Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
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float64 0
100
|
|---|---|---|---|
10,300 |
(1) Calculate: $\left( \frac{1}{8} \right)^{-\frac{1}{3}} - 3\log_{3}^{2}(\log_{3}4) \cdot (\log_{8}27) + 2\log_{\frac{1}{6}} \sqrt{3} - \log_{6}2$
(2) Calculate: $27^{\frac{2}{3}} - 2^{\log_{2}3} \times \log_{2}\frac{1}{8} + 2\lg \left( \sqrt{3+\sqrt{5}} + \sqrt{3-\sqrt{5}} \right)$
|
19
| 49.21875 |
10,301 |
Given that the product of the first $n$ terms of the sequence $\{a_{n}\}$ is $T_{n}$, where ${a_n}=\frac{n}{{2n-5}}$, determine the maximum value of $T_{n}$.
|
\frac{8}{3}
| 50 |
10,302 |
Solve the inequality
$$
(2+\sqrt{3})^x + 2 < 3(\sqrt{2-\sqrt{3}})^{2x}
$$
Find the sum of all integer values of \(x\) that satisfy this inequality and belong to the interval \((-20, 53)\).
|
-190
| 57.03125 |
10,303 |
Dasha added 158 numbers and obtained a sum of 1580. Then Sergey tripled the largest of these numbers and decreased another number by 20. The resulting sum remained unchanged. Find the smallest of the original numbers.
|
10
| 25.78125 |
10,304 |
Let \( f(x) \) be a function defined on \(\mathbf{R}\). Given that \( f(x) + x^{2} \) is an odd function and \( f(x) + 2^{x} \) is an even function, find the value of \( f(1) \).
|
-\frac{7}{4}
| 57.03125 |
10,305 |
The largest four-digit number whose digits add to 17 is 9800. The 5th largest four-digit number whose digits have a sum of 17 is:
|
9611
| 21.875 |
10,306 |
In an archery competition held in a certain city, after ranking the scores, the average score of the top seven participants is 3 points lower than the average score of the top four participants. The average score of the top ten participants is 4 points lower than the average score of the top seven participants. How many points more is the sum of the scores of the fifth, sixth, and seventh participants compared to the sum of the scores of the eighth, ninth, and tenth participants?
|
28
| 60.15625 |
10,307 |
Given a sequence $\{a_n\}$ that satisfies $a_{n+1}^2=a_na_{n+2}$, and $a_1= \frac{1}{3}$, $a_4= \frac{1}{81}$.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let $f(x)=\log_3x$, $b_n=f(a_1)+f(a_2)+\ldots+f(a_n)$, $T_n= \frac{1}{b_1}+ \frac{1}{b_2}+\ldots+ \frac{1}{b_n}$, find $T_{2017}$.
|
\frac{-2017}{1009}
| 0 |
10,308 |
The shortest distance from a point on the parabola $x^2=y$ to the line $y=2x+m$ is $\sqrt{5}$. Find the value of $m$.
|
-6
| 21.875 |
10,309 |
A semicircle has diameter $XY$ . A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$ , $U$ on the semicircle, and $V$ on $XY$ . What is the area of $STUV$ ?
|
36
| 10.9375 |
10,310 |
Given a function $f(x)$ defined on $\mathbb{R}$, for any real numbers $x_1$, $x_2$, it satisfies $f(x_1+x_2)=f(x_1)+f(x_2)+2$. The sequence $\{a_n\}$ satisfies $a_1=0$, and for any $n\in\mathbb{N}^*$, $a_n=f(n)$. Find the value of $f(2010)$.
|
4018
| 97.65625 |
10,311 |
Shift the graph of the function $f(x)=2\sin(2x+\frac{\pi}{6})$ to the left by $\frac{\pi}{12}$ units, and then shift it upwards by 1 unit to obtain the graph of $g(x)$. If $g(x_1)g(x_2)=9$, and $x_1, x_2 \in [-2\pi, 2\pi]$, then find the maximum value of $2x_1-x_2$.
|
\frac {49\pi}{12}
| 13.28125 |
10,312 |
Let $g(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $g(2)=0$, $90<g(9)<100$, $120<g(10)<130$, $7000k<g(150)<7000(k+1)$ for some integer $k$. What is $k$?
|
k=6
| 37.5 |
10,313 |
From 1 to 100, take a pair of integers (repetitions allowed) so that their sum is greater than 100. How many ways are there to pick such pairs?
|
5050
| 80.46875 |
10,314 |
Let \( M = 42 \cdot 43 \cdot 75 \cdot 196 \). Find the ratio of the sum of the odd divisors of \( M \) to the sum of the even divisors of \( M \).
|
\frac{1}{14}
| 25 |
10,315 |
Let $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{2} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]
Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Determine $\cos \theta.$
|
-\frac{1}{2}
| 82.03125 |
10,316 |
The number $17!$ has a certain number of positive integer divisors. What is the probability that one of them is odd?
|
\frac{1}{16}
| 28.90625 |
10,317 |
Consider the sequence
$$
a_{n}=\cos (\underbrace{100 \ldots 0^{\circ}}_{n-1})
$$
For example, $a_{1}=\cos 1^{\circ}, a_{6}=\cos 100000^{\circ}$.
How many of the numbers $a_{1}, a_{2}, \ldots, a_{100}$ are positive?
|
99
| 59.375 |
10,318 |
Compute the limit of the function:
$$
\lim _{x \rightarrow \pi} \frac{\ln (2+\cos x)}{\left(3^{\sin x}-1\right)^{2}}
$$
|
\frac{1}{2 \ln^2 3}
| 14.0625 |
10,319 |
In a right triangle $ABC$ with equal legs $AC$ and $BC$, a circle is constructed with $AC$ as its diameter, intersecting side $AB$ at point $M$. Find the distance from vertex $B$ to the center of this circle if $BM = \sqrt{2}$.
|
\sqrt{5}
| 46.875 |
10,320 |
Let $P(x) = x^2 + ax + b$ be a quadratic polynomial. For how many pairs $(a, b)$ of positive integers where $a, b < 1000$ do the quadratics $P(x+1)$ and $P(x) + 1$ have at least one root in common?
|
30
| 18.75 |
10,321 |
Two numbers \( x \) and \( y \) satisfy the equation \( 280x^{2} - 61xy + 3y^{2} - 13 = 0 \) and are respectively the fourth and ninth terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.
|
-5
| 3.90625 |
10,322 |
Solve the equation \(\sqrt{8x+5} + 2 \{x\} = 2x + 2\). Here, \(\{x\}\) denotes the fractional part of \(x\), i.e., \(\{x\} = x - \lfloor x \rfloor\). Write down the sum of all solutions.
|
0.75
| 0.78125 |
10,323 |
Given that vectors $\overrightarrow{α}$ and $\overrightarrow{β}$ are two mutually perpendicular unit vectors in a plane, and $(5\overrightarrow{α} - 2\overrightarrow{γ}) \cdot (12\overrightarrow{β} - 2\overrightarrow{γ}) = 0$, find the maximum value of $|\overrightarrow{γ}|$.
|
\frac{13}{2}
| 53.90625 |
10,324 |
The circle centered at $(3,-2)$ and with radius $5$ intersects the circle centered at $(3,4)$ and with radius $\sqrt{13}$ at two points $C$ and $D$. Find $(CD)^2$.
|
36
| 67.96875 |
10,325 |
Suppose \( A, B, C \) are three angles such that \( A \geq B \geq C \geq \frac{\pi}{8} \) and \( A + B + C = \frac{\pi}{2} \). Find the largest possible value of the product \( 720 \times (\sin A) \times (\cos B) \times (\sin C) \).
|
180
| 19.53125 |
10,326 |
Find the greatest positive number $\lambda$ such that for any real numbers $a$ and $b$, the inequality $\lambda a^{2} b^{2}(a+b)^{2} \leqslant\left(a^{2}+ab+b^{2}\right)^{3}$ holds.
|
\frac{27}{4}
| 97.65625 |
10,327 |
How many times during a day does the angle between the hour and minute hands measure exactly $17^{\circ}$?
|
44
| 67.1875 |
10,328 |
The angle bisectors \( A L_{1} \) and \( B L_{2} \) of triangle \( A B C \) intersect at point \( I \). It is known that \( A I : I L_{1} = 3 \) and \( B I : I L_{2} = 2 \). Find the ratio of the sides of triangle \( A B C \).
|
3:4:5
| 25.78125 |
10,329 |
Find all 4-digit numbers $\overline{abcd}$ that are multiples of $11$ , such that the 2-digit number $\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$ .
|
3454
| 96.875 |
10,330 |
Given two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ with respective sums of the first $n$ terms $S_n$ and $T_n$, if $\frac{S_n}{T_n} = \frac{2n-3}{4n-3}$ holds for any natural number $n$, find the value of $\frac{a_9}{b_5+b_7} + \frac{a_3}{b_8+b_4}$.
|
\frac{19}{41}
| 25 |
10,331 |
From a barrel, 4 liters of wine are drawn, and this is replaced with 4 liters of water. From the resulting mixture, 4 liters are drawn again and replaced with 4 liters of water. This operation is repeated a total of three times, and the final result is that there are 2.5 liters more water than wine. How many liters of wine were originally in the barrel?
|
16
| 24.21875 |
10,332 |
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$, respectively, and $\overrightarrow{m}=(a,−\sqrt {3}b)$, $\overrightarrow{n}=(\sin B,\cos A)$, if $a= \sqrt {7}$, $b=2$, and $\overrightarrow{m} \perp \overrightarrow{n}$, then the area of $\triangle ABC$ is ______.
|
\frac{3\sqrt{3}}{2}
| 65.625 |
10,333 |
Find all values of \( a \) for which the equation \( x^{2} + 2ax = 8a \) has two distinct integer roots. Record the product of all such \( a \), rounding to the nearest hundredth if necessary.
|
506.25
| 17.1875 |
10,334 |
A right octagonal pyramid has two cross sections obtained by slicing the pyramid with planes parallel to the octagonal base. The area of the smaller cross section is $256\sqrt{2}$ square feet and the area of the larger cross section is $576\sqrt{2}$ square feet. The distance between the two planes is $12$ feet. Determine the distance from the apex of the pyramid to the plane of the larger cross section.
|
36
| 89.0625 |
10,335 |
In the diagram, $AB$ is perpendicular to $BC$, and $CD$ is perpendicular to $AD$. Also, $AC = 625$ and $AD = 600$. If $\angle BAC = 2 \angle DAC$, what is the length of $BC$?
|
336
| 32.03125 |
10,336 |
If I choose four cards from a standard $52$-card deck, without replacement, what is the probability that I will end up with one card from each suit, in a sequential order (e.g., clubs, diamonds, hearts, spades)?
|
\frac{2197}{499800}
| 3.90625 |
10,337 |
Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean's scoring average. How many points, in total, did Ruth score?
|
171
| 70.3125 |
10,338 |
A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a heart, a spade, or a king?
|
\frac{133}{169}
| 61.71875 |
10,339 |
If $a$, $b\in \{-1,1,2\}$, then the probability that the function $f\left(x\right)=ax^{2}+2x+b$ has a zero point is ______.
|
\frac{2}{3}
| 84.375 |
10,340 |
Calculate the definite integral:
$$
\int_{0}^{4} e^{\sqrt{(4-x) /(4+x)}} \cdot \frac{d x}{(4+x) \sqrt{16-x^{2}}}
$$
|
\frac{1}{4}(e-1)
| 3.125 |
10,341 |
Identify a six-digit number \( N \) composed of distinct digits such that the numbers \( 2N, 3N, 4N, 5N, \) and \( 6N \) are permutations of its digits.
|
142857
| 85.15625 |
10,342 |
The exterior angles of a triangle are proportional to the numbers 5: 7: 8. Find the angle between the altitudes of this triangle drawn from the vertices of its smaller angles.
|
90
| 78.125 |
10,343 |
Calculate:<br/>$(Ⅰ)(1\frac{9}{16})^{0.5}+0.01^{-1}+(2\frac{10}{27})^{-\frac{2}{3}}-2π^0+\frac{3}{16}$;<br/>$(Ⅱ)\log _{2}(4\times 8^{2})+\log _{3}18-\log _{3}2+\log _{4}3\times \log _{3}16$.
|
12
| 96.875 |
10,344 |
Given a finite increasing sequence \(a_{1}, a_{2}, \ldots, a_{n}\) of natural numbers (with \(n \geq 3\)), and the recurrence relation \(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\) holds for all \(\kappa \leq n-2\). The sequence must contain \(a_{k} = 2022\). Determine the maximum number of three-digit numbers that are multiples of 4 that this sequence can contain.
|
225
| 12.5 |
10,345 |
In the tetrahedron $ABCD$, $\triangle ABC$ is an equilateral triangle, $AD = BD = 2$, $AD \perp BD$, and $AD \perp CD$. Find the distance from point $D$ to the plane $ABC$.
|
\frac{2\sqrt{3}}{3}
| 4.6875 |
10,346 |
In the trapezoid \(ABCD\) (\(AD \parallel BC\)), a perpendicular \(EF\) is drawn from point \(E\) (the midpoint of \(CD\)) to line \(AB\). Find the area of the trapezoid if \(AB = 5\) and \(EF = 4\).
|
20
| 33.59375 |
10,347 |
Two distinct natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum.
|
70000000
| 3.125 |
10,348 |
From a class, 5 students were selected to measure their heights (in cm), and the data obtained were 160, 162, 159, 160, 159. The variance $s^2$ of this group of data is \_\_\_\_\_\_.
|
\frac{6}{5}
| 6.25 |
10,349 |
Eighty bricks, each measuring \(3'' \times 8'' \times 15''\), are to be stacked to form a column. Each brick can be oriented to contribute \(3''\), \(8''\), or \(15''\) to the total column height. Determine how many different total column heights can be achieved using all eighty bricks.
|
961
| 8.59375 |
10,350 |
Farm A has a total of 625 chickens and ducks, and Farm B has a total of 748 chickens and ducks. The number of chickens in Farm B is 24% more than in Farm A, and the number of ducks in Farm A is 15% less than in Farm B. How many chickens does Farm B have?
|
248
| 22.65625 |
10,351 |
A certain company, in response to the national call for garbage classification, has launched a project with the support of the research department to innovate technologies. The project processes kitchen waste into reusable chemical products. It is known that the daily processing capacity $x$ (unit: tons) of the company is at least 70 tons and at most 100 tons. The function relationship between the total daily processing cost $y$ (unit: yuan) and the daily processing capacity $x$ can be approximately represented as $y=\frac{1}{2}x^2+40x+3200$, and the selling price of the chemical products obtained from processing 1 ton of kitchen waste is 110 yuan.
$(1)$ For how many tons of daily processing capacity does the company have the lowest average cost per ton of kitchen waste processed? At this point, is the company processing 1 ton of kitchen waste at a loss or profit?
$(2)$ In order for the company to achieve sustainable development, the government has decided to provide financial subsidies to the company, with two subsidy schemes:
① A fixed amount of financial subsidy of 2300 yuan per day;
② A financial subsidy based on the daily processing capacity, with an amount of 30x yuan. If you are the decision-maker of the company, which subsidy scheme would you choose to maximize profit? Why?
|
1800
| 38.28125 |
10,352 |
Given that -9, a_{1}, a_{2}, -1 are four real numbers forming an arithmetic sequence, and -9, b_{1}, b_{2}, b_{3}, -1 are five real numbers forming a geometric sequence, find the value of b_{2}(a_{2}-a_{1}).
|
-8
| 95.3125 |
10,353 |
Find the smallest positive integer \( n \) such that for any set of \( n \) distinct integers \( a_{1}, a_{2}, \ldots, a_{n} \), the product of all differences \( a_{i} - a_{j} \) for \( i < j \) is divisible by 1991.
|
182
| 33.59375 |
10,354 |
In a class of 60 students, a sample of size 5 is to be drawn using systematic sampling. The students are randomly assigned numbers from 01 to 60, and then divided into 5 groups in order of their numbers (1-5, 6-10, etc.). If the second number drawn is 16, the number drawn from the fourth group will be $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$.
|
40
| 89.84375 |
10,355 |
In a Cartesian coordinate system, the points where both the x-coordinate and y-coordinate are integers are called lattice points. How many lattice points (x, y) satisfy the inequality \((|x|-1)^{2}+(|y|-1)^{2}<2\)?
|
16
| 83.59375 |
10,356 |
Given a parabola $C$: $y^{2}=2px (p > 0)$ with focus $F$, directrix $x=-1$, and a point $M$ on the directrix below the $x$-axis. A line $l$ passing through $M$ and $F$ intersects with $C$ at point $N$, and $F$ is the midpoint of segment $MN$.
1. Find the equations of parabola $C$ and line $l$.
2. If line $l$ intersects with parabola $C$ at another point $P$ (different from $N$), find the length of segment $PN$.
|
\frac {16}{3}
| 74.21875 |
10,357 |
Given that $f'(x_0)=2$, find the value of $\lim_{k\rightarrow 0} \frac{f(x_0-k)-f(x_0)}{2k}$.
|
-1
| 82.8125 |
10,358 |
Let \( f(x) = \frac{1}{x^3 + 3x^2 + 2x} \). Determine the smallest positive integer \( n \) such that
\[ f(1) + f(2) + f(3) + \cdots + f(n) > \frac{503}{2014}. \]
|
44
| 11.71875 |
10,359 |
On a plate, there are different candies of three types: 2 lollipops, 3 chocolate candies, and 5 jelly candies. Sveta ate all of them one by one, choosing each next candy at random. Find the probability that the first and last candies she ate were of the same type.
|
14/45
| 64.0625 |
10,360 |
Arrange the letters a, a, b, b, c, c into three rows and two columns, with the requirement that each row has different letters and each column also has different letters, and find the total number of different arrangements.
|
12
| 13.28125 |
10,361 |
A bag contains 6 red balls and 8 white balls. If 5 balls are randomly placed into Box $A$ and the remaining 9 balls are placed into Box $B$, what is the probability that the sum of the number of white balls in Box $A$ and the number of red balls in Box $B$ is not a prime number? (Answer with a number)
|
213/1001
| 7.8125 |
10,362 |
Timur and Alexander are counting the trees growing around the house. They move in the same direction but start counting from different trees. How many trees are growing around the house if the tree that Timur counted as the 12th, Alexander counted as the 33rd, and the tree that Timur counted as the 105th, Alexander counted as the 8th?
|
118
| 34.375 |
10,363 |
In a trapezoid \(ABCD\) with bases \(AD=12\) and \(BC=8\), circles constructed on the sides \(AB\), \(BC\), and \(CD\) as diameters intersect at one point. The length of diagonal \(AC\) is 12. Find the length of \(BD\).
|
16
| 2.34375 |
10,364 |
Let $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$. If $S_{4}=-5$ and $S_{6}=21S_{2}$, calculate the value of $S_{8}$.
|
-85
| 49.21875 |
10,365 |
Find the smallest natural number that starts with the digit five, which becomes four times smaller if this five is removed from the beginning of its decimal representation and appended to the end.
|
512820
| 85.15625 |
10,366 |
(1) Use the Horner's method to calculate the polynomial $f(x) = 3x^6 + 5x^5 + 6x^4 + 79x^3 - 8x^2 + 35x + 12$ when $x = -4$, find the value of $v_3$.
(2) Convert the hexadecimal number $210_{(6)}$ into a decimal number.
|
78
| 82.03125 |
10,367 |
Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at $16 \mathrm{~h} 30 \mathrm{~m}$. If she pedals at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at $17 \mathrm{~h} 15 \mathrm{~m}$. At what speed should she pedal to arrive home at $17 \mathrm{~h}$?
|
12
| 39.0625 |
10,368 |
Sergey, while being a student, worked part-time at a student cafe throughout the year. Sergey's salary was 9000 rubles per month. In the same year, Sergey paid 100000 rubles for his treatment at a healthcare institution and purchased medication prescribed by a doctor for 20000 rubles (eligible for deduction). The following year, Sergey decided to claim a social tax deduction. What amount of personal income tax paid is subject to refund to Sergey from the budget? (Answer as a whole number, without spaces or units of measurement.)
|
14040
| 7.8125 |
10,369 |
Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list?
|
871
| 38.28125 |
10,370 |
Let \( a, b, c, d \) be real numbers defined by
$$
a=\sqrt{4-\sqrt{5-a}}, \quad b=\sqrt{4+\sqrt{5-b}}, \quad c=\sqrt{4-\sqrt{5+c}}, \quad d=\sqrt{4+\sqrt{5+d}}
$$
Calculate their product.
|
11
| 94.53125 |
10,371 |
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is given by
$$
\begin{cases}
x = 1 + t\cos\alpha \\
y = 2 + t\sin\alpha
\end{cases}
$$
where $t$ is the parameter. In the polar coordinate system, which uses the same unit length as $xOy$ and has the origin $O$ as the pole and the positive $x$-axis as the polar axis, the equation of circle $C$ is $\rho = 6\sin\theta$.
(I) Find the Cartesian coordinate equation of circle $C$.
(II) Suppose circle $C$ intersects line $l$ at points A and B. If point P has coordinates (1, 2), find the minimum value of $|PA|+|PB|$.
|
2\sqrt{7}
| 74.21875 |
10,372 |
An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?
Note: The 1 at the top is often labelled the "zeroth" row of Pascal's Triangle, by convention. So to count a total of 20 rows, use rows 0 through 19.
|
\frac{39}{210}
| 0 |
10,373 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{3^{5 x}-2^{x}}{x-\sin 9 x}$$
|
\frac{1}{8} \ln \frac{2}{243}
| 0 |
10,374 |
A "pass level game" has the following rules: On the \( n \)-th level, a die is thrown \( n \) times. If the sum of the points that appear in these \( n \) throws is greater than \( 2^{n} \), then the player passes the level.
1. What is the maximum number of levels a person can pass in this game?
2. What is the probability of passing the first three levels in a row?
(Note: The die is a fair six-faced cube with numbers \( 1, 2, 3, 4, 5, \) and \( 6 \). After the die is thrown and comes to rest, the number on the top face is the outcome.)
|
\frac{100}{243}
| 4.6875 |
10,375 |
Evaluate $\lfloor-5.77\rfloor+\lceil-3.26\rceil+\lfloor15.93\rfloor+\lceil32.10\rceil$.
|
39
| 8.59375 |
10,376 |
In the tetrahedron \( P-ABC \), edges \( PA \), \( AB \), and \( AC \) are mutually perpendicular, and \( PA = AB = AC \). Let \( E \) and \( F \) be the midpoints of \( AB \) and \( PC \) respectively. Find the sine of the angle \(\theta\) between \( EF \) and the plane \( PBC \).
|
\frac{1}{3}
| 57.8125 |
10,377 |
There are 7 volunteers to be arranged to participate in the promotional activities for the Shanghai World Expo on Saturday and Sunday, with 6 people participating each day and 3 people arranged for each day. There are __________ different arrangements possible (answer with a number).
|
140
| 6.25 |
10,378 |
Point \( M \) is the midpoint of side \( BC \) of triangle \( ABC \), where \( AB = 17 \), \( AC = 30 \), and \( BC = 19 \). A circle is constructed with diameter \( AB \). A point \( X \) is chosen arbitrarily on this circle. What is the minimum possible value of the length of segment \( MX \)?
|
6.5
| 10.9375 |
10,379 |
A point $ M$ is taken on the perpendicular bisector of the side $ AC$ of an acute-angled triangle $ ABC$ so that $ M$ and $ B$ are on the same side of $ AC$ . If $ \angle BAC\equal{}\angle MCB$ and $ \angle ABC\plus{}\angle MBC\equal{}180^{\circ}$ , find $ \angle BAC.$
|
30
| 23.4375 |
10,380 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $b\sin 2C=c\sin B$.
$(1)$ Find the value of angle $C$;
$(2)$ If $\sin \left(B- \frac {\pi}{3}\right)= \frac {3}{5}$, find the value of $\sin A$.
|
\frac {4 \sqrt {3}-3}{10}
| 0 |
10,381 |
Coins are arranged in a row from left to right. It is known that two of them are fake, they lie next to each other, the left one weighs 9 grams, the right one weighs 11 grams, and all the remaining are genuine and each weighs 10 grams. The coins are weighed on a balance scale, which either shows which of the two sides is heavier, or is in balance indicating the weights on both sides are equal. What is the maximum number \( n \) of coins for which it is possible to find the 11-gram coin in three weighings?
|
28
| 0 |
10,382 |
If in the expansion of the binomial \((x- \frac {2}{ \sqrt {x}})^{n}\), only the coefficient of the fifth term is the largest, then the coefficient of the term containing \(x^{2}\) is ______.
|
1120
| 52.34375 |
10,383 |
Yasmine makes her own chocolate beverage by mixing volumes of milk and syrup in the ratio \(5: 2\). Milk comes in \(2 \text{ L}\) bottles and syrup comes in \(1.4 \text{ L}\) bottles. Yasmine has a limitless supply of full bottles of milk and of syrup. Determine the smallest volume of chocolate beverage that Yasmine can make that uses only whole bottles of both milk and syrup.
|
19.6
| 71.09375 |
10,384 |
Given that $x$ and $y$ are positive numbers, $\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)$, and it satisfies $\frac{\sin\theta}{x} = \frac{\cos\theta}{y}$ and $\frac{\cos^2\theta}{x^2} + \frac{\sin^2\theta}{y^2} = \frac{10}{3(x^2+y^2)}$, determine the value of $\frac{x}{y}$.
|
\sqrt{3}
| 51.5625 |
10,385 |
Evaluate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\cos x \, dx}{(1+\cos x+\sin x)^{2}}
$$
|
\ln(2) - \frac{1}{2}
| 1.5625 |
10,386 |
A container is already filled with water. There are three lead balls: large, medium, and small. The first time, the small ball is submerged in the water; the second time, the small ball is removed, and the medium ball is submerged in the water; the third time, the medium ball is removed, and the large ball is submerged in the water. It is known that the water spilled the first time is 3 times the water spilled the second time, and the water spilled the third time is 3 times the water spilled the first time. Find the ratio of the volumes of the three balls.
|
3 : 4 : 13
| 28.125 |
10,387 |
How many two-digit numbers have digits whose sum is a perfect square less than or equal to 25?
|
17
| 95.3125 |
10,388 |
There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five boxes priced at 1 yuan, 3 yuan, 5 yuan, 7 yuan, and 9 yuan. Each gift is paired with one box. How many different total prices are possible?
|
19
| 80.46875 |
10,389 |
Let $p, q, r, s, t, u, v, w$ be distinct elements in the set $\{-8, -6, -4, -1, 1, 3, 5, 14\}$. What is the minimum possible value of $(p+q+r+s)^2 + (t+u+v+w)^2$?
|
10
| 15.625 |
10,390 |
The centers of faces $ABCD$ and $ADD'A'$ of the cube $ABCD-A'B'C'D'$ are $M$ and $N$ respectively. Find the sine of the angle between the skew lines $MN$ and $BD'$.
|
\frac{\sqrt{3}}{3}
| 71.875 |
10,391 |
The average of a set of data 1, 3, 2, 5, $x$ is 3. What is the standard deviation of this set of data?
|
\sqrt{2}
| 82.8125 |
10,392 |
Find the smallest integer $n$ such that an $n\times n$ square can be partitioned into $40\times 40$ and $49\times 49$ squares, with both types of squares present in the partition, if
a) $40|n$ ; b) $49|n$ ; c) $n\in \mathbb N$ .
|
1960
| 76.5625 |
10,393 |
Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 60^\circ$, and $DF = 6$. Find the radius of the incircle and the circumference of the circumscribed circle around $\triangle DEF$.
|
12\pi
| 71.09375 |
10,394 |
Let's call any natural number "very prime" if any number of consecutive digits (in particular, a digit or number itself) is a prime number. For example, $23$ and $37$ are "very prime" numbers, but $237$ and $357$ are not. Find the largest "prime" number (with justification!).
|
373
| 85.9375 |
10,395 |
Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is $\leq d$
|
\frac{1}{2\sqrt{2}}
| 1.5625 |
10,396 |
Each day 289 students are divided into 17 groups of 17. No two students are ever in the same group more than once. What is the largest number of days that this can be done?
|
18
| 46.875 |
10,397 |
Points \( K \) and \( N \) are located on the sides \( AB \) and \( AC \) of triangle \( ABC \) respectively, such that \( AK = BK \) and \( AN = 2NC \).
In what ratio does the segment \( KN \) divide the median \( AM \) of triangle \( ABC \)?
|
4:3
| 42.1875 |
10,398 |
Given the function
$$
f(x)=
\begin{cases}
1 & (1 \leq x \leq 2) \\
\frac{1}{2}x^2 - 1 & (2 < x \leq 3)
\end{cases}
$$
define $h(a) = \max\{f(x) - ax \mid x \in [1, 3]\} - \min\{f(x) - ax \mid x \in [1, 3]\}$ for any real number $a$.
1. Find the value of $h(0)$.
2. Find the expression for $h(a)$ and its minimum value.
|
\frac{5}{4}
| 1.5625 |
10,399 |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^{2}}{16}{+}\frac{y^{2}}{b^{2}}{=}1$, and the line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $10$, find the eccentricity of the ellipse.
|
\frac{1}{2}
| 0.78125 |
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