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int64 0
40.3k
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stringlengths 10
5.15k
| ground_truth
stringlengths 1
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float64 0
100
|
|---|---|---|---|
9,800 |
In a closed right triangular prism ABC-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub> there is a sphere with volume $V$. If $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$, then the maximum value of $V$ is \_\_\_\_\_\_.
|
\frac{9\pi}{2}
| 37.5 |
9,801 |
Let \( f(x) = x^2 + px + q \) and \( g(x) = x^2 + rx + s \) be two distinct quadratic polynomials where the \( x \)-coordinate of the vertex of \( f \) is a root of \( g \), and the \( x \)-coordinate of the vertex of \( g \) is a root of \( f \), also both \( f \) and \( g \) have the same minimum value. If the graphs of the two quadratic polynomials intersect at the point \( (50,-200), \) what is the value of \( p + r \)?
|
-200
| 36.71875 |
9,802 |
In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). If \(AD = 52\), \(BC = 65\), \(AB = 20\), and \(CD = 11\), find the area of the trapezoid.
|
594
| 67.96875 |
9,803 |
There are five students: A, B, C, D, and E.
1. In how many different ways can they line up in a row such that A and B must be adjacent, and C and D cannot be adjacent?
2. In how many different ways can these five students be distributed into three classes, with each class having at least one student?
|
150
| 89.84375 |
9,804 |
The front view of a cone is an equilateral triangle with a side length of 4. Find the surface area of the cone.
|
12\pi
| 53.90625 |
9,805 |
Among the following numbers
① $111111_{(2)}$
② $210_{(6)}$
③ $1000_{(4)}$
④ $81_{(8)}$
The largest number is \_\_\_\_\_\_\_\_, and the smallest number is \_\_\_\_\_\_\_\_.
|
111111_{(2)}
| 40.625 |
9,806 |
The number \( 20! = 1 \cdot 2 \cdot \ldots \cdot 20 = 2432902008176640000 \) has 41,040 natural divisors. How many of them are odd?
|
2160
| 79.6875 |
9,807 |
Chloe wants to buy a jacket that costs $45.50$. She has two $20$ bills, five quarters, a few nickels, and a pile of dimes in her wallet. What is the minimum number of dimes she needs if she also has six nickels?
|
40
| 67.1875 |
9,808 |
Let $m$ be the smallest positive three-digit integer congruent to 7 (mod 13). Let $n$ be the smallest positive four-digit integer congruent to 7 (mod 13). What is the value of $n - m$?
|
897
| 14.84375 |
9,809 |
An aluminum cube with an edge length of \( l = 10 \) cm is heated to a temperature of \( t_{1} = 100^{\circ} \mathrm{C} \). After this, it is placed on ice, which has a temperature of \( t_{2} = 0^{\circ} \mathrm{C} \). Determine the maximum depth to which the cube can sink. The specific heat capacity of aluminum is \( c_{a} = 900 \) J/kg\(^\circ \mathrm{C} \), the specific latent heat of fusion of ice is \( \lambda = 3.3 \times 10^{5} \) J/kg, the density of aluminum is \( \rho_{a} = 2700 \) kg/m\(^3 \), and the density of ice is \( \rho_{n} = 900 \) kg/m\(^3 \).
|
0.0818
| 4.6875 |
9,810 |
Let \(a_n\) be the sequence defined by \(a_1 = 3\) and \(a_{n+1} = 3^{k}\), where \(k = a_n\). Let \(b_n\) be the remainder when \(a_n\) is divided by 100. Which values \(b_n\) occur for infinitely many \(n\)?
|
87
| 76.5625 |
9,811 |
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $n$ ?
|
50
| 86.71875 |
9,812 |
The 200-digit number \( M \) is composed of 200 ones. What is the sum of the digits of the product \( M \times 2013 \)?
|
1200
| 91.40625 |
9,813 |
20 balls numbered 1 through 20 are placed in a bin. In how many ways can 4 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn and each drawn ball has a consecutive number as the previous one?
|
17
| 80.46875 |
9,814 |
The maximum value of the function \( y = \frac{\sin x \cos x}{1 + \sin x + \cos x} \) is $\quad$ .
|
\frac{\sqrt{2} - 1}{2}
| 63.28125 |
9,815 |
The value of the product \(\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cdots \cos \frac{7 \pi}{15}\).
|
\frac{1}{128}
| 46.875 |
9,816 |
Find the measure of the angle
$$
\delta=\arccos \left(\left(\sin 2905^{\circ}+\sin 2906^{\circ}+\cdots+\sin 6505^{\circ}\right)^{\cos } 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}\right)
$$
|
65
| 11.71875 |
9,817 |
Define a function $g(x),$ for positive integer values of $x,$ by \[g(x) = \left\{\begin{aligned} \log_3 x & \quad \text{ if } \log_3 x \text{ is an integer} \\ 1 + g(x + 1) & \quad \text{ otherwise}. \end{aligned} \right.\]Compute $g(200).$
|
48
| 7.03125 |
9,818 |
In a right triangle $ABC$ with legs $AB = 3$ and $BC = 4$, a circle is drawn through the midpoints of sides $AB$ and $AC$, touching the leg $BC$. Find the length of the segment of the hypotenuse $AC$ that lies inside this circle.
|
11/10
| 3.125 |
9,819 |
Calculate the definite integral:
$$
\int_{0}^{2} e^{\sqrt{(2-x) /(2+x)}} \cdot \frac{d x}{(2+x) \sqrt{4-x^{2}}}
$$
|
\frac{e-1}{2}
| 64.0625 |
9,820 |
The value of $(\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}$ is:
(a) $\sqrt{2}+\sqrt{3}$;
(b) $\frac{1}{2}(7+3 \sqrt{5})$;
(c) $1+2 \sqrt{3}$;
(d) 3 ;
(e) $3+2 \sqrt{2}$.
|
3 + 2 \sqrt{2}
| 84.375 |
9,821 |
If the fractional equation $\frac{3}{{x-4}}+\frac{{x+m}}{{4-x}}=1$ has a root, determine the value of $m$.
|
-1
| 3.90625 |
9,822 |
15 boys and 20 girls sat around a round table. It turned out that the number of pairs of boys sitting next to each other is one and a half times less than the number of pairs of girls sitting next to each other. Find the number of boy-girl pairs sitting next to each other.
|
10
| 42.96875 |
9,823 |
A clothing store buys 600 pairs of gloves at a price of 12 yuan per pair. They sell 470 pairs at a price of 14 yuan per pair, and the remaining gloves are sold at a price of 11 yuan per pair. What is the total profit made by the clothing store from selling this batch of gloves?
|
810
| 93.75 |
9,824 |
Let \(a\) and \(b\) be constants. The parabola \(C: y = (t^2 + t + 1)x^2 - 2(a + t)^2 x + t^2 + 3at + b\) passes through a fixed point \(P(1,0)\) for any real number \(t\). Find the value of \(t\) such that the chord obtained by intersecting the parabola \(C\) with the x-axis is the longest.
|
-1
| 27.34375 |
9,825 |
A triangle with interior angles $60^{\circ}, 45^{\circ}$ and $75^{\circ}$ is inscribed in a circle of radius 2. What is the area of the triangle?
|
3 + \sqrt{3}
| 30.46875 |
9,826 |
If $x$ is an even number, then find the largest integer that always divides the expression \[(15x+3)(15x+9)(5x+10).\]
|
90
| 25 |
9,827 |
Masha and the Bear ate a basket of raspberries and 60 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the Bear ate pies, and then they switched at some point. The Bear ate raspberries 6 times faster than Masha and pies 3 times faster. How many pies did the Bear eat if the Bear ate twice as many raspberries as Masha?
|
54
| 3.125 |
9,828 |
Income from September 2019 to December 2019 is:
$$
(55000+45000+10000+17400) * 4 = 509600 \text{ rubles}
$$
Expenses from September 2019 to November 2019 are:
$$
(40000+20000+5000+2000+2000) * 4 = 276000 \text{ rubles}
$$
By 31.12.2019 the family will have saved $1147240 + 521600 - 276000 = 1340840$ rubles and will be able to buy a car.
|
1340840
| 60.9375 |
9,829 |
The number \( N \) is of the form \( p^{\alpha} q^{\beta} r^{\gamma} \), where \( p, q, r \) are prime numbers, and \( p q - r = 3, p r - q = 9 \). Additionally, the numbers \( \frac{N}{p}, \frac{N}{q}, \frac{N}{r} \) respectively have 20, 12, and 15 fewer divisors than the number \( N \). Find the number \( N \).
|
857500
| 1.5625 |
9,830 |
9 judges score a gymnast in artistic gymnastics, with each giving an integer score. One highest score and one lowest score are removed, and the average of the remaining scores determines the gymnast's score. If the score is rounded to one decimal place using the rounding method, the gymnast scores 8.4 points. What would the gymnast's score be if it were accurate to two decimal places?
|
8.43
| 60.9375 |
9,831 |
The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous. For every real number \( x \), the equation \( f(x) \cdot f(f(x)) = 1 \) holds. It is known that \( f(1000) = 999 \). Find \( f(500) \).
|
\frac{1}{500}
| 69.53125 |
9,832 |
Let \( S_{n} \) be the sum of the first \( n \) terms of an arithmetic sequence \( \{a_{n}\} \). Given \( S_{6}=36 \) and \( S_{n}=324 \). If \( S_{n-6} = 144 \) for \( n > 6 \), then \( n \) equals \(\qquad\).
|
18
| 96.09375 |
9,833 |
How many three-digit whole numbers contain at least one digit 6 or at least one digit 8?
|
452
| 98.4375 |
9,834 |
If a natural number \(N\) can be expressed as the sum of 3 consecutive natural numbers, the sum of 11 consecutive natural numbers, and the sum of 12 consecutive natural numbers, then the smallest value of \(N\) is \(\qquad\). (Note: The smallest natural number is 0.)
|
66
| 75.78125 |
9,835 |
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$ . Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by $\triangle '$ . Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$ , and denote its area by $\triangle ''$ . Given that $\triangle ' = 30$ and $\triangle '' = 20$ , find $\triangle$ .
|
45
| 14.84375 |
9,836 |
Five guys are eating hamburgers. Each one puts a top half and a bottom half of a hamburger bun on the grill. When the buns are toasted, each guy randomly takes two pieces of bread off of the grill. What is the probability that each guy gets a top half and a bottom half?
|
8/63
| 21.875 |
9,837 |
Determine the largest prime number less than 5000 of the form \( a^n - 1 \), where \( a \) and \( n \) are positive integers, and \( n \) is greater than 1.
|
127
| 60.9375 |
9,838 |
The solutions to the equation \( x^3 - 4 \lfloor x \rfloor = 5 \), where \( x \) is a real number, are denoted by \( x_1, x_2, x_3, \ldots, x_k \) for some positive integer \( k \). Find \( \sum_{i=1}^{k} x_{i}^{3} \).
|
10
| 10.15625 |
9,839 |
In triangle $XYZ$, $XY=15$, $YZ=18$, and $ZX=21$. Point $G$ is on $\overline{XY}$, $H$ is on $\overline{YZ}$, and $I$ is on $\overline{ZX}$. Let $XG = p \cdot XY$, $YH = q \cdot YZ$, and $ZI = r \cdot ZX$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of triangle $GHI$ to the area of triangle $XYZ$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
|
41
| 44.53125 |
9,840 |
Inside triangle \( ABC \), a random point \( M \) is chosen. What is the probability that the area of one of the triangles \( ABM \), \( BCM \), or \( CAM \) will be greater than the sum of the areas of the other two?
|
0.75
| 0 |
9,841 |
In triangle $ABC,$ $\angle B = 45^\circ,$ $AB = 100,$ and $AC = 100$. Find the sum of all possible values of $BC$.
|
100 \sqrt{2}
| 54.6875 |
9,842 |
Given \( S_{n} = \sum_{i=1}^{10} i|n-i| \left( n \in \mathbf{Z}_{+} \right) \), find the minimum value of \( S_{n} \).
|
112
| 85.9375 |
9,843 |
Calculate:
$$
\left(10^{4}-9^{4}+8^{4}-7^{4}+\cdots+2^{4}-1^{4}\right)+\left(10^{2}+9^{2}+5 \times 8^{2}+5 \times 7^{2}+9 \times 6^{2}+9 \times 5^{2}+13 \times 4^{2}+13 \times 3^{2}\right) =
$$
|
7615
| 48.4375 |
9,844 |
Define the sequence $(b_i)$ by $b_{n+2} = \frac{b_n + 2011}{1 + b_{n+1}}$ for $n \geq 1$ with all terms being positive integers. Determine the minimum possible value of $b_1 + b_2$.
|
2012
| 5.46875 |
9,845 |
There are 4 different colors of light bulbs, with each color representing a different signal. Assuming there is an ample supply of each color, we need to install one light bulb at each vertex $P, A, B, C, A_{1}, B_{1}, C_{1}$ of an airport signal tower (as shown in the diagram), with the condition that the two ends of the same line segment must have light bulbs of different colors. How many different installation methods are there?
|
2916
| 12.5 |
9,846 |
How many pairs of values \( p, q \in \mathbf{N} \), not exceeding 100, exist for which the equation
\[ x^{5} + p x + q = 0 \]
has solutions in rational numbers?
|
133
| 67.96875 |
9,847 |
Masha has an integer multiple of toys compared to Lena, and Lena has the same multiple of toys compared to Katya. Masha gave 3 toys to Lena, and Katya gave 2 toys to Lena. After that, the number of toys each girl had formed an arithmetic progression. How many toys did each girl originally have? Provide the total number of toys the girls had initially.
|
105
| 44.53125 |
9,848 |
Find the value of \(\tan \left(\tan^{-1} \frac{1}{2} + \tan^{-1} \frac{1}{2 \times 2^2} + \tan^{-1} \frac{1}{2 \times 3^2} + \cdots + \tan^{-1} \frac{1}{2 \times 2009^2}\right)\).
|
\frac{2009}{2010}
| 35.15625 |
9,849 |
How many positive integers less than or equal to 5689 contain either the digit '6' or the digit '0'?
|
2545
| 3.125 |
9,850 |
Given that $\cos(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{10}$, with $x \in (\frac{\pi}{2}, \frac{3\pi}{4})$.
(1) Find the value of $\sin x$;
(2) Find the value of $\cos(2x - \frac{\pi}{3})$.
|
-\frac{7 + 24\sqrt{3}}{50}
| 60.15625 |
9,851 |
There were 100 cards lying face up with the white side on the table. Each card has one white side and one black side. Kostya flipped 50 cards, then Tanya flipped 60 cards, and after that Olya flipped 70 cards. As a result, all 100 cards ended up lying with the black side up. How many cards were flipped three times?
|
40
| 35.15625 |
9,852 |
In the cartesian coordinate system $(xOy)$, curve $({C}_{1})$ is defined by the parametric equations $\begin{cases}x=t+1,\ y=1-2t\end{cases}$ and curve $({C}_{2})$ is defined by the parametric equations $\begin{cases}x=a\cos θ,\ y=3\sin θ\end{cases}$ where $a > 0$.
1. If curve $({C}_{1})$ and curve $({C}_{2})$ have a common point on the $x$-axis, find the value of $a$.
2. When $a=3$, curves $({C}_{1})$ and $({C}_{2})$ intersect at points $A$ and $B$. Find the distance between points $A$ and $B$.
|
\frac{12\sqrt{5}}{5}
| 33.59375 |
9,853 |
A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box each time. Winning is defined as drawing two balls of the same color, while losing is defined as drawing two balls of different colors.
$(1)$ If it is specified that after the first draw, the balls are put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $X$.
$(2)$ If it is specified that after the first draw, the balls are not put back into the lottery box for the second draw, find the distribution and mathematical expectation of the number of wins $Y$.
|
\frac{6}{7}
| 3.90625 |
9,854 |
At a variety show, there are seven acts: dance, comic dialogue, sketch, singing, magic, acrobatics, and opera. When arranging the program order, the conditions are that dance, comic dialogue, and sketch cannot be adjacent to each other. How many different arrangements of the program are possible?
|
1440
| 6.25 |
9,855 |
Out of 10 distinct positive integers, the product of any 5 of them is even, and the sum of all 10 numbers is odd. What is the minimum sum of these 10 positive integers?
|
65
| 25 |
9,856 |
A certain organism starts with 4 cells. Each cell splits into two cells at the end of three days. However, at the end of each 3-day period, 10% of the cells die immediately after splitting. This process continues for a total of 9 days. How many cells are there at the end of the $9^\text{th}$ day?
|
23
| 8.59375 |
9,857 |
In $\triangle ABC$, if $a + c = 2b$, then find the value of $\tan \frac{A}{2} \cdot \tan \frac{C}{2}$.
|
1/3
| 45.3125 |
9,858 |
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfies the condition: for any real numbers \( x \) and \( y \),
\[ f(2x) + f(2y) = f(x+y) f(x-y) \]
and given that \( f(\pi) = 0 \) and \( f(x) \) is not identically zero, determine the period of \( f(x) \).
|
4\pi
| 21.875 |
9,859 |
The hares are cutting the log again, but now both ends of the log are fixed. Ten middle logs fell, and the two end ones remained fixed. How many cuts did the hares make?
|
11
| 49.21875 |
9,860 |
Calculate the definite integral:
$$
\int_{2 \operatorname{arctg} \frac{1}{3}}^{2 \operatorname{arctg} \frac{1}{2}} \frac{d x}{\sin x(1-\sin x)}
$$
|
\ln 3 - \ln 2 + 1
| 0 |
9,861 |
In parallelogram $ABCD$, $BE$ is the height from vertex $B$ to side $AD$, and segment $ED$ is extended from $D$ such that $ED = 8$. The base $BC$ of the parallelogram is $14$. The entire parallelogram has an area of $126$. Determine the area of the shaded region $BEDC$.
|
99
| 15.625 |
9,862 |
Burattino got on a train. After travelling half of the total distance, he fell asleep and slept until there was only half of the distance he slept left to travel. What fraction of the total journey did Burattino travel awake?
|
\frac{2}{3}
| 62.5 |
9,863 |
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value?
|
\frac{113}{648}
| 1.5625 |
9,864 |
Given that there are 4 tiers of horses, with Tian Ji's top-tier horse being better than King Qi's middle-tier horse, worse than King Qi's top-tier horse, Tian Ji's middle-tier horse being better than King Qi's bottom-tier horse, worse than King Qi's middle-tier horse, and Tian Ji's bottom-tier horse being worse than King Qi's bottom-tier horse, determine the probability that Tian Ji's selected horse wins the race.
|
\frac{1}{3}
| 36.71875 |
9,865 |
Let \( A \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + x - 110 < 0 \) and \( B \) be the set of real numbers \( x \) satisfying the inequality \( x^{2} + 10x - 96 < 0 \). Suppose that the set of integer solutions of the inequality \( x^{2} + ax + b < 0 \) is exactly the set of integers contained in \( A \cap B \). Find the maximum value of \( \lfloor |a - b| \rfloor \).
|
71
| 30.46875 |
9,866 |
Given the number 826,000,000, express it in scientific notation.
|
8.26\times 10^{8}
| 0 |
9,867 |
Let $G$ be the centroid of triangle $PQR.$ If $GP^2 + GQ^2 + GR^2 = 22,$ then find $PQ^2 + PR^2 + QR^2.$
|
66
| 96.09375 |
9,868 |
If \( a \) is the smallest cubic number divisible by 810, find the value of \( a \).
|
729000
| 79.6875 |
9,869 |
Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:
$ (i)$ $ a\plus{}c\equal{}d;$
$ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
$ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$ .
Determine $ n$ .
|
2002
| 64.0625 |
9,870 |
Xiaoming saw a tractor pulling a rope slowly on the road. Xiaoming decided to measure the length of the rope. If Xiaoming walks in the same direction as the tractor, it takes 140 steps to walk from one end of the rope to the other end; if Xiaoming walks in the opposite direction of the tractor, it takes 20 steps. The speed of the tractor and Xiaoming remain constant, and Xiaoming can walk 1 meter per step. What is the length of the rope in meters?
|
35
| 24.21875 |
9,871 |
Let $a$ be the sum of the numbers: $99 \times 0.9$ $999 \times 0.9$ $9999 \times 0.9$ $\vdots$ $999\cdots 9 \times 0.9$ where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$ .
Find the sum of the digits in the number $a$ .
|
891
| 21.09375 |
9,872 |
The set $\{[x]+[2x]+[3x] \mid x \in \mathbf{R}\} \bigcap \{1, 2, \cdots, 100\}$ contains how many elements, where $[x]$ represents the greatest integer less than or equal to $x$.
|
67
| 54.6875 |
9,873 |
A factory has a fixed daily cost of 20,000 yuan, and the maximum daily production capacity is 360 units. The cost increases by 100 yuan for each unit produced. The revenue function for producing $x$ units of product per day is $R(x) = -\frac{1}{2}x^2 + 400x$. Let $L(x)$ and $P(x)$ represent the daily profit and average profit (average profit = $\frac{\text{total profit}}{\text{total quantity}}$) for producing $x$ units of product, respectively.
(1) At what daily production quantity $x$ does the profit $L(x)$ reach its maximum value, and what is this maximum value?
(2) At what daily production quantity $x$ does the average profit $P(x)$ reach its maximum value, and what is this maximum value?
(3) Due to an economic crisis, the factory had to lay off workers, causing the maximum daily production capacity to drop to 160 units. At what daily production quantity $x$ does the average profit $P(x)$ reach its maximum value, and what is this maximum value?
|
95
| 50.78125 |
9,874 |
What is the largest \( x \) such that \( x^2 \) divides \( 24 \cdot 35 \cdot 46 \cdot 57 \)?
|
12
| 93.75 |
9,875 |
Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board?
|
13
| 82.03125 |
9,876 |
A table consisting of 1861 rows and 1861 columns is filled with natural numbers from 1 to 1861 such that each row contains all numbers from 1 to 1861. Find the sum of the numbers on the diagonal that connects the top left and bottom right corners of the table if the filling of the table is symmetric with respect to this diagonal.
|
1732591
| 86.71875 |
9,877 |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a=2b$. Also, $\sin A$, $\sin C$, $\sin B$ form an arithmetic sequence.
$(I)$ Find the value of $\cos (B+C)$;
$(II)$ If the area of $\triangle ABC$ is $\frac{8\sqrt{15}}{3}$, find the value of $c$.
|
4 \sqrt {2}
| 0 |
9,878 |
Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$ .
|
13
| 53.125 |
9,879 |
If you want to form a rectangular prism with a surface area of 52, what is the minimum number of small cubes with edge length 1 needed?
|
16
| 33.59375 |
9,880 |
A shooter fires at a target until the first hit, with a hit rate of 0.6 for each shot. If there are 4 bullets in total, what is the expected number of remaining bullets $\xi$?
|
2.376
| 22.65625 |
9,881 |
The altitudes of an acute-angled triangle \( ABC \) drawn from vertices \( B \) and \( C \) are 7 and 9, respectively, and the median \( AM \) is 8. Points \( P \) and \( Q \) are symmetric to point \( M \) with respect to sides \( AC \) and \( AB \), respectively. Find the perimeter of the quadrilateral \( APMQ \).
|
32
| 75 |
9,882 |
In triangle $\triangle ABC$, it is known that $AB=2$, $AC=3$, $\angle A=60^{\circ}$, and $\overrightarrow{BM}=\frac{1}{3}\overrightarrow{BC}$, $\overrightarrow{AN}=\overrightarrow{NB}$. Find $\overrightarrow{AC}•\overrightarrow{NM}$.
|
\frac{7}{2}
| 55.46875 |
9,883 |
In the country of Anchuria, a unified state exam takes place. The probability of guessing the correct answer to each question on the exam is 0.25. In 2011, to receive a certificate, one needed to answer correctly 3 questions out of 20. In 2012, the School Management of Anchuria decided that 3 questions were too few. Now, one needs to correctly answer 6 questions out of 40. The question is, if one knows nothing and simply guesses the answers, in which year is the probability of receiving an Anchurian certificate higher - in 2011 or in 2012?
|
2012
| 91.40625 |
9,884 |
Let the function $f(x)=\ln x-\frac{1}{2} ax^{2}-bx$.
$(1)$ When $a=b=\frac{1}{2}$, find the maximum value of the function $f(x)$;
$(2)$ Let $F(x)=f(x)+\frac{1}{2} x^{2}+bx+\frac{a}{x} (0 < x\leqslant 3)$. If the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on its graph is always less than or equal to $\frac{1}{2}$, find the range of the real number $a$;
$(3)$ When $a=0$, $b=-1$, the equation $x^{2}=2mf(x)$ (where $m > 0$) has a unique real solution, find the value of $m$.
|
\frac{1}{2}
| 14.0625 |
9,885 |
Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals?
|
11
| 7.03125 |
9,886 |
Given two circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-2y-6=0$, find the length of their common chord.
|
2\sqrt{3}
| 83.59375 |
9,887 |
Find the distance from the point \( M_{0} \) to the plane passing through the three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(1, 3, 0) \)
\( M_{2}(4, -1, 2) \)
\( M_{3}(3, 0, 1) \)
\( M_{0}(4, 3, 0) \)
|
\sqrt{6}
| 82.8125 |
9,888 |
Sixty students went on a trip to the zoo. Upon returning to school, it turned out that 55 of them forgot gloves at the zoo, 52 forgot scarves, and 50 managed to forget hats. Find the smallest number of the most scatterbrained students - those who lost all three items.
|
37
| 9.375 |
9,889 |
The older brother and the younger brother each bought several apples. The older brother said to the younger brother, "If I give you one apple, we will have the same number of apples." The younger brother thought for a moment and said to the older brother, "If I give you one apple, the number of apples you have will be twice as much as mine." How many apples did they buy in total?
|
12
| 99.21875 |
9,890 |
If $2 - \sin^{2}(x + 2y - 1) = \frac{x^{2} + y^{2} - 2(x + 1)(y - 1)}{x - y + 1}$, then the minimum value of the product $xy$ is $\qquad$ .
|
1/9
| 67.1875 |
9,891 |
The four zeros of the polynomial \(x^4 + px^2 + qx - 144\) are distinct real numbers in arithmetic progression. Compute the value of \(p.\)
|
-40
| 84.375 |
9,892 |
Given the hyperbola \( C_{1}: 2x^{2} - y^{2} = 1 \) and the ellipse \( C_{2}: 4x^{2} + y^{2} = 1 \), let \( M \) and \( N \) be moving points on the hyperbola \( C_{1} \) and the ellipse \( C_{2} \) respectively, with \( O \) as the origin. If \( O M \) is perpendicular to \( O N \), find the distance from point \( O \) to the line \( M N \).
|
\frac{\sqrt{3}}{3}
| 1.5625 |
9,893 |
Find the maximum possible area of a section passing through the diagonal $XY$ of an arbitrary parallelepiped with edges $a \leq b \leq c$. The section is a parallelogram $Z X T Y$ with vertices lying on opposite edges of the parallelepiped. The area of the parallelogram equals the product of the length of the diagonal $XY$ and the distance from point $Z$ to $XY$.
The largest area of a possible section, considering diagonal sections, is determined, with the given sides $a$, $b$, and $c$ in order of $a \le b \le c$. The areas of potential sections are:
$$
S_{1}=a \sqrt{b^{2}+c^{2}}, S_{2}=b \sqrt{a^{2}+c^{2}}, \text { and } S_{3}=c \sqrt{b^{2}+a^{2}} .
$$
Given the condition $a \leq b \leq c$, it is apparent that $S_{1} \leq S_{3}$ and $S_{2} \leq S_{3}$. Therefore, the maximum area section passes through the longest edge.
Thus, the maximum area section is $10 \sqrt{4^{2}+3^{2}}=50$.
|
50
| 85.9375 |
9,894 |
For what real value of $u$ is $\frac{-15+\sqrt{205}}{8}$ a root of $4x^2 + 15x + u$?
|
\frac{5}{4}
| 78.90625 |
9,895 |
Given that $P$ is a point inside rectangle $ABCD$, the distances from $P$ to the vertices of the rectangle are $PA = 5$ inches, $PD = 12$ inches, and $PC = 13$ inches. Find $PB$, which is $x$ inches.
|
5\sqrt{2}
| 76.5625 |
9,896 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given that $a=4$, $b=6$, and $C=60^\circ$:
1. Calculate $\overrightarrow{BC} \cdot \overrightarrow{CA}$;
2. Find the projection of $\overrightarrow{CA}$ onto $\overrightarrow{BC}$.
|
-3
| 14.0625 |
9,897 |
Point \(C\) divides diameter \(AB\) in the ratio \(AC:BC = 2:1\). A point \(P\) is selected on the circle. Determine the possible values that the ratio \(\tan \angle PAC: \tan \angle APC\) can take. Specify the smallest such value.
|
1/2
| 12.5 |
9,898 |
A one-meter gas pipe has rusted through in two places. Determine the probability that all three resulting parts can be used as connectors to gas stoves, given that according to regulations, the stove must not be closer than 25 cm to the main gas pipe.
|
1/16
| 4.6875 |
9,899 |
The center of a balloon is observed by two ground observers at angles of elevation of $45^{\circ}$ and $22.5^{\circ}$, respectively. The first observer is to the south, and the second one is to the northwest of the point directly under the balloon. The distance between the two observers is 1600 meters. How high is the balloon floating above the horizontal ground?
|
500
| 0 |
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