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24,500 | In a sequence, if for all $n \in \mathbb{N}^0$, we have $a_na_{n+1}a_{n+2} = k$ (where $k$ is a constant), then this sequence is called a geometric sequence, and $k$ is called the common product. Given that the sequence $a_n$ is a geometric sequence, and $a_1 = 1$, $a_2 = 2$, with a common product of 8, find the sum $a_1 + a_2 + a_3 + \ldots + a_{12}$. | 28 | 83.59375 |
24,501 | In a triangle with sides of lengths 13, 14, and 15, the orthocenter is denoted by \( H \). The altitude from vertex \( A \) to the side of length 14 is \( A D \). What is the ratio \( \frac{H D}{H A} \)? | 5:11 | 0 |
24,502 | Given the number 105,000, express it in scientific notation. | 1.05\times 10^{5} | 0 |
24,503 | What is the sum of all two-digit positive integers whose squares end with the digits 25? | 644 | 0 |
24,504 | We define $|\begin{array}{l}{a}&{c}\\{b}&{d}\end{array}|=ad-bc$. For example, $|\begin{array}{l}{1}&{3}\\{2}&{4}\end{array}|=1\times 4-2\times 3=4-6=-2$. If $x$ and $y$ are integers, and satisfy $1 \lt |\begin{array}{l}{2}&{y}\\{x}&{3}\end{array}| \lt 3$, then the minimum value of $x+y$ is ____. | -5 | 52.34375 |
24,505 | Given Alex, Jamie, and Casey play a game over 8 rounds, and for each round, the probability Alex wins is $\frac{1}{3}$, and Jamie is three times as likely to win as Casey, calculate the probability that Alex wins four rounds, Jamie wins three rounds, and Casey wins one round. | \frac{35}{486} | 11.71875 |
24,506 | Given the function $f(x) = -x^3 + ax^2 - 4$ has an extremum at $x = 2$, and $m, n \in [-1, 1]$, then the minimum value of $f(m) + f'(n)$ is \_\_\_\_\_\_\_\_. | -13 | 30.46875 |
24,507 | In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, how far is point $F$ from the midpoint of segment $DE$? | 7.5 | 16.40625 |
24,508 | Triangles $\triangle DEF$ and $\triangle D'E'F'$ lie in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(15,0)$, $D'(15,25)$, $E'(25,25)$, $F'(15,10)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\triangle DEF$ to $\triangle D'E'F'$. Find $n+u+v$. | 115 | 7.8125 |
24,509 | Given circle $C_1$: $(x-1)^2+(y-2)^2=1$
(1) Find the equation of the tangent line to circle $C_1$ passing through point $P(2,4)$.
(2) If circle $C_1$ intersects with circle $C_2$: $(x+1)^2+(y-1)^2=4$ at points $A$ and $B$, find the length of segment $AB$. | \frac {4 \sqrt {5}}{5} | 0 |
24,510 | Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are $S_n$ and $T_n$, respectively. For any positive integer $n$, it holds that $$\frac {S_{n}}{T_{n}} = \frac {3n+5}{2n+3}$$, then $$\frac {a_{7}}{b_{7}} = \_\_\_\_\_\_ ．$$ | \frac {44}{29} | 37.5 |
24,511 | Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 8.$ Find the maximum value of
\[\sqrt{3x + 2} + \sqrt{3y + 2} + \sqrt{3z + 2}.\] | 3\sqrt{10} | 65.625 |
24,512 | Given two lines $ l_1: x + my + 6 = 0 $ and $ l_2: (m-2)x + 3y + 2m = 0 $, if $ l_1 \parallel l_2 $, then the distance between $ l_1 $ and $ l_2 $ is __________. | \frac{8\sqrt{2}}{3} | 86.71875 |
24,513 | 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 | 78.90625 |
24,514 | The number $n$ is a three-digit integer and is the product of two distinct prime factors $x$ and $10x+y$, where $x$ and $y$ are each less than 10, with no restrictions on $y$ being prime. What is the largest possible value of $n$? | 553 | 0.78125 |
24,515 | Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 15x\]for all integers $x.$ Find the smallest possible value of $a.$ | 15 | 83.59375 |
24,516 | School A and School B each have 3 teachers signed up for a teaching support program, with School A having 2 males and 1 female, and School B having 1 male and 2 females.
(1) If one teacher is randomly selected from each of the schools, list all possible outcomes and calculate the probability that the two selected teachers are of the same gender;
(2) If 2 teachers are randomly selected from the total of 6 teachers, list all possible outcomes and calculate the probability that the two selected teachers come from the same school. | \frac{2}{5} | 88.28125 |
24,517 |
A secret facility is in the shape of a rectangle measuring $200 \times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by the shortest paths along the external perimeter (while the intruder remained in place). Three guards ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder? | 150 | 83.59375 |
24,518 | For all values of $x$ for which it is defined, let $g(x) = \cot \frac{x}{2} - \cot 2x$. This expression can be written as
\[g(x) = \frac{\sin kx}{\sin \frac{x}{2} \sin 2x}.\]
Find the value of $k$. | \frac{3}{2} | 95.3125 |
24,519 | Given the expansion of $(\sqrt{x} + \frac{2}{x^2})^n$, the ratio of the coefficient of the fifth term to the coefficient of the third term is 56:3.
(Ⅰ) Find the constant term in the expansion;
(Ⅱ) When $x=4$, find the term with the maximum binomial coefficient in the expansion. | \frac{63}{256} | 19.53125 |
24,520 | Simplify $\sqrt{9800}$. | 70\sqrt{2} | 54.6875 |
24,521 | The table below lists the air distances in kilometers between several cities. If two different cities from the table are chosen at random, what is the probability that the distance between them is less than 9000 kilometers? Express your answer as a common fraction.
\begin{tabular}{|c|c|c|c|c|}
\hline
& Sydney & New York & Tokyo & Paris \\ \hline
Sydney & & 16000 & 7800 & 16900 \\ \hline
New York & 16000 & & 10800 & 5860 \\ \hline
Tokyo & 7800 & 10800 & & 9700 \\ \hline
Paris & 16900 & 5860 & 9700 & \\ \hline
\end{tabular} | \frac{1}{3} | 19.53125 |
24,522 | A net for hexagonal pyramid is constructed by placing a triangle with side lengths $x$ , $x$ , and $y$ on each side of a regular hexagon with side length $y$ . What is the maximum volume of the pyramid formed by the net if $x+y=20$ ? | 128\sqrt{15} | 0.78125 |
24,523 | Given $a, b \in \mathbb{R}$ and $a^{2}+2b^{2}=6$, find the minimum value of $a+ \sqrt{2}b$. | -2\sqrt{3} | 85.9375 |
24,524 | In the Cartesian coordinate system $(xOy)$, an ellipse $(C)$ is defined by the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \frac{\sqrt{2}}{2}$. The point $P(2, 1)$ lies on the ellipse $(C)$.
(1) Find the equation of the ellipse $(C)$;
(2) If points $A$ and $B$ both lie on the ellipse $(C)$ and the midpoint $M$ of $AB$ lies on the line segment $OP$ (excluding endpoints).
$\quad\quad$ (a) Find the slope of the line $AB$;
$\quad\quad$ (b) Find the maximum area of $\triangle AOB$. | \frac{3 \sqrt{2}}{2} | 32.8125 |
24,525 | In triangle $ABC$, the sides opposite to the angles $A$, $B$, and $C$ are labeled as $a$, $b$, and $c$ respectively. Given $a=5, B= \frac {\pi}{3},$ and $\cos A= \frac {11}{14}$, find the area $S$ of the triangle $ABC$. | 10 \sqrt {3} | 0 |
24,526 | Calculate the area of the smallest square that can completely contain a circle with a radius of 7 units. | 196 | 99.21875 |
24,527 | The graph of $y = \frac{p(x)}{q(x)}$ where $p(x)$ is quadratic and $q(x)$ is quadratic is given conceptually (imagine a graph with necessary features). The function has vertical asymptotes at $x = -4$ and $x = 1$. The graph passes through the point $(0,0)$ and $(2,-1)$. Determine $\frac{p(-1)}{q(-1)}$ if $q(x) = (x+4)(x-1)$ and $p(x) = kx + m$. | -\frac{1}{2} | 82.8125 |
24,528 | Let P be a moving point on the line $3x+4y+3=0$, and through point P, two tangents are drawn to the circle $C: x^2+y^2-2x-2y+1=0$, with the points of tangency being A and B, respectively. Find the minimum value of the area of quadrilateral PACB. | \sqrt{3} | 59.375 |
24,529 | Calculate $4535_6 + 23243_6$. Express your answer in base $6$. | 32222_6 | 40.625 |
24,530 | Given a circle $C: (x-3)^{2}+(y-4)^{2}=1$, and points $A(-1,0)$, $B(1,0)$, let $P$ be a moving point on the circle, then the maximum and minimum values of $d=|PA|^{2}+|PB|^{2}$ are \_\_\_\_\_\_ and \_\_\_\_\_\_ respectively. | 34 | 55.46875 |
24,531 | Find $b^2$ if the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{b^2} = 1$ and the foci of the hyperbola
\[\frac{x^2}{196} - \frac{y^2}{121} = \frac{1}{49}\] coincide. | \frac{908}{49} | 82.03125 |
24,532 | Find the length of the common chord of the circle $x^{2}+y^{2}=50$ and $x^{2}+y^{2}-12x-6y+40=0$. | 2\sqrt{5} | 63.28125 |
24,533 | Given positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{8}$, find the least possible value of $xy$. | 96 | 32.8125 |
24,534 | The number of children in the families $A$, $B$, $C$, $D$, and $E$ are shown in the table below:
| | $A$ | $B$ | $C$ | $D$ | $E$ |
|---------|-----|-----|-----|-----|-----|
| Boys | $0$ | $1$ | $0$ | $1$ | $1$ |
| Girls | $0$ | $0$ | $1$ | $1$ | $2$ |
$(1)$ If a girl is randomly selected from these children, given that the selected child is a girl, find the probability that the girl is from family $E$;
$(2)$ If three families are selected randomly from these $5$ families, and $X$ represents the number of families where girls outnumber boys, find the probability distribution and expectation of $X$. | \frac{6}{5} | 43.75 |
24,535 | In $\triangle ABC$, $B(-\sqrt{5}, 0)$, $C(\sqrt{5}, 0)$, and the sum of the lengths of the medians on sides $AB$ and $AC$ is $9$.
(Ⅰ) Find the equation of the trajectory of the centroid $G$ of $\triangle ABC$.
(Ⅱ) Let $P$ be any point on the trajectory found in (Ⅰ), find the minimum value of $\cos\angle BPC$. | -\frac{1}{9} | 5.46875 |
24,536 | Let the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ be $a$, $b$, $c$ respectively, and $\cos B= \frac {3}{5}$, $b=2$
(Ⅰ) When $A=30^{\circ}$, find the value of $a$;
(Ⅱ) When the area of $\triangle ABC$ is $3$, find the value of $a+c$. | 2 \sqrt {7} | 0 |
24,537 | A factory's total cost of producing $x$ units of a product is $c(x) = 1200 + \frac{2}{75}x^{3}$ (in ten thousand yuan). It is known that the unit price $P$ (in ten thousand yuan) of the product and the number of units $x$ satisfy: $P^{2} = \frac{k}{x}$. The unit price for producing 100 units of this product is 50 ten thousand yuan. How many units should be produced to maximize the total profit? | 25 | 82.8125 |
24,538 | The sum of 36 consecutive integers is $6^4$. What is their median? | 36 | 42.96875 |
24,539 | a) Vanya flips a coin 3 times, and Tanya flips a coin 2 times. What is the probability that Vanya gets more heads than Tanya?
b) Vanya flips a coin $n+1$ times, and Tanya flips a coin $n$ times. What is the probability that Vanya gets more heads than Tanya? | \frac{1}{2} | 80.46875 |
24,540 | Calculate the value of $(-3 \frac{3}{8})^{- \frac{2}{3}}$. | \frac{4}{9} | 65.625 |
24,541 | Determine the seventh element in Row 20 of Pascal's triangle. | 38760 | 94.53125 |
24,542 | A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 307,307,307 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form? | 1001001 | 46.875 |
24,543 | A store sells a batch of football souvenir books, with a cost price of $40$ yuan per book and a selling price of $44$ yuan per book. The store can sell 300 books per day. The store decides to increase the selling price, and after investigation, it is found that for every $1$ yuan increase in price, the daily sales decrease by 10 books. Let the new selling price after the increase be $x$ yuan $\left(44\leqslant x\leqslant 52\right)$, and let the daily sales be $y$ books.
$(1)$ Write down the functional relationship between $y$ and $x$;
$(2)$ At what price per book will the store maximize the profit from selling the souvenir books each day, and what is the maximum profit in yuan? | 2640 | 70.3125 |
24,544 | Cara is sitting at a circular table with her seven friends. Two of her friends, Alice and Bob, insist on sitting together but not next to Cara. How many different possible pairs of people could Cara be sitting between? | 10 | 5.46875 |
24,545 | Given the distribution of the random variable $\xi$ is $P(\xi=x)= \dfrac{xk}{15}$, where $x$ takes values $(1,2,3,4,5)$, find the value of $P\left( \left. \dfrac{1}{2} < \xi < \dfrac{5}{2} \right. \right)$. | \dfrac{1}{5} | 97.65625 |
24,546 | Throw 6 dice at a time, find the probability, in the lowest form, such that there will be exactly four kinds of the outcome. | 325/648 | 32.8125 |
24,547 | In $\triangle ABC$, $\sqrt {2}csinAcosB=asinC$.
(I) Find the measure of $\angle B$;
(II) If the area of $\triangle ABC$ is $a^2$, find the value of $cosA$. | \frac {3 \sqrt {10}}{10} | 0 |
24,548 | Given a randomly selected number $x$ in the interval $[0,\pi]$, determine the probability of the event "$-1 \leqslant \tan x \leqslant \sqrt {3}$". | \dfrac{7}{12} | 67.96875 |
24,549 | Calculate the value of $\text{rem} \left(\frac{5}{7}, \frac{3}{4}\right)$ and then multiply the result by $-2$. | -\frac{10}{7} | 82.8125 |
24,550 | Without using a calculator, find the largest prime factor of $17^4 + 2 \times 17^2 + 1 - 16^4$. | 17 | 82.03125 |
24,551 | Given the sequence \(\{a_{n}\}\) with the sum of the first \(n\) terms \(S_{n} = n^{2} - 1\) \((n \in \mathbf{N}_{+})\), find \(a_{1} + a_{3} + a_{5} + a_{7} + a_{9} = \). | 44 | 82.8125 |
24,552 | Given sets $A=\{-1,1,2\}$ and $B=\{-2,1,2\}$, a number $k$ is randomly selected from set $A$ and a number $b$ is randomly selected from set $B$. The probability that the line $y=kx+b$ does not pass through the third quadrant is $\_\_\_\_\_\_$. | P = \frac{2}{9} | 0.78125 |
24,553 | Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$ respectively. It is known that $\left(\sin C+\sin B\right)\left(c-b\right)=a\left(\sin A-\sin B\right)$.
$(1)$ Find the measure of angle $C$.
$(2)$ If the angle bisector of $\angle ACB$ intersects $AB$ at point $D$ and $CD=2$, $AD=2DB$, find the area of triangle $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 66.40625 |
24,554 |
A firecracker was thrown vertically upwards with a speed of 20 m/s. Three seconds after the start of its flight, it exploded into two unequal parts, the mass ratio of which is $1: 2$. The smaller fragment immediately after the explosion flew horizontally at a speed of $16 \mathrm{~m}/\mathrm{s}$. Find the magnitude of the speed of the second fragment (in m/s) immediately after the explosion. Assume the acceleration due to gravity to be $10 \mathrm{~m}/\mathrm{s}^{2}$. | 17 | 65.625 |
24,555 | Let \( X \), \( Y \), and \( Z \) be nonnegative integers such that \( X+Y+Z = 15 \). What is the maximum value of \[ X\cdot Y\cdot Z + X\cdot Y + Y\cdot Z + Z\cdot X ? \] | 200 | 96.09375 |
24,556 | If $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \ldots + a_{11}(x+2)^{11}$, then the value of $a_0 + a_1 + \ldots + a_{11}$ is. | -2 | 96.09375 |
24,557 | Given the function $f(x) = \frac{x^2}{1+x^2}$.
$(1)$ Calculate the values of $f(2) + f\left(\frac{1}{2}\right)$, $f(3) + f\left(\frac{1}{3}\right)$, $f(4) + f\left(\frac{1}{4}\right)$, and infer a general conclusion (proof not required);
$(2)$ Calculate the value of $2f(2) + 2f(3) + \ldots + 2f(2017) + f\left(\frac{1}{2}\right) + f\left(\frac{1}{3}\right) + \ldots + f\left(\frac{1}{2017}\right) + \frac{1}{2^2}f(2) + \frac{1}{3^2}f(3) + \ldots + \frac{1}{2017^2} \cdot f(2017)$. | 4032 | 48.4375 |
24,558 | If the minute hand is moved back by 5 minutes, the number of radians it has turned is __________. | \frac{\pi}{6} | 85.9375 |
24,559 | In the triangular pyramid $SABC$, the height $SO$ passes through point $O$ - the center of the circle inscribed in the base $ABC$ of the pyramid. It is known that $\angle SAC = 60^\circ$, $\angle SCA = 45^\circ$, and the ratio of the area of triangle $AOB$ to the area of triangle $ABC$ is $\frac{1}{2 + \sqrt{3}}$. Find the angle $\angle BSC$. | 75 | 8.59375 |
24,560 | Given $y_1 = x^2 - 7x + 6$, $y_2 = 7x - 3$, and $y = y_1 + xy_2$, find the value of $y$ when $x = 2$. | 18 | 74.21875 |
24,561 | In the Cartesian coordinate system, the equation of circle C is $x^2 + y^2 - 4x = 0$, and its center is point C. Consider the polar coordinate system with the origin as the pole and the non-negative half of the x-axis as the polar axis. Curve $C_1: \rho = -4\sqrt{3}\sin\theta$ intersects circle C at points A and B.
(1) Find the polar equation of line AB.
(2) If line $C_2$ passing through point C(2, 0) is parameterized by $\begin{cases} x = 2 + \frac{\sqrt{3}}{2}t \\ y = \frac{1}{2}t \end{cases}$ (where t is a parameter) and meets line AB at point D and the y-axis at point E, find the value of $|CD|:|CE|$. | 1:2 | 1.5625 |
24,562 | In the expansion of \((x+y+z)^{8}\), what is the sum of the coefficients for all terms of the form \(x^{2} y^{a} z^{b}\) (where \(a, b \in \mathbf{N})\)? | 1792 | 78.90625 |
24,563 | What is the greatest product obtainable from two integers whose sum is 2016? | 1016064 | 53.90625 |
24,564 | Given the coefficient of determination R^2 for four different regression models, where the R^2 values are 0.98, 0.67, 0.85, and 0.36, determine which model has the best fitting effect. | 0.98 | 73.4375 |
24,565 | Given vectors $\overrightarrow{m} = (a\cos x, \cos x)$ and $\overrightarrow{n} = (2\cos x, b\sin x)$, with a function $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$ that satisfies $f(0) = 2$ and $f\left(\frac{\pi}{3}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2}$,
(1) If $x \in \left[0, \frac{\pi}{2}\right]$, find the maximum and minimum values of $f(x)$;
(2) If $f\left(\frac{\theta}{2}\right) = \frac{3}{2}$, and $\theta$ is an internal angle of a triangle, find $\tan \theta$. | -\frac{4 + \sqrt{7}}{3} | 9.375 |
24,566 | If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$ , compute the value of the expression
\[
\left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
\left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
\left(
\frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}}
- \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}}
\right).
\] | -8 | 86.71875 |
24,567 | Given a geometric sequence $\{a_n\}$ with positive terms, the sum of the first $n$ terms is $S_n$. If $-3$, $S_5$, and $S_{10}$ form an arithmetic sequence, calculate the minimum value of $S_{15} - S_{10}$. | 12 | 20.3125 |
24,568 | Let $x = (2 + \sqrt{2})^6,$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$ Find
\[
x(1 - f).
\] | 64 | 4.6875 |
24,569 | There are four balls in a bag, each with the same shape and size, and their numbers are \\(1\\), \\(2\\), \\(3\\), and \\(4\\).
\\((1)\\) Draw two balls randomly from the bag. Calculate the probability that the sum of the numbers on the balls drawn is no greater than \\(4\\).
\\((2)\\) First, draw a ball randomly from the bag, and its number is \\(m\\). Put the ball back into the bag, then draw another ball randomly, and its number is \\(n\\). Calculate the probability that \\(n < m + 2\\). | \dfrac{13}{16} | 72.65625 |
24,570 | Given the expression \[2 - (-3) - 4 - (-5) - 6 - (-7) \times 2,\] calculate its value. | -14 | 0 |
24,571 | A basketball team has 15 available players. Initially, 5 players start the game, and the other 10 are available as substitutes. The coach can make up to 4 substitutions during the game, under the same rules as the soccer game—no reentry for substituted players and each substitution is distinct. Calculate the number of ways the coach can make these substitutions and find the remainder when divided by 100. | 51 | 22.65625 |
24,572 | Find an approximate value of $0.998^6$ such that the error is less than $0.001$. | 0.988 | 35.15625 |
24,573 | There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$ . What is the maximum possible number of cups in the kitchen? | 29 | 89.84375 |
24,574 | There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer "Yes" or "No" to the question: "Are there more knights on the island than liars?". They answered in turn such that everyone else could hear. Knights told the truth, liars lied. Each follower gave the same answer as the majority of those who answered before them, and if "Yes" and "No" answers were equal, they gave either answer. It turned out that the number of "Yes" answers was exactly 1009. What is the maximum number of followers that could have been among the islanders? | 1009 | 42.1875 |
24,575 | Find all irreducible fractions \( \frac{a}{b} \) that can be represented in the form \( b, a \) (comma separates the decimal representations of natural numbers \( b \) and \( a \)). | \frac{5}{2} | 7.8125 |
24,576 | Given the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ $(a > b > 0)$, where $M$ and $N$ are the left and right vertices of the ellipse, and $P$ is any point on the ellipse. The slopes of the lines $PM$ and $PN$ are ${k_{1}}$ and ${k_{2}}$ respectively, and ${k_{1}}{k_{2}} \neq 0$. If the minimum value of $|{k_{1}}| + |{k_{2}}|$ is $1$, find the eccentricity of the ellipse. | \frac{\sqrt{3}}{2} | 92.96875 |
24,577 | Jenny places a total of 30 red Easter eggs in several green baskets and a total of 45 orange Easter eggs in some blue baskets. Each basket contains the same number of eggs and there are at least 5 eggs in each basket. How many eggs did Jenny put in each basket? | 15 | 55.46875 |
24,578 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2B=A+C$ and $a+\sqrt{2}b=2c$, find the value of $\sin C$. | \frac{\sqrt{6}+\sqrt{2}}{4} | 28.125 |
24,579 | The region consisting of all points in three-dimensional space within $4$ units of line segment $\overline{CD}$ has volume $544\pi$. Calculate the length of $CD$. | \frac{86}{3} | 63.28125 |
24,580 | The base nine numbers $125_9$ and $33_9$ need to be multiplied and the result expressed in base nine. What is the base nine sum of the digits of their product? | 16 | 71.09375 |
24,581 | Add 78.1563 to 24.3981 and round to the nearest hundredth. | 102.55 | 68.75 |
24,582 | What is the sum of all the even integers between $200$ and $400$? | 30300 | 7.03125 |
24,583 | Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$. | 40 | 9.375 |
24,584 | A wooden block is 5 inches long, 5 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into twenty-five 1-inch cubes. How many of the resulting cubes each have a total number of red faces that is an odd number? | 13 | 17.96875 |
24,585 | In triangle $ABC$, where $\angle A = 90^\circ$, $BC = 20$, and $\tan C = 4\cos B$. Find the length of $AB$. | 5\sqrt{15} | 69.53125 |
24,586 | Let \( a \) and \( b \) be integers such that \( ab = 144 \). Find the minimum value of \( a + b \). | -145 | 64.0625 |
24,587 | The value of \( 444 - 44 - 4 \) is | 396 | 94.53125 |
24,588 | Given the function $f(x)=\ln x+ax (a\in R)$.
(1) When $a=- \frac{1}{3}$, find the extreme values of the function $f(x)$ in the interval $[e,e^{2}]$;
(2) When $a=1$, the function $g(x)=f(x)- \frac{2}{t}x^{2}$ has only one zero point. Find the value of the positive number $t$. | t=2 | 29.6875 |
24,589 | Given a function $f(x) = m\ln{x} + nx$ whose tangent at point $(1, f(1))$ is parallel to the line $x + y - 2 = 0$, and $f(1) = -2$, where $m, n \in \mathbb{R}$,
(Ⅰ) Find the values of $m$ and $n$, and determine the intervals of monotonicity for the function $f(x)$;
(Ⅱ) Let $g(x)= \frac{1}{t}(-x^{2} + 2x)$, for a positive real number $t$. If there exists $x_0 \in [1, e]$ such that $f(x_0) + x_0 \geq g(x_0)$ holds, find the maximum value of $t$. | \frac{e(e - 2)}{e - 1} | 2.34375 |
24,590 | In a circuit with two independently controlled automatic switches connected in parallel, the probability that the circuit can operate normally is given that one of the switches can close, and the probabilities of the switches being able to close are $0.5$ and $0.7$ respectively, calculate this probability. | 0.85 | 96.875 |
24,591 | Sally's salary in 2006 was $\$ 37,500 $. For 2007 she got a salary increase of $ x $ percent. For 2008 she got another salary increase of $ x $ percent. For 2009 she got a salary decrease of $ 2x $ percent. Her 2009 salary is $ \ $34,825$ . Suppose instead, Sally had gotten a $2x$ percent salary decrease for 2007, an $x$ percent salary increase for 2008, and an $x$ percent salary increase for 2009. What would her 2009 salary be then? | 34825 | 10.9375 |
24,592 | What is the least five-digit positive integer which is congruent to 7 (mod 12)? | 10,003 | 0 |
24,593 | A container holds $47\frac{2}{3}$ cups of sugar. If one recipe requires $1\frac{1}{2}$ cups of sugar, how many full recipes can be made with the sugar in the container? Express your answer as a mixed number. | 31\frac{7}{9} | 72.65625 |
24,594 | Find the equation of the circle that is tangent to the x-axis, has its center on the line $3x - y = 0$, and the chord cut by the line $x - y = 0$ has a length of 2. | \frac{9}{7} | 0 |
24,595 | Simplify $({\frac{3}{{a+1}}-a+1})÷\frac{{{a^2}-4}}{{{a^2}+2a+1}}$, then choose a suitable number from $-1$, $2$, $3$ to substitute and evaluate. | -4 | 46.09375 |
24,596 | Paul wrote the list of all four-digit numbers such that the hundreds digit is $5$ and the tens digit is $7$ . For example, $1573$ and $7570$ are on Paul's list, but $2754$ and $571$ are not. Find the sum of all the numbers on Pablo's list. $Note$ . The numbers on Pablo's list cannot start with zero. | 501705 | 42.1875 |
24,597 | Given the function $$f(x)= \begin{cases} \sqrt {x}+3, & x\geq0 \\ ax+b, & x<0\end{cases}$$ satisfies the condition: $y=f(x)$ is a monotonic function on $\mathbb{R}$ and $f(a)=-f(b)=4$, then the value of $f(-1)$ is \_\_\_\_\_\_. | -3 | 62.5 |
24,598 | Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? | \frac{9\pi - 18}{2} | 17.96875 |
24,599 | Find the maximum value of the expression \(\cos(x+y)\) given that \(\cos x - \cos y = \frac{1}{4}\). | 31/32 | 10.9375 |
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