimized? What is the minimum total construction cost? | 13000 | 50 |
23,416 | Let $\mathbf{v}$ be a vector such that
\[\left\| \mathbf{v} + \begin{pmatrix} 4 \\ -2 \end{pmatrix} \right\| = 10.\]
Find the smallest possible value of $\|\mathbf{v}\|$. | 10 - 2\sqrt{5} | 43.75 |
23,417 | Wei decides to modify the design of his logo by using a larger square and three tangent circles instead. Each circle remains tangent to two sides of the square and to one adjacent circle. If each side of the square is now 24 inches, calculate the number of square inches that will be shaded. | 576 - 108\pi | 61.71875 |
23,418 | If for any $x\in R$, $2x+2\leqslant ax^{2}+bx+c\leqslant 2x^{2}-2x+4$ always holds, then the maximum value of $ab$ is ______. | \frac{1}{2} | 4.6875 |
23,419 | Given that $cos\left( \frac{\pi}{6}-\alpha\right) = \frac{\sqrt{3}}{2}$, find the value of $cos\left( \frac{5\pi}{6}+\alpha\right) - sin^2\left(-\alpha+\frac{7\pi}{6}\right)$.
If $cos\alpha = \frac{2}{3}$ and $\alpha$ is an angle in the fourth quadrant, find the value of $\frac{sin(\alpha-2\pi)+sin(-\alpha-3\pi)cos(-\alpha-3\pi)}{cos(\pi -\alpha)-cos(-\pi-\alpha)cos(\alpha-4\pi)}$. | \frac{\sqrt{5}}{2} | 37.5 |
23,420 | A rental company owns 100 cars. When the monthly rent for each car is 3000 yuan, all of them can be rented out. For every 50 yuan increase in the monthly rent per car, there will be one more car that is not rented out. The maintenance cost for each rented car is 150 yuan per month, and for each car that is not rented out, the maintenance cost is 50 yuan per month. When the monthly rent per car is set to x yuan, the monthly income of the rental company is y yuan.
(1) Please write down the function relationship between x and y (no need to specify the domain).
(2) If the rental company rented out 88 cars in a certain month, how much is the monthly income of the rental company? | 303000 | 45.3125 |
23,421 | In convex quadrilateral \(EFGH\), \(\angle E = \angle G\), \(EF = GH = 150\), and \(EH \neq FG\). The perimeter of \(EFGH\) is 580. Find \(\cos E\). | \frac{14}{15} | 75 |
23,422 | What is the measure of an orthogonal trihedral angle? What is the sum of the measures of polyhedral angles that share a common vertex, have no common interior points, and together cover the entire space? | 4\pi | 7.8125 |
23,423 | The two roots of the equation $x^{2}-9x+18=0$ are the base and one of the legs of an isosceles triangle. Find the perimeter of this isosceles triangle. | 15 | 68.75 |
23,424 | The volume of a regular octagonal prism is $8 \, \mathrm{m}^{3}$, and its height is $2.2 \, \mathrm{m}$. Find the lateral surface area of the prism. | 16 \sqrt{2.2 \cdot (\sqrt{2}-1)} | 0 |
23,425 | A hospital's internal medicine ward has 15 nurses, who work in pairs, rotating shifts every 8 hours. After two specific nurses work the same shift together, calculate the maximum number of days required for them to work the same shift again. | 35 | 10.15625 |
23,426 | Given $f\left(x\right)=e^{x}-ax+\frac{1}{2}{x}^{2}$, where $a \gt -1$.<br/>$(Ⅰ)$ When $a=0$, find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(0,f\left(0\right)\right)$;<br/>$(Ⅱ)$ When $a=1$, find the extreme values of the function $f\left(x\right)$;<br/>$(Ⅲ)$ If $f(x)≥\frac{1}{2}x^2+x+b$ holds for all $x\in R$, find the maximum value of $b-a$. | 1 + \frac{1}{e} | 39.84375 |
23,427 | Lucia needs to save 35 files onto disks, each with a capacity of 1.6 MB. 5 of the files are 0.9 MB each, 10 of the files are 0.8 MB each, and the rest are 0.5 MB each. Calculate the smallest number of disks needed to store all 35 files. | 15 | 25 |
23,428 | Two types of anti-inflammatory drugs must be selected from $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$, with the restriction that $X_{1}$ and $X_{2}$ must be used together, and one type of antipyretic drug must be selected from $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, with the further restriction that $X_{3}$ and $T_{4}$ cannot be used at the same time. Calculate the number of different test schemes. | 14 | 0.78125 |
23,429 | Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a focal length of $2$, and point $Q( \frac{a^{2}}{ \sqrt{a^{2}-b^{2}}},0)$ on the line $l$: $x=2$.
(1) Find the standard equation of the ellipse $C$;
(2) Let $O$ be the coordinate origin, $P$ a moving point on line $l$, and $l'$ a line passing through point $P$ that is tangent to the ellipse at point $A$. Find the minimum value of the area $S$ of $\triangle POA$. | \frac{ \sqrt{2}}{2} | 29.6875 |
23,430 | Let the sequence \(b_1, b_2, b_3, \dots\) be defined such that \(b_1 = 24\), \(b_{12} = 150\), and for all \(n \geq 3\), \(b_n\) is the arithmetic mean of the first \(n - 1\) terms. Find \(b_2\). | 276 | 81.25 |
23,431 | For point M on the ellipse $\frac {x^{2}}{9}+ \frac {y^{2}}{4}=1$, find the minimum distance from M to the line $x+2y-10=0$. | \sqrt{5} | 79.6875 |
23,432 | What is $8^{15} \div 64^3$? | 8^9 | 0 |
23,433 | A trapezoid field is uniformly planted with wheat. The trapezoid has one pair of parallel sides measuring 80 m and 160 m, respectively, with the longer side on the bottom. The other two non-parallel sides each measure 120 m. The angle between a slanted side and the longer base is $45^\circ$. At harvest, the wheat at any point 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? | \frac{1}{2} | 23.4375 |
23,434 | The distance \( AB \) is 100 km. Cyclists depart simultaneously from \( A \) and \( B \) towards each other with speeds of 20 km/h and 30 km/h respectively. Along with the first cyclist from \( A \), a fly departs with a speed of 50 km/h. The fly travels until it meets the cyclist from \( B \), then turns around and flies back to meet the cyclist from \( A \), and continues this pattern. How many kilometers will the fly travel in the direction from \( A \) to \( B \) before the cyclists meet? | 100 | 7.03125 |
23,435 | Given point $A(1,2)$ and circle $C: x^{2}+y^{2}+2mx+2y+2=0$.
$(1)$ If there are two tangents passing through point $A$, find the range of $m$.
$(2)$ When $m=-2$, a point $P$ on the line $2x-y+3=0$ is chosen to form two tangents $PM$ and $PN$ to the circle. Find the minimum area of quadrilateral $PMCN$. | \frac{7\sqrt{15}}{5} | 45.3125 |
23,436 | Given $\{a_{n}\}$ is a geometric sequence, $a_{2}a_{4}a_{5}=a_{3}a_{6}$, $a_{9}a_{10}=-8$, then $a_{7}=\_\_\_\_\_\_$. | -2 | 54.6875 |
23,437 | Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | 110.8333 | 3.125 |
23,438 |
A construction company built a cottage village consisting of three identical streets. The houses in the village are identical according to the plan and are up for sale at the same price. The business plan of the company, which includes expenses for all stages of construction, contains an estimate of the total cost of the entire village.
It is known that the business plan includes the following expenses (total: for the entire village):
- Foundation - 150 million units
- Walls/roofing/floor structures - 105 million units
- Engineering communications (gas, water, electricity, fiber optic) - 225 million units
- Finishing works - 45 million units
Estimate the price at which each house in the village is sold, given that the markup of the construction company is 20 percent of the cost, and each stage of construction for one house according to the business plan is expressed as an integer million. | 42 | 28.90625 |
23,439 | Thirty tiles are numbered 1 through 30 and are placed into box $C$. Thirty other tiles numbered 21 through 50 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 20 and the tile from box $D$ is either odd or greater than 40? Express your answer as a common fraction. | \frac{19}{45} | 78.125 |
23,440 | Given that $\{a\_n\}$ is a geometric sequence, $a\_2=2$, $a\_6=162$, find $a\_{10}$ = $\_\_\_\_\_\_$ . | 13122 | 89.0625 |
23,441 | Given an arithmetic sequence $\{a\_n\}$ with a common difference $d > 0$, and $a\_2$, $a\_5-1$, $a\_{10}$ form a geometric sequence. If $a\_1=5$, and $S\_n$ represents the sum of the first $n$ terms of the sequence, find the minimum value of $\frac{2S\_n+n+32}{a\_n+1}$. | \frac{20}{3} | 3.125 |
23,442 | Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers $x$ .
*2019 CCA Math Bonanza Lightning Round #3.4* | 576 | 72.65625 |
23,443 | From 5 differently colored balls, select 4 balls to place into 3 distinct boxes, with the requirement that no box is left empty. The total number of different ways to do this is ______. (Answer with a number) | 180 | 89.0625 |
23,444 | If the inequality $x^2+ax+1\geqslant 0$ holds for all $x\in\left(0, \frac{1}{2}\right]$, then the minimum value of $a$ is ______. | -\frac{5}{2} | 80.46875 |
23,445 | All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:
(a) There is no right triangle
(b) There is no acute triangle
having all vertices in the vertices of the 2016-gon that are still white? | 1008 | 9.375 |
23,446 | A moving point $A$ is on the circle $C$: $(x-1)^{2}+y^{2}=1$, and a moving point $B$ is on the line $l:x+y-4=0$. The coordinates of the fixed point $P$ are $P(-2,2)$. The minimum value of $|PB|+|AB|$ is ______. | \sqrt{37}-1 | 35.15625 |
23,447 | Two distinct integers, $x$ and $y$, are randomly chosen from the set $\{1,2,3,4,5,6,7,8,9,10,11,12\}$. What is the probability that $(x-1)(y-1)$ is odd? | \frac{5}{22} | 90.625 |
23,448 | In the Cartesian coordinate plane $(xOy)$, a point $A(2,0)$, a moving point $B$ on the curve $y= \sqrt {1-x^{2}}$, and a point $C$ in the first quadrant form an isosceles right triangle $ABC$ with $\angle A=90^{\circ}$. The maximum length of the line segment $OC$ is _______. | 1+2 \sqrt {2} | 0 |
23,449 | Find the area bounded by the graph of $y = \arccos(\cos x)$ and the $x$-axis on the interval $0 \leq x \leq 2\pi$. | \pi^2 | 96.09375 |
23,450 | Rectangle \( EFGH \) is 10 cm by 6 cm. \( P \) is the midpoint of \( \overline{EF} \), and \( Q \) is the midpoint of \( \overline{GH} \). Calculate the area of region \( EPGQ \).
** | 30 | 92.96875 |
23,451 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $\cos A= \frac {4}{5}$, $\cos B= -\frac { \sqrt {2}}{10}$.
$(1)$ Find $C$;
$(2)$ If $c=5$, find the area of $\triangle ABC$. | \frac {21}{2} | 68.75 |
23,452 | In a right triangular pyramid P-ABC, where PA, PB, and PC are mutually perpendicular and PA=1, the center of the circumscribed sphere is O. Find the distance from O to plane ABC. | \frac{\sqrt{3}}{6} | 46.875 |
23,453 | How many four-digit numbers divisible by 17 are also even? | 265 | 21.875 |
23,454 | Given the function $f(x)= \begin{cases} \sin \frac {π}{2}x,-4\leqslant x\leqslant 0 \\ 2^{x}+1,x > 0\end{cases}$, find the zero point of $y=f[f(x)]-3$. | x=-3 | 35.9375 |
23,455 | Simplify first, then evaluate: $(\frac{{2x}^{2}+2x}{{x}^{2}-1}-\frac{{x}^{2}-x}{{x}^{2}-2x+1})÷\frac{x}{x+1}$, where $x=|\sqrt{3}-2|+(\frac{1}{2})^{-1}-(π-3.14)^0-\sqrt[3]{27}+1$. | -\frac{2\sqrt{3}}{3} + 1 | 0 |
23,456 | Simplify the expression: $(a-\frac{3a}{a+1}$) $÷\frac{{a}^{2}-4a+4}{a+1}$, then choose a number you like from $-2$, $-1$, and $2$ to substitute for $a$ and calculate the value. | \frac{1}{2} | 24.21875 |
23,457 | Calculate $\lim_{n \to \infty} \frac{C_n^2}{n^2+1}$. | \frac{1}{2} | 61.71875 |
23,458 | $(1)$ Calculate: $|2-\sqrt{3}|+(\sqrt{2}+1)^0+3\tan30°+(-1)^{2023}-(\frac{1}{2})^{-1}$.
$(2)$ Simplify first and then find the value: $(\frac{x-1}{x}-\frac{x-2}{x+1})÷\frac{2x^2-x}{x^2+2x+1}$, where $x$ satisfies $x^{2}-2x-2=0$. | \frac{1}{2} | 72.65625 |
23,459 | In acute triangle $\triangle ABC$, $a$, $b$, $c$ are the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and $4a\sin B = \sqrt{7}b$.
$(1)$ If $a = 6$ and $b+c = 8$, find the area of $\triangle ABC$.
$(2)$ Find the value of $\sin (2A+\frac{2\pi}{3})$. | \frac{\sqrt{3}-3\sqrt{7}}{16} | 85.15625 |
23,460 | If the general term of the sequence \\(\{a_n\}\) is \\(a_n = (-1)^{n+1} \cdot (3n-2)\\), then \\(a_1 + a_2 + \cdots + a_{20} = \)____. | -30 | 96.09375 |
23,461 | If $x$ and $y$ are positive integers such that $xy - 2x + 5y = 111$, what is the minimal possible value of $|x - y|$? | 93 | 43.75 |
23,462 | A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in the display. | 24 | 0 |
23,463 | Julia's garden has a 3:7 ratio of tulips to daisies. She currently has 35 daisies. She plans to add 30 more daisies and wants to plant additional tulips to maintain the original ratio. How many tulips will she have after this addition? | 28 | 58.59375 |
23,464 | A point is randomly thrown onto the interval $[6, 10]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3k-10\right)x^{2}+(3k-8)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | 1/3 | 34.375 |
23,465 | A noodle shop offers classic specialty noodles to customers, who can either dine in at the shop (referred to as "dine-in" noodles) or purchase packaged fresh noodles with condiments (referred to as "fresh" noodles). It is known that the total price of 3 portions of "dine-in" noodles and 2 portions of "fresh" noodles is 31 yuan, and the total price of 4 portions of "dine-in" noodles and 1 portion of "fresh" noodles is 33 yuan.
$(1)$ Find the price of each portion of "dine-in" noodles and "fresh" noodles, respectively.
$(2)$ In April, the shop sold 2500 portions of "dine-in" noodles and 1500 portions of "fresh" noodles. To thank the customers, starting from May 1st, the price of each portion of "dine-in" noodles remains unchanged, while the price of each portion of "fresh" noodles decreases by $\frac{3}{4}a\%$. After analyzing the sales and revenue in May, it was found that the sales volume of "dine-in" noodles remained the same as in April, the sales volume of "fresh" noodles increased by $\frac{5}{2}a\%$ based on April, and the total sales of these two types of noodles increased by $\frac{1}{2}a\%$ based on April. Find the value of $a$. | \frac{40}{9} | 0.78125 |
23,466 | What is the product of all real numbers that are tripled when added to their reciprocals? | -\frac{1}{2} | 77.34375 |
23,467 | If the sum of the binomial coefficients of the odd terms in the expansion of ${(x-\frac{2}{x})}^{n}$ is $16$, then the coefficient of $x^{3}$ in the expansion is ______. | -10 | 64.0625 |
23,468 | Given that $\sin \theta$ and $\cos \theta$ are the two roots of the equation $4x^{2}-4mx+2m-1=0$, and $\frac {3\pi}{2} < \theta < 2\pi$, find the angle $\theta$. | \frac {5\pi}{3} | 6.25 |
23,469 | Given vectors $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, $|\overrightarrow{c}|=\sqrt{2}$, and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$, calculate $\cos 〈\overrightarrow{a}-\overrightarrow{c}$,$\overrightarrow{b}-\overrightarrow{c}〉$. | \frac{4}{5} | 84.375 |
23,470 | If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$ , $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$ , where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$ ? | 30 | 31.25 |
23,471 | In the Cartesian coordinate system $xOy$, triangle $\triangle ABC$ has vertices $A(-6, 0)$ and $C(6, 0)$. Vertex $B$ lies on the left branch of the hyperbola $\frac{x^2}{25} - \frac{y^2}{11} = 1$. Find the value of $\frac{\sin A - \sin C}{\sin B}$. | \frac{5}{6} | 24.21875 |
23,472 | A barrel with a height of 1.5 meters is completely filled with water and sealed with a lid. The mass of the water in the barrel is 1000 kg. A long, thin vertical tube with a cross-section of $1 \mathrm{~cm}^{2}$ is inserted into the lid of the barrel and completely filled with water. Find the length of the tube if it is known that after filling, the pressure at the bottom of the barrel increased by 2 times. The density of water is 1000 kg/m³. | 1.5 | 82.03125 |
23,473 | Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. | 110.8333333 | 0 |
23,474 | Given that $3\sin \alpha + 4\cos \alpha = 5$.
(1) Find the value of $\tan \alpha$;
(2) Find the value of $\cot (\frac{3\pi}{2} - \alpha) \cdot \sin^2 (\frac{3\pi}{2} + \alpha)$. | \frac{12}{25} | 86.71875 |
23,475 | Given the quadratic equation \( ax^2 + bx + c \) and the table of values \( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \) for a sequence of equally spaced increasing values of \( x \), determine the function value that does not belong to the table. | 6851 | 31.25 |
23,476 | A line $l$ does not pass through the origin $O$ and intersects an ellipse $\frac{x^{2}}{2}+y^{2}=1$ at points $A$ and $B$. $M$ is the midpoint of segment $AB$. Determine the product of the slopes of line $AB$ and line $OM$. | -\frac{1}{2} | 82.8125 |
23,477 | In the geometric sequence {a_n}, a_6 and a_{10} are the two roots of the equation x^2+6x+2=0. Determine the value of a_8. | -\sqrt{2} | 7.03125 |
23,478 | Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$.
(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the probability that these three segments can form an isosceles triangle. | \frac{7}{18} | 10.15625 |
23,479 | Given that $\frac {π}{2}＜α＜ \frac {3π}{2}$, points A, B, and C are in the same plane rectangular coordinate system with coordinates A(3, 0), B(0, 3), and C(cosα, sinα) respectively.
(1) If $| \overrightarrow {AC}|=| \overrightarrow {BC}|$, find the value of angle α;
(2) When $\overrightarrow {AC}\cdot \overrightarrow {BC}=-1$, find the value of $\frac {2sin^{2}α+sin(2α)}{1+tan\alpha }$. | - \frac {5}{9} | 39.84375 |
23,480 | Given that $a > 0$, $b > 0$, and $\frac{1}{a} + \frac{2}{b} = 2$.
(1) Find the minimum value of $ab$;
(2) Find the minimum value of $a + 2b$, and find the corresponding values of $a$ and $b$. | \frac{9}{2} | 35.15625 |
23,481 | In a joint maritime search and rescue exercise between China and foreign countries, the Chinese side participated with 4 ships and 3 airplanes; the foreign side with 5 ships and 2 airplanes. If 2 units are selected from each group (either 1 airplane or 1 ship counts as one unit, and all ships and airplanes are distinct), and among the four selected units exactly one is an airplane, the total number of different selection methods is ___. | 180 | 7.8125 |
23,482 | What is the constant term of the expansion of $\left(5x + \frac{2}{5x}\right)^8$? | 1120 | 99.21875 |
23,483 | Take one point $M$ on the curve $y=\ln x$ and another point $N$ on the line $y=2x+6$, respectively. The minimum value of $|MN|$ is ______. | \dfrac {(7+\ln 2) \sqrt {5}}{5} | 0 |
23,484 | Given that the random variable $\xi$ follows a normal distribution $N(0,\sigma^{2})$, and $P(\xi > 2)=0.023$, determine $P(-2\leqslant \xi\leqslant 2)$. | 0.954 | 81.25 |
23,485 | The common ratio of the geometric sequence $a+\log_{2}3$, $a+\log_{4}3$, $a+\log_{8}3$ is __________. | \frac{1}{3} | 31.25 |
23,486 | Given the polynomial $f(x) = 4x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$, find the value of $V_1$ when calculating $f(5)$ using the Horner's Method. | 22 | 64.0625 |
23,487 | Justine has two fair dice, one with sides labeled $1,2,\ldots, m$ and one with sides labeled $1,2,\ldots, n.$ She rolls both dice once. If $\tfrac{3}{20}$ is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible values of $m+n$ .
*Proposed by Justin Stevens* | 996 | 32.8125 |
23,488 | Seven thousand twenty-two can be expressed as the sum of a two-digit number and a four-digit number. | 7022 | 0 |
23,489 | Suppose $\cos S = 0.5$ in the diagram below. What is $ST$?
[asy]
pair P,S,T;
P = (0,0);
S = (6,0);
T = (0,6*tan(acos(0.5)));
draw(P--S--T--P);
draw(rightanglemark(S,P,T,18));
label("$P$",P,SW);
label("$S$",S,SE);
label("$T$",T,N);
label("$10$",S/2,S);
[/asy] | 20 | 51.5625 |
23,490 | Let \\(n\\) be a positive integer, and \\(f(n) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\\). It is calculated that \\(f(2) = \frac{3}{2}\\), \\(f(4) > 2\\), \\(f(8) > \frac{5}{2}\\), and \\(f(16) > 3\\). Observing the results above, according to the pattern, it can be inferred that \\(f(128) > \_\_\_\_\_\_\_\_． | \frac{9}{2} | 93.75 |
23,491 | In the Cartesian coordinate system $(xOy)$, a curve $C_{1}$ is defined by the parametric equations $x=\cos{\theta}$ and $y=\sin{\theta}$, and a line $l$ is defined by the polar equation $\rho(2\cos{\theta} - \sin{\theta}) = 6$.
1. Find the Cartesian equations for the curve $C_{1}$ and the line $l$.
2. Find a point $P$ on the curve $C_{1}$ such that the distance from $P$ to the line $l$ is minimized, and compute this minimum distance. | \frac{6\sqrt{5}}{5} - 1 | 3.125 |
23,492 | From 5 male doctors and 4 female doctors, 3 doctors are to be selected to form a medical team, with the requirement that both male and female doctors must be included. Calculate the total number of different team formation plans. | 70 | 99.21875 |
23,493 | Five students, labeled as A, B, C, D, and E, are standing in a row to participate in a literary performance. If student A does not stand at either end, calculate the number of different arrangements where students C and D are adjacent. | 24 | 26.5625 |
23,494 | What is the largest perfect square factor of 4410? | 441 | 97.65625 |
23,495 | Given the total number of stations is 6 and 3 of them are selected for getting off, calculate the probability that person A and person B get off at different stations. | \frac{2}{3} | 6.25 |
23,496 | Calculate the value of $333 + 33 + 3$. | 369 | 96.875 |
23,497 | How many digits are there in the number \(N\) if \(N=2^{12} \times 5^{8}\) ?
If \(\left(2^{48}-1\right)\) is divisible by two whole numbers between 60 and 70, find them.
Given \(2^{\frac{1}{2}} \times 9^{\frac{1}{9}}\) and \(3^{\frac{1}{3}} \times 8^{\frac{1}{8}}\), what is the greatest number? | 3^{\frac{1}{3}} \times 8^{\frac{1}{8}} | 19.53125 |
23,498 | In how many ways can the number 210 be factored into a product of four natural numbers? The order of the factors does not matter. | 15 | 47.65625 |
23,499 | A high school second-year class has a total of 1000 students. To investigate the vision condition of the students in this grade, if 50 samples are selected using the systematic sampling method, and all students are randomly numbered from 000, 001, 002, …, to 999, if it is determined that the second number is drawn in each group, then the student number drawn in the 6th group is. | 101 | 70.3125 |
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