Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
21,500 | Let $x_1,$ $x_2,$ $x_3$ be positive real numbers such that $x_1 + 3x_2 + 5x_3 = 100.$ Find the smallest possible value of
\[ x_1^2 + x_2^2 + x_3^2. \]
--- | \frac{2000}{7} | 54.6875 |
21,501 | Given any number a from the set {1, 2, 3, ..., 99, 100} and any number b from the same set, calculate the probability that the last digit of 3^a + 7^b is 8. | \frac{3}{16} | 59.375 |
21,502 | Given that an odd function \( f(x) \) satisfies the condition \( f(x+3) = f(x) \). When \( x \in [0,1] \), \( f(x) = 3^x - 1 \). Find the value of \( f\left(\log_1 36\right) \). | -1/3 | 0 |
21,503 | On a straight street, there are 5 buildings numbered from left to right as 1, 2, 3, 4, 5. The k-th building has exactly k (k=1, 2, 3, 4, 5) workers from Factory A, and the distance between two adjacent buildings is 50 meters. Factory A plans to build a station on this street. To minimize the total distance all workers from Factory A have to walk to the station, the station should be built at a distance of meters from Building 1. | 150 | 50 |
21,504 | Given that line $MN$ passes through the left focus $F$ of the ellipse $\frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse at points $M$ and $N$. Line $PQ$ passes through the origin $O$ and is parallel to $MN$, intersecting the ellipse at points $P$ and $Q$. Find the value of $\frac{|PQ|^{2}}{|MN|}$. | 2\sqrt{2} | 39.84375 |
21,505 | Among the natural numbers from 1 to 1000, there are a total of number 7s. | 300 | 93.75 |
21,506 | The café has enough chairs to seat $312_8$ people. If $3$ people are supposed to sit at one table, how many tables does the café have? | 67 | 45.3125 |
21,507 | Find the values of $a$ and $b$ such that $a + b^2$ can be calculated, where $x = a \pm b i$ are the solutions to the equation $5x^2 + 7 = 2x - 10$. Express your answer as a fraction. | \frac{89}{25} | 95.3125 |
21,508 | Let \( T = 3 \times ((1 + i)^{15} - (1 - i)^{15}) \), where \( i = \sqrt{-1} \). Calculate \( |T| \). | 768 | 76.5625 |
21,509 | Given $π < α < 2π$, $\cos (α-9π)=- \dfrac {3}{5}$, find the value of $\cos (α- \dfrac {11π}{2})$. | \dfrac{4}{5} | 85.9375 |
21,510 | Find a costant $C$ , such that $$ \frac{S}{ab+bc+ca}\le C $$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle.
(The maximal number of points is given for the best possible constant, with proof.) | \frac{1}{4\sqrt{3}} | 6.25 |
21,511 | A sphere has a volume of \( 288\pi \) cubic inches. Determine the surface area of the sphere. Also, if the sphere were to be perfectly cut in half, what would be the circumference of the flat circular surface of one of the halves? Express your answers in terms of \( \pi \). | 12\pi | 83.59375 |
21,512 | The minimum positive period of the function $f(x) = \sin \omega x + \sqrt{3}\cos \omega x + 1$ ($\omega > 0$) is $\pi$. When $x \in [m, n]$, $f(x)$ has at least 5 zeros. The minimum value of $n-m$ is \_\_\_\_\_\_. | 2\pi | 30.46875 |
21,513 | Suppose two distinct integers are chosen from between 1 and 29, inclusive. What is the probability that their product is neither a multiple of 2 nor 3? | \dfrac{45}{406} | 67.1875 |
21,514 | The points $(1, 3)$ and $(5, -1)$ are adjacent vertices of a square. What is the area of the square? | 32 | 87.5 |
21,515 | The opposite number of $2- \sqrt{3}$ is ______, and its absolute value is ______. | 2- \sqrt{3} | 40.625 |
21,516 | Given a shooter who has a probability of $\frac{3}{4}$ of hitting target A with a single shot and a probability of $\frac{2}{3}$ of hitting target B with each of two shots, determine the probability that the shooter hits exactly one of the three shots. | \frac{7}{36} | 74.21875 |
21,517 | After learning about functions, the mathematics team of a high school first grade conducted a mathematical modeling activity. Through a survey of the sales of a certain product in a supermarket near the school, it was found that the relationship between the daily sales price P(x) (in yuan per item) of the product in the past month (calculated as 30 days) and the time x (in days) approximately satisfies the function P(x) = 1 + $\frac{k}{x}$ (where k is a constant, and k > 0). The relationship between the daily sales quantity Q(x) (in items) of the product and the time x (days) is shown in the table below:
| x (days) | 10 | 20 | 25 | 30 |
|----------|-----|-----|-----|-----|
| Q(x) (items) | 110 | 120 | 125 | 120 |
It is known that the daily sales revenue of the product on the 10th day is 121, and the function relationship between the daily sales revenue and x is f(x) (unit: yuan).
(1) Find the value of k.
(2) Provide the following two function models: ① Q(x) = a<sup>x</sup> (a > 0, a ≠ 1), ② Q(x) = a|x-25|+b (a ≠ 0). Based on the data in the table, choose the most suitable function to describe the relationship between the daily sales quantity Q(x) of the product and the time x, and find the analytical expression of that function.
(3) Based on Q(x) in (2), find the minimum value of f(x) (1 ≤ x ≤ 30, x ∈ N*). | 121 | 50.78125 |
21,518 | In an equilateral triangle $ABC$ with side length $1$, let $\overrightarrow{BC} = \overrightarrow{a}$, $\overrightarrow{AC} = \overrightarrow{b}$, and $\overrightarrow{AB} = \overrightarrow{c}$. Evaluate the value of $\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{b} \cdot \overrightarrow{c} + \overrightarrow{c} \cdot \overrightarrow{a}$. | \dfrac{1}{2} | 9.375 |
21,519 | As shown in the diagram, $E$ is the midpoint of the leg $AB$ of trapezoid $ABCD$. $DF \perp EC$, $DF=10$, and $EC=24$. Find the area of trapezoid $ABCD$. | 240 | 10.9375 |
21,520 | Inside the square $ABCD$, a point $M$ is taken such that $\angle MAB = 60^{\circ}$ and $\angle MCD = 15^{\circ}$. Find $\angle MBC$. | 30 | 15.625 |
21,521 | Two circles with a radius of 15 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord (dotted segment) in centimeters between these two circles. Express your answer in simplest radical form. | 15\sqrt{3} | 60.15625 |
21,522 |
A polynomial $P(x)$ with integer coefficients possesses the properties
$$
P(1)=2019, \quad P(2019)=1, \quad P(k)=k,
$$
where $k$ is an integer. Find this integer $k$. | 1010 | 75 |
21,523 | Given a parabola $y^2 = 2px$ ($p > 0$) and a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) share a common focus $F$, and point $A$ is the intersection point of the two curves. If $AF \perp x$-axis, find the eccentricity of the hyperbola. | \sqrt{2} + 1 | 38.28125 |
21,524 | Solve for $R$ if $\sqrt[4]{R^3} = 64\sqrt[16]{4}$. | 256 \cdot 2^{1/6} | 2.34375 |
21,525 | A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger? | 672 | 76.5625 |
21,526 | Given the inequality $\frac{x-2}{ax-1} > 0$ with the solution set $(-1,2)$, determine the constant term in the expansion of the binomial $(ax-\frac{1}{x^{2}})^{6}$. | 15 | 93.75 |
21,527 | Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, find the maximum value of $x-y$. | 1+3\sqrt{2} | 84.375 |
21,528 | Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121. | 8100 | 60.15625 |
21,529 | The real roots of the equations \( x^{5} + x + 1 = 0 \) and \( x + \sqrt[5]{x} + 1 = 0 \) are \(\alpha\) and \(\beta\), respectively. What is the value of \(\alpha + \beta\)? | -1 | 31.25 |
21,530 | Given the function $f(x)=\cos (\omega x+\varphi)$ ($\omega > 0$, $|\varphi| \leqslant \frac {\pi}{2}$), when $x=- \frac {\pi}{4}$, the function $f(x)$ can achieve its minimum value, and when $x= \frac {\pi}{4}$, the function $y=f(x)$ can achieve its maximum value. Moreover, $f(x)$ is monotonic in the interval $( \frac {\pi}{18}, \frac {5\pi}{36})$. Find the value of $\varphi$ when $\omega$ takes its maximum value. | - \frac {\pi}{2} | 64.0625 |
21,531 | The number of games won by five baseball teams are displayed on a chart. The team names are missing, and we have several clues to help identify them:
1. The Sharks won fewer games than the Raptors.
2. The Royals won more games than the Dragons, but fewer games than the Knights.
3. The Dragons won more than 30 games.
How many games did the Royals win? The teams’ wins are from a chart showing the following numbers of wins: 45, 35, 40, 50, and 60 games. | 50 | 42.96875 |
21,532 | Two cubic dice are thrown in succession, where \\(x\\) represents the number shown by the first die, and \\(y\\) represents the number shown by the second die.
\\((1)\\) Find the probability that point \\(P(x,y)\\) lies on the line \\(y=x-1\\);
\\((2)\\) Find the probability that point \\(P(x,y)\\) satisfies \\(y^{2} < 4x\\). | \dfrac{17}{36} | 26.5625 |
21,533 | Given a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. It is known that $2a\sin (C+ \frac{\pi}{6})=b+c$.
1. Find the value of angle $A$.
2. If $B= \frac{\pi}{4}$ and $b-a= \sqrt{2}- \sqrt{3}$, find the area of $\triangle ABC$. | \frac{3 + \sqrt{3}}{4} | 52.34375 |
21,534 | In the expansion of $(x^{2}+1)^{2}(x-1)^{6}$, the coefficient of $x^{5}$ is ____. | -52 | 64.0625 |
21,535 | The line $\sqrt{2}ax+by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\triangle AOB$ is a right-angled triangle (where $O$ is the origin). The maximum distance between point $P(a,b)$ and point $(0,1)$ is ______. | \sqrt{2} + 1 | 17.96875 |
21,536 | In regular hexagon $ABCDEF$, diagonal $AD$ is drawn. Given that each interior angle of a regular hexagon measures 120 degrees, calculate the measure of angle $DAB$. | 30 | 67.96875 |
21,537 | Given that the function $y=f(x)+\sin \frac {π}{6}x$ is an even function, and $f(\log _{ \sqrt {2}}2)= \sqrt {3}$, determine $f(\log _{2} \frac {1}{4})$. | 2 \sqrt {3} | 0 |
21,538 | Given that the first term of the sequence $\{a_n\}$ is $a_{1}= \frac {1}{8}$, and the sequence $\{b_n\}$ is a geometric sequence with $b_{5}=2$. If $$b_{n}= \frac {a_{n+1}}{a_{n}}$$, find the value of $a_{10}$. | 64 | 71.875 |
21,539 | If \( \sqrt{\frac{3}{x} + 3} = \frac{5}{3} \), solve for \( x \). | -\frac{27}{2} | 80.46875 |
21,540 | I won a VIP trip for five to a music festival. I can bring four of my friends. I have 10 friends to choose from: 4 are musicians, and 6 are non-musicians. In how many ways can I form my music festival group so that at least one musician is in the group? | 195 | 67.1875 |
21,541 | A large supermarket purchased a popular disinfectant laundry detergent. Due to the rise in raw material prices, the cost price per bottle of detergent this year increased by $4$ compared to last year. The quantity of detergent purchased for $1440$ yuan this year is the same as the quantity purchased for $1200$ yuan last year. When the selling price per bottle of detergent is $36$ yuan, the supermarket can sell 600 bottles per week. In order to increase sales, the supermarket decides to reduce the price. Market research shows that for every $1$ reduction in price, the weekly sales volume can increase by 100 bottles. It is stipulated that the selling price of this disinfectant laundry detergent should not be lower than the cost price.<br/>$(1)$ Find the cost price per bottle of this disinfectant laundry detergent this year;<br/>$(2)$ When the selling price per bottle of this disinfectant laundry detergent is set at how much, the weekly sales profit of this detergent is maximized? What is the maximum profit in yuan? | 8100 | 52.34375 |
21,542 | Professor Lee has eleven different language books lined up on a bookshelf: three Arabic, four German, and four Spanish. Calculate the number of ways to arrange the eleven books on the shelf while keeping the Arabic books together. | 2,177,280 | 0 |
21,543 | Evaluate $\left|-1 + \frac{2}{3}i\right|$. | \frac{\sqrt{13}}{3} | 95.3125 |
21,544 | Calculate $7 \cdot 12\frac{1}{4}$. | 85\frac{3}{4} | 71.875 |
21,545 | If the graph of the function $f(x) = |x+m| + |nx+1|$ is symmetric about $x=2$, then the set $\{x | x = m+n\} = \quad$. | \{-4\} | 3.90625 |
21,546 | The function $y=\frac{x^3+11x^2+38x+35}{x+3}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$? | 20 | 65.625 |
21,547 | In the sequence $\{a_n\}$, if for all $n \in \mathbb{N}^*$, it holds that $a_n = -3a_{n+1}$, and $$\lim_{n \to \infty}(a_{2}+a_{4}+a_{6}+\ldots+a_{2n}) = \frac{9}{2},$$ then the value of $a_1$ is \_\_\_\_\_\_. | -12 | 71.875 |
21,548 | The product of four different positive integers is 360. What is the maximum possible sum of these four integers? | 66 | 25.78125 |
21,549 | The movie "Thirty Thousand Miles in Chang'an" allows the audience to experience the unique charm of Tang poetry that has been passed down for thousands of years and the beauty of traditional Chinese culture. In the film, Li Bai was born in the year $701$ AD. If we represent this as $+701$ years, then Confucius was born in the year ______ BC, given that he was born in the year $551$ BC. | -551 | 93.75 |
21,550 | Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, where $a_n > 0$. If $S_6 - 2S_3 = 5$, then the minimum value of $S_9 - S_6$ is ______. | 20 | 27.34375 |
21,551 | Complex numbers \(a\), \(b\), \(c\) form an equilateral triangle with side length 24 in the complex plane. If \(|a + b + c| = 48\), find \(|ab + ac + bc|\). | 768 | 55.46875 |
21,552 | Given that $sin(x- \frac {π}{4})= \frac {2}{3}$, find the value of $sin2x$. | \frac{1}{9} | 91.40625 |
21,553 | Evaluate $\left\lceil\sqrt{3}\,\right\rceil+\left\lceil\sqrt{33}\,\right\rceil+\left\lceil\sqrt{333}\,\right\rceil$. | 27 | 23.4375 |
21,554 | Find the area of triangle $ABC$ given below:
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$1$",(B+C)/2,NE);
label("$45^\circ$",(0,0.75),E);
[/asy] | \frac{1}{4} | 10.9375 |
21,555 | What is the largest six-digit number that can be obtained by removing nine digits from the number 778157260669103, without changing the order of its digits?
(a) 778152
(b) 781569
(c) 879103
(d) 986103
(e) 987776 | 879103 | 89.84375 |
21,556 | Given that $f(n) = \left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise,}\end{matrix}\right.$, evaluate the sum $\sum_{n = 1}^{1023}{f(n)}$. | 22.5 | 0.78125 |
21,557 | Find the focal length of the hyperbola that shares the same asymptotes with the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$ and passes through the point $A(-3, 3\sqrt{2})$. | \frac{5\sqrt{2}}{2} | 86.71875 |
21,558 | Find the minimum value of
\[\sqrt{x^2 + (x-2)^2} + \sqrt{(x-2)^2 + (x+2)^2}\] over all real numbers $x$. | 2\sqrt{5} | 47.65625 |
21,559 | The sum of the first 2015 digits of the decimal part of the repeating decimal \(0.0142857\) is $\qquad$ | 9065 | 29.6875 |
21,560 | Simplify $\sqrt{\frac{1}{{49}}}=$____; $|{2-\sqrt{5}}|=$____. | \sqrt{5}-2 | 98.4375 |
21,561 | On Arbor Day, a class at a certain school divided into 10 small groups to participate in tree planting activities. The number of trees planted by the 10 groups is shown in the table below:
| Number of Trees Planted | 5 | 6 | 7 |
|--------------------------|-----|-----|-----|
| Number of Groups | 3 | 4 | 3 |
The variance of the number of trees planted by these 10 groups is ______. | 0.6 | 97.65625 |
21,562 | Given $f(x)= \frac{1}{4^{x}+2}$, use the method of deriving the sum formula for an arithmetic sequence to find the value of $f( \frac{1}{10})+f( \frac{2}{10})+…+f( \frac{9}{10})$. | \frac{9}{4} | 83.59375 |
21,563 | In a three-dimensional Cartesian coordinate system, the vertices of triangle ∆ABC are A(3,4,1), B(0,4,5), and C(5,2,0). Find the value of tan A/2. | \sqrt{5} | 46.875 |
21,564 | Expand $(1+0.1)^{500}$ by the binomial theorem and calculate each term as
\[{500 \choose k}(0.1)^k\] for $k = 0, 1, 2, \ldots, 500.$ Determine the value of $k$ for which the term is the largest. | 45 | 87.5 |
21,565 | In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $(x-1)^{2}+y^{2}=1$, and the parametric equation of curve $C_{2}$ is:
$$
\begin{cases}
x= \sqrt {2}\cos \theta \\
y=\sin \theta
\end{cases}
$$
($\theta$ is the parameter), with $O$ as the pole and the positive half-axis of $x$ as the polar axis in the polar coordinate system.
(1) Find the polar equations of $C_{1}$ and $C_{2}$.
(2) The ray $y= \frac { \sqrt {3}}{3}x(x\geqslant 0)$ intersects with $C_{1}$ at a point $A$ different from the origin, and intersects with $C_{2}$ at point $B$. Find $|AB|$. | \sqrt {3}- \frac {2 \sqrt {10}}{5} | 0 |
21,566 | Given $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-3,2)$, for what value of $k$ does
(1) $k \overrightarrow{a}+ \overrightarrow{b}$ and $\overrightarrow{a}-3 \overrightarrow{b}$ are perpendicular?
(2) $k \overrightarrow{a}+ \overrightarrow{b}$ and $\overrightarrow{a}-3 \overrightarrow{b}$ are parallel? When they are parallel, are they in the same or opposite direction? | -\frac{1}{3} | 63.28125 |
21,567 | Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the bisector of \(\angle D\). | 10 | 7.8125 |
21,568 | 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, what is the distance (in inches) between the centers of the two pulleys? | 26 | 39.0625 |
21,569 | Calculate the percentage of five-digit numbers that have at least one repeated digit (the repeated digits do not need to be adjacent). Express your answer as a decimal to the nearest tenth. | 69.8 | 35.15625 |
21,570 | Arrange 3 male students and 4 female students in a row. Under the following different requirements, calculate the number of different arrangement methods:
(1) Person A and Person B must stand at the two ends;
(2) All male students must be grouped together;
(3) Male students must not stand next to each other;
(4) Exactly one person stands between Person A and Person B. | 1200 | 10.9375 |
21,571 | In $\triangle ABC$, $\cos A = \frac{5}{13}$, find $\sin 2A = \_\_\_\_\_\_\_\_, \cos 2A = \_\_\_\_\_\_\_\_, \tan 2A = \_\_\_\_\_\_\_\_.$ | -\frac{120}{119} | 73.4375 |
21,572 | A certain clothing factory produces jackets and $T$-shirts, with each jacket priced at $100$ yuan and each $T$-shirt priced at $60$ yuan. During a promotional period, the factory offers two discount options to customers:<br/>① Buy one jacket and get one $T$-shirt for free;<br/>② Both the jacket and $T$-shirt are paid at $80\%$ of the original price.<br/>Now, a customer wants to buy 30 jackets and $x$ $T$-shirts from the factory $\left(x \gt 30\right)$.<br/>$(1)$ If the customer chooses option ① to purchase, the customer needs to pay ______ yuan for the jackets and ______ yuan for the $T$-shirts (expressed in terms of $x$); if the customer chooses option ② to purchase, the customer needs to pay ______ yuan for the jackets and ______ yuan for the $T$-shirts (expressed in terms of $x$);<br/>$(2)$ If $x=40$, by calculation, determine which option, ① or ②, is more cost-effective?<br/>$(3)$ If both discount options can be used simultaneously, when $x=40$, can you provide a more cost-effective purchasing plan? Please write down your purchasing plan and explain the reason. | 3480 | 70.3125 |
21,573 | Calculate the value of the polynomial f(x) = 7x^7 + 6x^6 + 5x^5 + 4x^4 + 3x^3 + 2x^2 + x using the Qin Jiushao algorithm when x = 3. Find the value of V₄. | 789 | 62.5 |
21,574 | If 3400 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 38 | 5.46875 |
21,575 | From a group of 6 students, 4 are to be selected to participate in competitions for four subjects: mathematics, physics, chemistry, and biology. If two students, A and B, cannot participate in the biology competition, determine the number of different selection plans. | 240 | 57.8125 |
21,576 | Find the number of solutions to:
\[\sin x = \left(\frac{1}{3}\right)^x\]
on the interval $(0, 50\pi)$. | 50 | 69.53125 |
21,577 | Six bags contain 18, 19, 21, 23, 25, and 34 marbles, respectively. One of the bags contains marbles with cracks, while the remaining five bags contain marbles without cracks. Jenny took three of the bags, and George took two of the other bags, leaving the bag with the cracked marbles. If the number of marbles Jenny received is exactly twice the number of marbles George received, determine the number of marbles in the bag with cracks. | 23 | 66.40625 |
21,578 | A special six-sided die has its faces numbered 1 through 6 and has the property that rolling each number \( x \) is \( x \) times as likely as rolling a 1. For example, the probability of rolling a 5 is 5 times the probability of rolling a 1, while the probability of rolling a 2 is 2 times the probability of rolling a 1. Robbie and Francine play a game where they each roll this die three times, and the total of their three rolls is their score. The winner is the player with the highest score; if the two players are tied, neither player wins. After two rolls each, Robbie has a score of 8 and Francine has a score of 10. The probability that Robbie will win can be written in lowest terms as \(\frac{r}{400+s}\), where \( r \) and \( s \) are positive integers. What is the value of \( r+s \)? | 96 | 0.78125 |
21,579 | Given the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1$ with two foci $F_{1}$ and $F_{2}$. A point $P$ lies on the ellipse such that $| PF_{1} | - | PF_{2} | = 2$. Determine the area of $\triangle PF_{1}F_{2}$. | \sqrt{2} | 85.9375 |
21,580 | Translate the graph of $y= \sqrt {2}\sin (2x+ \frac {\pi}{3})$ to the right by $\phi(0 < \phi < \pi)$ units to obtain the graph of the function $y=2\sin x(\sin x-\cos x)-1$. Then, $\phi=$ ______. | \frac {13\pi}{24} | 76.5625 |
21,581 | A person rolls two dice simultaneously and gets the scores $a$ and $b$. The eccentricity $e$ of the ellipse $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$ satisfies $e \geq \frac{\sqrt{3}}{2}$. Calculate the probability that this event occurs. | \frac{1}{4} | 25 |
21,582 | Find $1 - 0.\overline{123}$. | \frac{292}{333} | 89.84375 |
21,583 | Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$ , $AD = 1000$ , $BD = 2000$ , $BC = 2001$ , and $DC = 1999$ . Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$ . Find $AE$ . | 1000 | 35.15625 |
21,584 | A geometric sequence of positive integers has its first term as 5 and its fourth term as 480. What is the second term of the sequence? | 20 | 34.375 |
21,585 | The coefficient of the $x^3$ term in the expansion of $(2-\sqrt{x})^8$ is $1120x^3$. | 112 | 51.5625 |
21,586 | Find the coefficient of the $x^3$ term in the expansion of the product $$(3x^3 + 2x^2 + 4x + 5)(4x^2 + 5x + 6).$$ | 44 | 79.6875 |
21,587 | The pentagon \(PQRST\) is divided into four triangles with equal perimeters. The triangle \(PQR\) is equilateral. \(PTU\), \(SUT\), and \(RSU\) are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon \(PQRST\) to the perimeter of the triangle \(PQR\)? | 5:3 | 0 |
21,588 | In triangle $ABC$, let $AB = 4$, $AC = 7$, $BC = 9$, and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Find $\cos \angle BAD$. | \sqrt{\frac{5}{14}} | 20.3125 |
21,589 | The numbers \(2, 3, 12, 14, 15, 20, 21\) may be divided into two sets so that the product of the numbers in each set is the same.
What is this product? | 2520 | 39.0625 |
21,590 | Calculate:<br/>$(1)-6-3+\left(-7\right)-\left(-2\right)$;<br/>$(2)\left(-1\right)^{2023}+5\times \left(-2\right)-12\div \left(-4\right)$. | -8 | 84.375 |
21,591 | Given that $\{a_n\}$ is a geometric sequence and $S_n$ is the sum of the first $n$ terms, with $S_3=2$ and $S_6=6$, find the value of $a_{13}+a_{14}+a_{15}$. | 32 | 43.75 |
21,592 | Xinyi is a beautiful city with abundant tourism resources. The Malingshan Scenic Area is a $4A$-rated tourist attraction in our city. The scenic area has developed a souvenir that costs $30$ yuan to produce and is sold. The selling price is set not to be lower than the cost and not higher than $54$ yuan. After selling for a period of time, it was found that when the product was priced at $40$ yuan, 80 units could be sold per day. Furthermore, for every $1$ yuan increase in the selling price, the daily sales volume would decrease by 2 units. If the goal is to make a profit of $1200$ yuan per day from selling this product, please determine the selling price at that time. | 50 | 39.84375 |
21,593 | Given that $α\in\mathbb{R}$ and $\sin α + 2\cos α = \frac{\sqrt{10}}{2}$, find the value of $\tan α$. | -\frac{1}{3} | 10.9375 |
21,594 | How many positive integers less than 2023 are congruent to 7 modulo 13? | 156 | 69.53125 |
21,595 | In right triangle $ABC$ with $\angle A = 90^\circ$, $AC = 3$, $AB = 4$, and $BC = 5$, point $D$ is on side $BC$. If the perimeters of $\triangle ACD$ and $\triangle ABD$ are equal, then what is the area of $\triangle ABD$? | $\frac{12}{5}$ | 0 |
21,596 | Find the largest $K$ satisfying the following:
Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$ , then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty. | 1011 | 25.78125 |
21,597 | During the New Year, Xiaoming's family bought many bottles of juice. On New Year's Eve, they drank half of the total amount minus 1 bottle. On the first day of the New Year, they drank half of the remaining amount again. On the second day of the New Year, they drank half of the remaining amount plus 1 bottle, leaving them with 2 bottles. How many bottles of juice did Xiaoming's family buy in total? | 22 | 67.96875 |
21,598 | Given $(3-2x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+…+a_{5}x^{5}$, find the value of $a_{0}+a_{1}+2a_{2}+…+5a_{5}$. | 233 | 9.375 |
21,599 | Given that the red light lasts for $45$ seconds, determine the probability that a pedestrian will have to wait at least $20$ seconds before the light turns green. | \dfrac{5}{9} | 77.34375 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.