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32,300 | Find the number of sets $A$ that satisfy the three conditions: $\star$ $A$ is a set of two positive integers $\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number $\star$ $A$ contains the number $30.$ | 129 | 3.125 |
32,301 | Reading material: For non-zero real numbers $a$ and $b$, if the value of the fraction $\frac{(x-a)(x-b)}{x}$ with respect to $x$ is zero, then the solutions are $x_{1}=a$ and $x_{2}=b$. Also, because $\frac{(x-a)(x-b)}{x}=\frac{{x}^{2}-(a+b)x+ab}{x}=x+\frac{ab}{x}-\left(a+b\right)$, the solutions to the equation $x+\frac{ab}{x}=a+b$ with respect to $x$ are $x_{1}=a$ and $x_{2}=b$.
$(1)$ Understanding and application: The solutions to the equation $\frac{{x}^{2}+2}{x}=5+\frac{2}{5}$ are: $x_{1}=$______, $x_{2}=$______;
$(2)$ Knowledge transfer: If the solutions to the equation $x+\frac{3}{x}=7$ are $x_{1}=a$ and $x_{2}=b$, find the value of $a^{2}+b^{2}$;
$(3)$ Extension and enhancement: If the solutions to the equation $\frac{6}{x-1}=k-x$ are $x_{1}=t+1$ and $x_{2}=t^{2}+2$, find the value of $k^{2}-4k+4t^{3}$. | 32 | 7.03125 |
32,302 | Given the function $f(x)=ax+b\sin x\ (0 < x < \frac {π}{2})$, if $a\neq b$ and $a, b\in \{-2,0,1,2\}$, the probability that the slope of the tangent line at any point on the graph of $f(x)$ is non-negative is ___. | \frac {7}{12} | 10.15625 |
32,303 | In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $C=\frac{\pi}{3}$.
1. If the area of $\triangle ABC$ is $\sqrt{3}$, find $a$ and $b$.
2. If $\sin B = 2\sin A$, find the area of $\triangle ABC$. | \frac{4\sqrt{3}}{3} | 0 |
32,304 | Given that points A and B lie on the graph of y = \frac{1}{x} in the first quadrant, ∠OAB = 90°, and AO = AB, find the area of the isosceles right triangle ∆OAB. | \frac{\sqrt{5}}{2} | 0 |
32,305 | In the rectangular coordinate system $xOy$, the equation of line $C_1$ is $y=-\sqrt{3}x$, and the parametric equations of curve $C_2$ are given by $\begin{cases}x=-\sqrt{3}+\cos\varphi\\y=-2+\sin\varphi\end{cases}$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis.
(I) Find the polar equation of $C_1$ and the rectangular equation of $C_2$;
(II) Rotate line $C_1$ counterclockwise around the coordinate origin by an angle of $\frac{\pi}{3}$ to obtain line $C_3$, which intersects curve $C_2$ at points $A$ and $B$. Find the length $|AB|$. | \sqrt{3} | 3.90625 |
32,306 | In isosceles trapezoid $ABCD$ where $AB$ (shorter base) is 10 and $CD$ (longer base) is 20. The non-parallel sides $AD$ and $BC$ are extended to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? | \frac{1}{3} | 50.78125 |
32,307 | Square $PQRS$ has sides of length 1. Points $T$ and $U$ are on $\overline{QR}$ and $\overline{RS}$, respectively, so that $\triangle PTU$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\overline{PT}.$ The length of a side of this smaller square is $\frac{a-\sqrt{b}}{c},$ where $a, b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b+c.$ | 12 | 16.40625 |
32,308 | Compute
$$\sum_{k=1}^{2000} k(\lceil \log_{\sqrt{3}}{k}\rceil - \lfloor\log_{\sqrt{3}}{k} \rfloor).$$ | 1999907 | 67.96875 |
32,309 | Given the function $f(x) = x + a\ln x$ has its tangent line at $x = 1$ perpendicular to the line $x + 2y = 0$, and the function $g(x) = f(x) + \frac{1}{2}x^2 - bx$,
(Ⅰ) Determine the value of the real number $a$;
(Ⅱ) Let $x_1$ and $x_2$ ($x_1 < x_2$) be two extreme points of the function $g(x)$. If $b \geq \frac{7}{2}$, find the minimum value of $g(x_1) - g(x_2)$. | \frac{15}{8} - 2\ln 2 | 10.9375 |
32,310 | In the sequence $\{a_n\}$, $a_1=1$, $a_2=2$, and $a_{n+2} - a_n = 1 + (-1)^n$ ($n \in \mathbb{N}^*$), then the sum $a_1 + a_2 + \ldots + a_{51} =$ ? | 676 | 57.03125 |
32,311 | Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 30^\circ, \sin 30^\circ), (\cos 45^\circ, \sin 45^\circ), \text{ and } (\cos s^\circ, \sin s^\circ)\] is isosceles and its area is greater than $0.1$.
A) 15
B) 30
C) 45
D) 60 | 60 | 5.46875 |
32,312 | Given the sets $A={1,4,x}$ and $B={1,2x,x^{2}}$, if $A \cap B={4,1}$, find the value of $x$. | -2 | 43.75 |
32,313 | As shown in the diagram, there are 12 points on the circumference of a circle, dividing the circumference into 12 equal parts. How many rectangles can be formed using these equally divided points as the four vertices? | 15 | 75 |
32,314 | Express $0.6\overline{03}$ as a common fraction. | \frac{104}{165} | 0 |
32,315 | A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 16$. With the exception of the bottom row, each square rests on two squares in the row immediately below. In each square of the sixteenth row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $5$? | 16384 | 0.78125 |
32,316 | A projectile is fired with an initial speed $v$ from the ground at an angle between $0^\circ$ and $90^\circ$ to the horizontal. The trajectory of the projectile can be described by the parametric equations
\[
x = vt \cos \theta, \quad y = vt \sin \theta - \frac{1}{2} gt^2,
\]
where $t$ is the time, $g$ is the acceleration due to gravity, and $\theta$ varies from $0^\circ$ to $90^\circ$. As $\theta$ varies within this range, the highest points of the projectile paths trace out a curve. Calculate the area enclosed by this curve, which can be expressed as $c \cdot \frac{v^4}{g^2}$. Find the value of $c$. | \frac{\pi}{16} | 35.15625 |
32,317 | The sides of triangle $EFG$ are in the ratio of $3:4:5$. Segment $FK$ is the angle bisector drawn to the shortest side, dividing it into segments $EK$ and $KG$. What is the length, in inches, of the longer subsegment of side $EG$ if the length of side $EG$ is $15$ inches? Express your answer as a common fraction. | \frac{60}{7} | 46.09375 |
32,318 | Harry Potter can do any of the three tricks arbitrary number of times: $i)$ switch $1$ plum and $1$ pear with $2$ apples $ii)$ switch $1$ pear and $1$ apple with $3$ plums $iii)$ switch $1$ apple and $1$ plum with $4$ pears
In the beginning, Harry had $2012$ of plums, apples and pears, each. Harry did some tricks and now he has $2012$ apples, $2012$ pears and more than $2012$ plums. What is the minimal number of plums he can have? | 2025 | 17.1875 |
32,319 | A circle is inscribed in a right triangle. The point of tangency divides the hypotenuse into two segments measuring 6 cm and 7 cm. Calculate the area of the triangle. | 42 | 50.78125 |
32,320 | Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+6)=f(x)$. For $x \in [-3,-1)$, $f(x)=-(x+2)^{2}$, and for $x \in [-1,3)$, $f(x)=x$. Calculate the sum $f(1)+f(2)+f(3)+\ldots+f(2015)$. | 336 | 48.4375 |
32,321 | Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$ . Find the value of $2^{-(1+\log_23)x}$ | 216 | 66.40625 |
32,322 | Seventy percent of a train's passengers are women, and fifteen percent of those women are in the luxury compartment. What is the number of women in the luxury compartment if the train is carrying 300 passengers? | 32 | 92.96875 |
32,323 |
Find the root of the following equation to three significant digits:
$$
(\sqrt{5}-\sqrt{2})(1+x)=(\sqrt{6}-\sqrt{3})(1-x)
$$ | -0.068 | 35.9375 |
32,324 | Given that $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}+\overrightarrow{b})\perp\overrightarrow{a}$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 98.4375 |
32,325 | Gerard cuts a large rectangle into four smaller rectangles. The perimeters of three of these smaller rectangles are 16, 18, and 24. What is the perimeter of the fourth small rectangle?
A) 8
B) 10
C) 12
D) 14
E) 16 | 10 | 32.8125 |
32,326 | A high school math preparation group consists of six science teachers and two liberal arts teachers. During a three-day period of smog-related class suspensions, they need to arrange teachers to be on duty for question-answering sessions. The requirement is that each day, there must be one liberal arts teacher and two science teachers on duty. Each teacher should be on duty for at least one day and at most two days. How many different arrangements are possible? | 540 | 12.5 |
32,327 | Which of the following multiplication expressions has a product that is a multiple of 54? (Fill in the serial number).
$261 \times 345$
$234 \times 345$
$256 \times 345$
$562 \times 345$ | $234 \times 345$ | 0 |
32,328 | If the space vectors $\overrightarrow{{e_1}}$ and $\overrightarrow{{e_2}}$ satisfy $|\overrightarrow{{e_1}}|=|2\overrightarrow{{e_1}}+\overrightarrow{{e_2}}|=3$, determine the maximum value of the projection of $\overrightarrow{{e_1}}$ in the direction of $\overrightarrow{{e_2}}$. | -\frac{3\sqrt{3}}{2} | 1.5625 |
32,329 | Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$. | 2889 | 14.84375 |
32,330 | Roger collects the first 18 U.S. state quarters released in the order that the states joined the union. Five states joined the union during the decade 1790 through 1799. What fraction of Roger's 18 quarters represents states that joined the union during this decade? Express your answer as a common fraction. | \frac{5}{18} | 67.1875 |
32,331 | Given that the lateral surface of a cone is the semicircle with a radius of $2\sqrt{3}$, find the radius of the base of the cone. If the vertex of the cone and the circumference of its base lie on the surface of a sphere $O$, determine the volume of the sphere. | \frac{32\pi}{3} | 28.90625 |
32,332 | An ant starts from vertex \( A \) of rectangular prism \( ABCD-A_1B_1C_1D_1 \) and travels along the surface to reach vertex \( C_1 \) with the shortest distance being 6. What is the maximum volume of the rectangular prism? | 12\sqrt{3} | 1.5625 |
32,333 | Let the polynomial be defined as $$Q(x) = \left(\frac{x^{20} - 1}{x-1}\right)^2 - x^{20}.$$ Calculate the sum of the first five distinct $\alpha_k$ values where each zero of $Q(x)$ can be expressed in the complex form $z_k = r_k [\cos(2\pi \alpha_k) + i\sin(2\pi \alpha_k)]$, with $\alpha_k \in (0, 1)$ and $r_k > 0$. | \frac{3}{4} | 50.78125 |
32,334 | The average age of 8 people in a room is 35 years. A 22-year-old person leaves the room. Calculate the average age of the seven remaining people. | \frac{258}{7} | 3.90625 |
32,335 | Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$ , $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$ . A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$ , and $G_0\in \triangle A_1A_2A_3$ ; what is $s$ ? | 180/71 | 42.1875 |
32,336 | In the expansion of the binomial ${({(\frac{1}{x}}^{\frac{1}{4}}+{{x}^{2}}^{\frac{1}{3}})}^{n})$, the coefficient of the third last term is $45$. Find the coefficient of the term containing $x^{3}$. | 210 | 69.53125 |
32,337 | Line $m$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $30^{\circ}$ counterclockwise about the point $(10, 10)$ to form line $n$. Find the $x$-coordinate of the $x$-intercept of line $n$. | \frac{20\sqrt{3} + 20}{2\sqrt{3} + 3} | 0 |
32,338 | Given an ellipse in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A\left( 1,\frac{1}{2} \right)$.
(Ⅰ) Find the standard equation of the ellipse;
(Ⅱ) If a line passing through the origin $O$ intersects the ellipse at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$. | \sqrt {2} | 0 |
32,339 | In recent years, the food delivery industry in China has been developing rapidly, and delivery drivers shuttling through the streets of cities have become a beautiful scenery. A certain food delivery driver travels to and from $4$ different food delivery stores (numbered $1, 2, 3, 4$) every day. The rule is: he first picks up an order from store $1$, called the first pick-up, and then he goes to any of the other $3$ stores for the second pick-up, and so on. Assuming that starting from the second pick-up, he always goes to one of the other $3$ stores that he did not pick up from last time. Let event $A_{k}=\{$the $k$-th pick-up is exactly from store $1\}$, $P(A_{k})$ is the probability of event $A_{k}$ occurring. Obviously, $P(A_{1})=1$, $P(A_{2})=0$. Then $P(A_{3})=$______, $P(A_{10})=$______ (round the second answer to $0.01$). | 0.25 | 61.71875 |
32,340 | $D$ is a point on side $AB$ of triangle $ABC$, satisfying $AD=2$ and $DB=8$. Let $\angle ABC=\alpha$ and $\angle CAB=\beta$. <br/>$(1)$ When $CD\perp AB$ and $\beta =2\alpha$, find the value of $CD$; <br/>$(2)$ If $α+\beta=\frac{π}{4}$, find the maximum area of triangle $ACD$. | 5(\sqrt{2} - 1) | 5.46875 |
32,341 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, where $|\overrightarrow{a}|= \sqrt {2}$, $|\overrightarrow{b}|=2$, and $(\overrightarrow{a}-\overrightarrow{b})\perp \overrightarrow{a}$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} | 92.96875 |
32,342 | In the following two equations, the same Chinese character represents the same digit, and different Chinese characters represent different digits:
数字花园 + 探秘 = 2015,
探秘 + 1 + 2 + 3 + ... + 10 = 花园
So the four-digit 数字花园 = ______ | 1985 | 0 |
32,343 | Given that a rectangular room is 15 feet long and 108 inches wide, calculate the area of the new extended room after adding a 3 feet wide walkway along the entire length of one side, in square yards, where 1 yard equals 3 feet and 1 foot equals 12 inches. | 20 | 20.3125 |
32,344 | 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} | 25.78125 |
32,345 |
A printer received an annual order to print 10,000 posters each month. Each month's poster should have the month's name printed on it. Thus, he needs to print 10,000 posters with the word "JANUARY", 10,000 posters with the word "FEBRUARY", 10,000 posters with the word "MARCH", and so on. The typefaces used to print the month names were made on special order and were very expensive. Therefore, the printer wanted to buy as few typefaces as possible, with the idea that some of the typefaces used for one month's name could be reused for the names of other months. The goal is to have a sufficient stock of typefaces to print the names of all months throughout the year.
How many different typefaces does the printer need to buy? All words are printed in uppercase letters, as shown above. | 22 | 6.25 |
32,346 | Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$ . He writes the number $0$ on the board and invites Dima to play a game. Dima tells the integer $0 < p < 10, p$ can vary from round to round. Sasha chooses $p$ cards before which he puts a “ $+$ ” sign, and before the other cards he puts a “ $-$ " sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves. | 1023 | 57.8125 |
32,347 | What is the value of \(b\) such that the graph of the equation \[ 3x^2 + 9y^2 - 12x + 27y = b\] represents a non-degenerate ellipse? | -\frac{129}{4} | 53.90625 |
32,348 | In the right triangle $ABC$, where $\angle B = \angle C$, the length of $AC$ is $8\sqrt{2}$. Calculate the area of triangle $ABC$. | 64 | 6.25 |
32,349 | Two cards are placed in each of three different envelopes, and cards 1 and 2 are placed in the same envelope. Calculate the total number of different arrangements. | 18 | 14.84375 |
32,350 | Jennifer is hiking in the mountains. She walks northward for 3 miles, then turns 45 degrees eastward and walks 5 miles. How far is she from her starting point? Express your answer in simplest radical form. | \sqrt{34 + 15\sqrt{2}} | 70.3125 |
32,351 | Given the function $f\left(x\right)=\left(x+1\right)e^{x}$.
$(1)$ Find the intervals where the function $f\left(x\right)$ is monotonic.
$(2)$ Find the maximum and minimum values of $f\left(x\right)$ on the interval $\left[-4,0\right]$. | -\frac{1}{e^2} | 0 |
32,352 | A belt is installed on two pulleys with radii of 14 inches and 4 inches respectively. The belt is taut and does not intersect itself. If the distance between the points where the belt touches the two pulleys is 24 inches, find the distance between the centers of the two pulleys. | 26 | 45.3125 |
32,353 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have an angle of $30^{\circ}$ between them, and $|\overrightarrow{a}|=\sqrt{3}$, $|\overrightarrow{b}|=1$,
$(1)$ Find the value of $|\overrightarrow{a}-2\overrightarrow{b}|$
$(2)$ Let vector $\overrightarrow{p}=\overrightarrow{a}+2\overrightarrow{b}$, $\overrightarrow{q}=\overrightarrow{a}-2\overrightarrow{b}$, find the projection of vector $\overrightarrow{p}$ in the direction of $\overrightarrow{q}$ | -1 | 6.25 |
32,354 | In the diagram, each of the two circles has center \(O\). Also, \(O P: P Q = 1:2\). If the radius of the larger circle is 9, what is the area of the shaded region? | 72 \pi | 92.1875 |
32,355 | The numbers which contain only even digits in their decimal representations are written in ascending order such that \[2,4,6,8,20,22,24,26,28,40,42,\dots\] What is the $2014^{\text{th}}$ number in that sequence? | 62048 | 0.78125 |
32,356 | Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king? | \frac{17}{884} | 0 |
32,357 | Given positive numbers $x$ and $y$ satisfying $2x+y=2$, the minimum value of $\frac{1}{x}-y$ is achieved when $x=$ ______, and the minimum value is ______. | 2\sqrt{2}-2 | 38.28125 |
32,358 | Cynthia and Lynnelle are collaborating on a problem set. Over a $24$ -hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$ -hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time. | 4/9 | 24.21875 |
32,359 | There are 5 people seated at a circular table. What is the probability that Angie and Carlos are seated directly opposite each other? | \frac{1}{2} | 4.6875 |
32,360 | The line $y = a$ intersects the curves $y = 2(x + 1)$ and $y = x + \ln x$ at points $A$ and $B$, respectively. Find the minimum value of $|AB|$. | \frac{3}{2} | 5.46875 |
32,361 | Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? | 1333 | 17.96875 |
32,362 | Billy starts his hike in a park and walks eastward for 7 miles. Then, he turns $45^{\circ}$ northward and hikes another 8 miles. Determine how far he is from his starting point. Express your answer in simplest radical form. | \sqrt{113 + 56\sqrt{2}} | 60.15625 |
32,363 | A machine that records the number of visitors to a museum shows 1,879,564. Note that this number has all distinct digits. What is the minimum number of additional visitors needed for the machine to register another number that also has all distinct digits?
(a) 35
(b) 36
(c) 38
(d) 47
(e) 52 | 38 | 82.8125 |
32,364 | Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted to exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$ . | 21 | 84.375 |
32,365 | The height of the pyramid $P-ABCD$ with a square base of side length $2\sqrt{2}$ is $1$. If the radius of the circumscribed sphere of the pyramid is $2\sqrt{2}$, then the distance between the center of the square $ABCD$ and the point $P$ is ______. | 2\sqrt{2} | 2.34375 |
32,366 | Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 24 | 71.875 |
32,367 | In the rectangular coordinate system $(xOy)$, the slope angle of line $l$ passing through point $M(2,1)$ is $\frac{\pi}{4}$. Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, using the same unit length for both coordinate systems. The polar equation of circle $C$ is $\rho = 4\sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right)$.
(I) Find the parametric equations of line $l$ and the rectangular form of the equation of circle $C$.
(II) Suppose circle $C$ intersects line $l$ at points $A$ and $B$. Find the value of $\frac{1}{|MA|} + \frac{1}{|MB|}$. | \frac{\sqrt{30}}{7} | 11.71875 |
32,368 | From 50 products, 10 are selected for inspection. The total number of items is \_\_\_\_\_\_\_, and the sample size is \_\_\_\_\_\_. | 10 | 55.46875 |
32,369 | As shown in the diagram, \(FGHI\) is a trapezium with side \(GF\) parallel to \(HI\). The lengths of \(FG\) and \(HI\) are 50 and 20 respectively. The point \(J\) is on the side \(FG\) such that the segment \(IJ\) divides the trapezium into two parts of equal area. What is the length of \(FJ\)? | 35 | 3.125 |
32,370 | How many distinct arrangements of the letters in the word "balloon" are there, given that it contains repeating letters? | 1260 | 68.75 |
32,371 | Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $D$ and $E$ are the upper and right vertices of the ellipse $C$, and $S_{\triangle DEF_2} = \frac{\sqrt{3}}{2}$, eccentricity $e = \frac{1}{2}$.
(1) Find the standard equation of ellipse $C$;
(2) Let line $l$ pass through $F\_2$ and intersect ellipse $C$ at points $A$ and $B$. Find the minimum value of $\frac{|F\_2A| \cdot |F\_2B|}{S_{\triangle OAB}}$ (where point $O$ is the coordinate origin). | \frac{3}{2} | 42.96875 |
32,372 | Given an arithmetic sequence ${\_{a\_n}}$ with a non-zero common difference $d$, and $a\_7$, $a\_3$, $a\_1$ are three consecutive terms of a geometric sequence ${\_{b\_n}}$.
(1) If $a\_1=4$, find the sum of the first 10 terms of the sequence ${\_{a\_n}}$, denoted as $S_{10}$;
(2) If the sum of the first 100 terms of the sequence ${\_{b\_n}}$, denoted as $T_{100}=150$, find the value of $b\_2+b\_4+b\_6+...+b_{100}$. | 50 | 6.25 |
32,373 | Two equilateral triangles with perimeters of 12 and 15 are positioned such that their sides are respectively parallel. Find the perimeter of the resulting hexagon. | 27 | 49.21875 |
32,374 | Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.138 | 0 |
32,375 | Given the numbers 2 and 8, find the product of three numbers that form a geometric sequence with these two numbers. | 64 | 65.625 |
32,376 | A building contractor needs to pay his $108$ workers $\$ 200 $ each. He is carrying $ 122 $ one hundred dollar bills and $ 188 $ fifty dollar bills. Only $ 45 $ workers get paid with two $ \ $100$ bills. Find the number of workers who get paid with four $\$ 50$ bills. | 31 | 33.59375 |
32,377 | Define the operation: $\begin{vmatrix} a_{1} & a_{2} \\ a_{3} & a_{4}\end{vmatrix} =a_{1}a_{4}-a_{2}a_{3}$, and consider the function $f(x)= \begin{vmatrix} \sqrt {3} & \sin \omega x \\ 1 & \cos \omega x\end{vmatrix} (\omega > 0)$. If the graph of $f(x)$ is shifted to the left by $\dfrac {2\pi}{3}$ units, and the resulting graph corresponds to an even function, then determine the minimum value of $\omega$. | \dfrac{5}{4} | 67.96875 |
32,378 | A fifteen-digit integer is formed by repeating a positive five-digit integer three times. For example, 42563,42563,42563 or 60786,60786,60786 are integers of this form. What is the greatest common divisor of all fifteen-digit integers in this form? | 10000100001 | 22.65625 |
32,379 | In the tetrahedron S-ABC, the lateral edge SA is perpendicular to the plane ABC, and the base ABC is an equilateral triangle with a side length of $\sqrt{3}$. If SA = $2\sqrt{3}$, then the volume of the circumscribed sphere of the tetrahedron is \_\_\_\_\_\_. | \frac{32}{3}\pi | 5.46875 |
32,380 | The diagram shows a shaded region inside a regular hexagon. The shaded region is divided into equilateral triangles. What fraction of the area of the hexagon is shaded?
A) $\frac{3}{8}$
B) $\frac{2}{5}$
C) $\frac{3}{7}$
D) $\frac{5}{12}$
E) $\frac{1}{2}$ | \frac{1}{2} | 85.15625 |
32,381 | Given a triangle $\triangle ABC$ with an area of $S$, and $\overrightarrow{AB} \cdot \overrightarrow{AC} = S$.
(I) Find the value of $\tan 2A$;
(II) If $\cos C = \frac{3}{5}$, and $|\overrightarrow{AC} - \overrightarrow{AB}| = 2$, find the area $S$ of $\triangle ABC$. | \frac{8}{5} | 24.21875 |
32,382 | In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer. | 110 | 76.5625 |
32,383 | Given the ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its right focus at $F(\sqrt{3}, 0)$, and point $M(-\sqrt{3}, \dfrac{1}{2})$ on ellipse $C$.
(Ⅰ) Find the standard equation of ellipse $C$;
(Ⅱ) Line $l$ passes through point $F$ and intersects ellipse $C$ at points $A$ and $B$. A perpendicular line from the origin $O$ to line $l$ meets at point $P$. If the area of $\triangle OAB$ is $\dfrac{\lambda|AB| + 4}{2|OP|}$ ($\lambda$ is a real number), find the value of $\lambda$. | -1 | 7.03125 |
32,384 | In triangle $\triangle ABC$, with side lengths $AB=2$ and $BC=4$, and angle $A=60^{\circ}$, calculate the length of side $AC$. | 1+\sqrt{13} | 38.28125 |
32,385 | Let the real numbers \( x_{1}, x_{2}, \cdots, x_{2008} \) satisfy the condition \( \left|x_{1} - x_{2}\right| + \left|x_{2} - x_{3}\right| + \cdots + \left|x_{2007} - x_{2008}\right| = 2008 \). Define \( y_{k} = \frac{1}{k} (x_{1} + x_{2} + \cdots + x_{k}) \) for \( k = 1, 2, \cdots, 2008 \). Find the maximum value of \( T = \left|y_{1} - y_{2}\right| + \left|y_{2} - y_{3}\right| + \cdots + \left|y_{2007} - y_{2008}\right| \). | 2007 | 48.4375 |
32,386 | A point is chosen at random on the number line between 0 and 1, and the point is colored red. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored blue. What is the probability that the number of the blue point is greater than the number of the red point, but less than three times the number of the red point? | \frac{1}{2} | 4.6875 |
32,387 | Xiao Wang places some equilateral triangle paper pieces on the table. The first time he places 1 piece; the second time he places three more pieces around the first triangle; the third time he places more pieces around the shape formed in the second placement, and so on. The requirement is: each piece placed in each subsequent placement must share at least one edge with a piece placed in the previous placement, and apart from sharing edges, there should be no other overlaps (see diagram). After the 20th placement, the total number of equilateral triangle pieces used is: | 571 | 32.03125 |
32,388 | Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2020$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? | 1010 | 6.25 |
32,389 | A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston that has moved inside the cylinder by $h$ meters.
Hint: The equation of state for the gas is given by $\rho V=$ const, where $\rho$ is pressure and $V$ is volume.
Given:
$$
H=0.4 \text{ m}, \ h=0.2 \text{ m}, \ R=0.1 \text{ m}
$$ | 900 | 2.34375 |
32,390 | We call a pair $(a,b)$ of positive integers, $a<391$ , *pupusa* if $$ \textup{lcm}(a,b)>\textup{lcm}(a,391) $$ Find the minimum value of $b$ across all *pupusa* pairs.
Fun Fact: OMCC 2017 was held in El Salvador. *Pupusa* is their national dish. It is a corn tortilla filled with cheese, meat, etc. | 18 | 35.9375 |
32,391 | To estimate the consumption of disposable wooden chopsticks, in 1999, a sample of 10 restaurants from a total of 600 high, medium, and low-grade restaurants in a certain county was taken. The daily consumption of disposable chopstick boxes in these restaurants was as follows:
0.6, 3.7, 2.2, 1.5, 2.8, 1.7, 1.2, 2.1, 3.2, 1.0
(1) Estimate the total consumption of disposable chopstick boxes in the county for the year 1999 by calculating the sample (assuming 350 business days per year);
(2) In 2001, another survey on the consumption of disposable wooden chopsticks was conducted in the same manner, and the result was that the average daily use of disposable chopstick boxes in the 10 sampled restaurants was 2.42 boxes. Calculate the average annual growth rate of the consumption of disposable wooden chopsticks for the years 2000 and 2001 (the number of restaurants in the county and the total number of business days in the year remained the same as in 1999);
(3) Under the conditions of (2), if producing a set of desks and chairs for primary and secondary school students requires 0.07 cubic meters of wood, calculate how many sets of student desks and chairs can be produced with the wood used for disposable chopsticks in the county in 2001. The relevant data needed for the calculation are: 100 pairs of chopsticks per box, each pair of chopsticks weighs 5 grams, and the density of the wood used is 0.5×10<sup>3</sup> kg/m<sup>3</sup>;
(4) If you were asked to estimate the amount of wood consumed by disposable chopsticks in your province for a year, how would you use statistical knowledge to do so? Briefly describe in words. | 7260 | 10.15625 |
32,392 | In the plane of equilateral triangle $PQR$, points $S$, $T$, and $U$ are such that triangle $PQS$, $QRT$, and $RUP$ are also equilateral triangles. Given the side length of $PQR$ is 4 units, find the area of hexagon $PQURTS$. | 16\sqrt{3} | 33.59375 |
32,393 | Given that $\alpha$ and $\beta$ are the roots of the equation $x^2 - 3x - 2 = 0,$ find the value of $5 \alpha^4 + 12 \beta^3.$ | 672.5 + 31.5\sqrt{17} | 0 |
32,394 | Find the least upper bound for the set of values \((x_1 x_2 + 2x_2 x_3 + x_3 x_4) / (x_1^2 + x_2^2 + x_3^2 + x_4^2)\), where \(x_i\) are real numbers, not all zero. | \frac{\sqrt{2}+1}{2} | 0 |
32,395 | How many pairs of positive integers \( (m, n) \) satisfy \( m^2 \cdot n < 30 \)? | 41 | 9.375 |
32,396 | Let the function \( f(x) = x^3 + a x^2 + b x + c \) (where \( a, b, c \) are all non-zero integers). If \( f(a) = a^3 \) and \( f(b) = b^3 \), then the value of \( c \) is | 16 | 17.96875 |
32,397 | A store sells two suits at the same time, both priced at 168 yuan. One suit makes a 20% profit, while the other incurs a 20% loss. Calculate the net profit or loss of the store. | 14 | 0.78125 |
32,398 | Seven standard dice are glued together to make a solid. The pairs of faces of the dice that are glued together have the same number of dots on them. How many dots are on the surface of the solid? | 105 | 27.34375 |
32,399 | Given that $\sin x - \cos x < 0$, determine the range of the function $y = \frac{\sin x}{|\sin x|} + \frac{\cos x}{|\cos x|} + \frac{\tan x}{|\tan x|}$. | -1 | 19.53125 |
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