Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
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float64 0
100
|
|---|---|---|---|
11,400 |
A natural number is called lucky if all its digits are equal to 7. For example, 7 and 7777 are lucky, but 767 is not. João wrote down the first twenty lucky numbers starting from 7, and then added them. What is the remainder of that sum when divided by 1000?
|
70
| 39.84375 |
11,401 |
Five doctors A, B, C, D, and E are assigned to four different service positions located in the Sichuan disaster zone, labeled A, B, C, and D. Each position must be filled by at least one doctor. Calculate the total number of ways doctors A and B can serve independently in different positions.
|
72
| 17.96875 |
11,402 |
On the game show $\text{\emph{Wheel of Fortunes}}$, you encounter a spinner divided equally into 6 regions labeled as follows: "Bankrupt," "$600", "$100", "$2000", "$150", "$700". What is the probability that you will accumulate exactly $1450$ in your first three spins without landing on "Bankrupt"? Assume each spin is independent and all outcomes have equal likelihood.
|
\frac{6}{125}
| 25.78125 |
11,403 |
Given that the complex number $z_1$ corresponds to the point $(-1,1)$ on the complex plane, and the complex number $z_2$ satisfies $z_1z_2=-2$, find the value of $|z_2+2i|$.
|
\sqrt{10}
| 92.96875 |
11,404 |
If \( p \) is the smallest positive prime number such that for some integer \( n \), \( p \) divides \( n^{2} + 5n + 23 \), then \( p = \)
|
17
| 43.75 |
11,405 |
Several oranges (not necessarily of equal mass) were picked from a tree. On weighing them, it turned out that the mass of any three oranges taken together is less than 5% of the total mass of the remaining oranges. What is the minimum number of oranges that could have been picked?
|
64
| 79.6875 |
11,406 |
The ratio of boys to girls in Mr. Smith's class is 3:4, and there are 42 students in total. What percent of the students are boys.
|
42.86\%
| 89.0625 |
11,407 |
Given a monotonically increasing sequence of positive integers $\left\{a_{n}\right\}$ that satisfies the recurrence relation $a_{n+2}=3 a_{n+1}-a_{n}$, with $a_{6}=280$, find the value of $a_{7}$.
|
733
| 23.4375 |
11,408 |
Observe:
$$
\begin{array}{l}
1 \times 2 \times 3 \times 4 + 1 = 5^{2} \\
2 \times 3 \times 4 \times 5 + 1 = 11^{2} \\
3 \times 4 \times 5 \times 6 + 1 = 19^{2} \\
\ldots \ldots
\end{array}
$$
Calculate $\sqrt{2020 \times 2021 \times 2022 \times 2023 + 1}=$
|
4086461
| 83.59375 |
11,409 |
Given that the sum of the first $n$ terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ are $(S_n)$ and $(T_n)$, respectively. If for any positive integer $n$, $\frac{S_n}{T_n}=\frac{2n-5}{3n-5}$, determine the value of $\frac{a_7}{b_2+b_8}+\frac{a_3}{b_4+b_6}$.
|
\frac{13}{22}
| 38.28125 |
11,410 |
On the board are written 5 integers. By adding them in pairs, the following set of 10 numbers was obtained: $5, 8, 9, 13, 14, 14, 15, 17, 18, 23$. Determine which numbers are written on the board. In the answer, write their product.
|
4752
| 28.125 |
11,411 |
The base $ABCD$ of a tetrahedron $P-ABCD$ is a convex quadrilateral with diagonals $AC$ and $BD$ intersecting at $O$. If the area of $\triangle AOB$ is 36, the area of $\triangle COD$ is 64, and the height of the tetrahedron is 9, what is the minimum volume of such a tetrahedron?
|
588
| 6.25 |
11,412 |
A point \((x, y)\) is randomly picked from inside the rectangle with vertices \((0,0)\), \((5,0)\), \((5,2)\), and \((0,2)\). What is the probability that \(x < 2y\)?
|
\frac{2}{5}
| 66.40625 |
11,413 |
The price of Type A remote control car is 46.5 yuan, and the price of Type B remote control car is 54.5 yuan. Lele has 120 yuan. If he buys both types of remote control cars, will he have enough money? If so, how much money will he have left after the purchase?
|
19
| 6.25 |
11,414 |
Find the minimum value, for \(a, b > 0\), of the expression
\[
\frac{|a + 3b - b(a + 9b)| + |3b - a + 3b(a - b)|}{\sqrt{a^{2} + 9b^{2}}}
\]
|
\frac{\sqrt{10}}{5}
| 12.5 |
11,415 |
Four girls and eight boys came for a class photograph. Children approach the photographer in pairs and take a joint photo. Among how many minimum photos must there necessarily be either a photo of two boys, a photo of two girls, or two photos with the same children?
|
33
| 31.25 |
11,416 |
Let \[g(x) =
\begin{cases}
2x - 4 &\text{if } x < 0, \\
5 - 3x &\text{if } x \geq 0.
\end{cases}\]
Find $g(-2)$ and $g(3)$.
|
-4
| 57.03125 |
11,417 |
Three Graces each had the same number of fruits and met 9 Muses. Each Grace gave an equal number of fruits to each Muse. After that, each Muse and each Grace had the same number of fruits. How many fruits did each Grace have before meeting the Muses?
|
12
| 67.1875 |
11,418 |
In the plane quadrilateral \(ABCD\), points \(E\) and \(F\) are the midpoints of sides \(AD\) and \(BC\) respectively. Given that \(AB = 1\), \(EF = \sqrt{2}\), and \(CD = 3\), and that \(\overrightarrow{AD} \cdot \overrightarrow{BC} = 15\), find \(\overrightarrow{AC} \cdot \overrightarrow{BD}\).
|
16
| 3.125 |
11,419 |
Suppose that \( X \) and \( Y \) are angles with \( \tan X = \frac{1}{m} \) and \( \tan Y = \frac{a}{n} \) for some positive integers \( a, m, \) and \( n \). Determine the number of positive integers \( a \leq 50 \) for which there are exactly 6 pairs of positive integers \( (m, n) \) with \( X + Y = 45^{\circ} \).
(Note: The formula \( \tan (X + Y) = \frac{\tan X + \tan Y}{1 - \tan X \tan Y} \) may be useful.)
|
12
| 28.90625 |
11,420 |
Experts and Viewers play "What? Where? When?" until one side wins six rounds—the first to win six rounds wins the game. The probability of the Experts winning a single round is 0.6, and there are no ties. Currently, the Experts are losing with a score of $3:4$. Find the probability that the Experts will still win.
|
0.4752
| 28.90625 |
11,421 |
There are two schools, A and B, each sending 5 students to participate in a long-distance race. The rule is: the student who finishes in the \( K \)-th place receives \( K \) points (no two students finish at the same time). The school with the lower total score wins. How many possible scores can the winning team have?
|
13
| 78.125 |
11,422 |
How can we connect 50 cities with the minimum number of flight routes so that it's possible to travel from any city to any other city with no more than two layovers?
|
49
| 91.40625 |
11,423 |
Li Shuang rides a bike from location $A$ to location $B$ at a speed of 320 meters per minute. On the way, due to a bike malfunction, he pushes the bike and continues walking for 5 minutes to a location 1800 meters from $B$ to repair the bike. Fifteen minutes later, he resumes riding towards $B$ at 1.5 times his original cycling speed. Upon reaching $B$, he is 17 minutes later than the estimated time. What is Li Shuang's speed while pushing the bike in meters per minute?
|
72
| 6.25 |
11,424 |
Find $\left(\sqrt[4]{(\sqrt{5})^5}\right)^2$.
|
5 \sqrt[4]{5}
| 1.5625 |
11,425 |
Determine the number of intersection points of 10 lines, given that only two of them are parallel and exactly three of these lines intersect at one point.
|
42
| 53.125 |
11,426 |
Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to points $P_{2}$ on $CD$, $P_{3}$ on $DA$, and $P_{4}$ on $AB$, with the reflection angle equal to the incidence angle. If $P_{4}$ coincides with $P_{0}$, find $\tan \theta$.
|
\frac{1}{2}
| 25.78125 |
11,427 |
Given two real numbers \( p > 1 \) and \( q > 1 \) such that \( \frac{1}{p} + \frac{1}{q} = 1 \) and \( pq = 9 \), what is \( q \)?
|
\frac{9 + 3\sqrt{5}}{2}
| 46.09375 |
11,428 |
(1) Given $\frac{\sin\alpha + 3\cos\alpha}{3\cos\alpha - \sin\alpha} = 5$, find the value of $\sin^2\alpha - \sin\alpha\cos\alpha$.
(2) Given a point $P(-4, 3)$ on the terminal side of angle $\alpha$, determine the value of $\frac{\cos\left(\frac{\pi}{2} + \alpha\right)\sin\left(-\pi - \alpha\right)}{\cos\left(\frac{11\pi}{2} - \alpha\right)\sin\left(\frac{9\pi}{2} + \alpha\right)}$.
|
\frac{3}{4}
| 35.15625 |
11,429 |
Given a positive integer \( n \) such that \( n \leq 2016 \) and \(\left\{\frac{n}{2}\right\}+\left\{\frac{n}{4}\right\}+\left\{\frac{n}{6}\right\}+\left\{\frac{n}{12}\right\}=3\), where \(\{x\} = x - \lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \), find the number of such integers \( n \).
|
168
| 73.4375 |
11,430 |
Group the set of positive odd numbers $\{1, 3, 5, \cdots\}$ in increasing order such that the $n$-th group has $(2n-1)$ odd numbers:
\[
\{1\}, \quad \{3, 5, 7\}, \quad \{9, 11, 13, 15, 17\}, \cdots
\]
(first group)(second group)(third group)
Determine which group 1991 belongs to.
|
32
| 53.125 |
11,431 |
Let \( p, q, r, s, t, u \) be positive real numbers such that \( p + q + r + s + t + u = 10 \). Find the minimum value of
\[ \frac{1}{p} + \frac{9}{q} + \frac{4}{r} + \frac{16}{s} + \frac{25}{t} + \frac{36}{u}. \]
|
44.1
| 14.84375 |
11,432 |
Name the number that consists of 11 hundreds, 11 tens, and 11 units.
|
1221
| 94.53125 |
11,433 |
In $\triangle ABC$, it is given that $\cos A= \frac{5}{13}$, $\tan \frac{B}{2}+\cot \frac{B}{2}= \frac{10}{3}$, and $c=21$.
1. Find the value of $\cos (A-B)$;
2. Find the area of $\triangle ABC$.
|
126
| 60.9375 |
11,434 |
Calculate the volume of an octahedron which has an inscribed sphere of radius 1.
|
4\sqrt{3}
| 93.75 |
11,435 |
Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$.
|
27
| 10.9375 |
11,436 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $(a+b)(\sin A-\sin B)=c(\sin C-\sin B)$.
$(1)$ Find $A$.
$(2)$ If $a=4$, find the maximum value of the area $S$ of $\triangle ABC$.
|
4 \sqrt {3}
| 0 |
11,437 |
Evaluate the value of the expression \((5(5(5(5+1)+1)+1)+1)\).
|
781
| 64.0625 |
11,438 |
Given a rectangle with dimensions \(100 \times 101\), divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal.
|
200
| 66.40625 |
11,439 |
$BL$ is the angle bisector of triangle $ABC$. Find its area, given that $|AL| = 2$, $|BL| = 3\sqrt{10}$, and $|CL| = 3$.
|
\frac{15 \sqrt{15}}{4}
| 10.9375 |
11,440 |
Find the minimum value of the function \( f(x)=\cos 4x + 6\cos 3x + 17\cos 2x + 30\cos x \) for \( x \in \mathbb{R} \).
|
-18
| 7.8125 |
11,441 |
There are 3 rods with several golden disks of different sizes placed on them. Initially, 5 disks are arranged on the leftmost rod (A) in descending order of size. According to the rule that only one disk can be moved at a time and a larger disk can never be placed on top of a smaller one, the goal is to move all 5 disks to the rightmost rod (C). What is the minimum number of moves required to achieve this?
|
31
| 100 |
11,442 |
The first term of a sequence is 2, the second term is 3, and each subsequent term is formed such that each term is 1 less than the product of its two neighbors. What is the sum of the first 1095 terms of the sequence?
|
1971
| 68.75 |
11,443 |
There are 1000 lights and 1000 switches. Each switch simultaneously controls all lights whose numbers are multiples of the switch's number. Initially, all lights are on. Now, if switches numbered 2, 3, and 5 are pulled, how many lights will remain on?
|
499
| 81.25 |
11,444 |
Given \( S = x^{2} + y^{2} - 2(x + y) \), where \( x \) and \( y \) satisfy \( \log_{2} x + \log_{2} y = 1 \), find the minimum value of \( S \).
|
4 - 4\sqrt{2}
| 10.9375 |
11,445 |
Chinese mathematician Hua Luogeng saw a brain teaser in a magazine that the passenger next to him was reading while on a trip abroad: find the cube root of $59319$. Hua Luogeng blurted out the answer, astonishing everyone. They quickly asked about the calculation's mystery. Do you know how he calculated the result quickly and accurately? Below is Xiaochao's exploration process, please complete it:
$(1)$ Find $\sqrt[3]{59319}$.
① From $10^{3}=1000$ and $100^{3}=1000000$, we can determine that $\sqrt[3]{59319}$ has ____ digits;
② Since the units digit of $59319$ is $9$, we can determine that the units digit of $\sqrt[3]{59319}$ is ____;
③ If we subtract the last three digits $319$ from $59319$ to get $59$, and $3^{3}=27$, $4^{3}=64$, we can determine that the tens digit of $\sqrt[3]{59319}$ is ____, thus obtaining $\sqrt[3]{59319}=\_\_\_\_\_\_$.
$(2)$ Given that $195112$ is also a cube of an integer, using a similar method, we can obtain $\sqrt[3]{195112}=\_\_\_\_\_\_$.
|
58
| 89.0625 |
11,446 |
What is the largest $2$-digit prime factor of the integer $n = {300\choose 150}$?
|
97
| 9.375 |
11,447 |
If \(a^{2} - 1 = 123 \times 125\) and \(a > 0\), find \(a\).
If the remainder of \(x^{3} - 16x^{2} - 9x + a\) when divided by \(x - 2\) is \(b\), find \(b\).
If an \(n\)-sided polygon has \((b + 4)\) diagonals, find \(n\).
If the points \((3, n), (5,1)\), and \((7, d)\) are collinear, find \(d\).
|
-10
| 34.375 |
11,448 |
Suppose the domain of function $y=f(x)$ is $D$. If for any $x_{1}, x_{2} \in D$, when $x_{1} + x_{2} = 2a$, it always holds that $f(x_{1}) + f(x_{2}) = 2b$, then the point $(a,b)$ is called the center of symmetry of the graph of the function $y=f(x)$. Investigate a center of symmetry for the function $f(x) = 2x + 3\cos\left(\frac{\pi}{2}x\right) - 3$ and use the definition of the center of symmetry to find the value of $f\left(\frac{1}{2018}\right) + f\left(\frac{2}{2018}\right) + \ldots + f\left(\frac{4034}{2018}\right) + f\left(\frac{4035}{2018}\right)$.
|
-4035
| 80.46875 |
11,449 |
The cost of four pencils and one pen is $\$2.60$, and the cost of one pencil and three pens is $\$2.15$. Find the cost of three pencils and two pens.
|
2.63
| 68.75 |
11,450 |
The digits of a certain three-digit number form a geometric progression. If the digits of the hundreds and units places are swapped, the new three-digit number will be 594 less than the original number. If, in the original number, the hundreds digit is removed and the remaining two-digit number has its digits swapped, the resulting two-digit number will be 18 less than the number formed by the last two digits of the original number. Find the original number.
|
842
| 69.53125 |
11,451 |
Using the digits 1, 2, 3, 4, 5, how many even four-digit numbers less than 4000 can be formed if each digit can be used more than once?
|
150
| 100 |
11,452 |
Evaluate the polynomial \[ p(x) = x^4 - 3x^3 - 9x^2 + 27x - 8, \] where $x$ is a positive number such that $x^2 - 3x - 9 = 0$.
|
\frac{65 + 81\sqrt{5}}{2}
| 25.78125 |
11,453 |
A list of $2023$ positive integers has a unique mode, which occurs exactly $11$ times. Determine the least number of distinct values that can occur in the list.
|
203
| 58.59375 |
11,454 |
Three positive reals \( x \), \( y \), and \( z \) are such that
\[
\begin{array}{l}
x^{2}+2(y-1)(z-1)=85 \\
y^{2}+2(z-1)(x-1)=84 \\
z^{2}+2(x-1)(y-1)=89
\end{array}
\]
Compute \( x + y + z \).
|
18
| 41.40625 |
11,455 |
Appending three digits at the end of 2007, one obtains an integer \(N\) of seven digits. In order to get \(N\) to be the minimal number which is divisible by 3, 5, and 7 simultaneously, what are the three digits that one would append?
|
075
| 18.75 |
11,456 |
Yan is at a point between his house and a park. He has two options to reach the park: He can either walk directly to the park or he can walk back to his house and use his scooter to reach the park. He scoots 10 times faster than he walks, and both routes take the same amount of time. Determine the ratio of Yan's distance from his house to his distance to the park.
|
\frac{9}{11}
| 68.75 |
11,457 |
Two balls are randomly chosen from a box containing 20 balls numbered from 1 to 20. Calculate the probability that the sum of the numbers on the two balls is divisible by 3.
|
\frac{32}{95}
| 48.4375 |
11,458 |
Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards?
|
1680
| 93.75 |
11,459 |
Let \( a, b, c \) be the roots of the cubic equation
\[
x^3 + 3x^2 + 5x + 7 = 0
\]
The cubic polynomial \( P \) satisfies the following conditions:
\[
\begin{array}{l}
P(a) = b + c, \quad P(b) = a + c, \\
P(c) = a + b, \quad P(a + b + c) = -16.
\end{array}
\]
Determine the value of \( P(0) \).
|
11
| 54.6875 |
11,460 |
In a clock workshop, there are several digital clocks (more than one), displaying time in a 12-hour format (the number of hours on the clock screen ranges from 1 to 12). All clocks run at the same speed but show completely different times: the number of hours on the screen of any two different clocks is different, and the number of minutes as well.
One day, the master added up the number of hours on the screens of all available clocks, then added up the number of minutes on the screens of all available clocks, and remembered the two resulting numbers. After some time, he did the same thing again and found that both the total number of hours and the total number of minutes had decreased by 1. What is the maximum number of digital clocks that could be in the workshop?
|
11
| 17.1875 |
11,461 |
Find the least positive integer \( x \) that satisfies both \( x + 7219 \equiv 5305 \pmod{17} \) and \( x \equiv 4 \pmod{7} \).
|
109
| 55.46875 |
11,462 |
There are 7 volunteers, among which 3 people only speak Russian, and 4 people speak both Russian and English. From these, 4 people are to be selected to serve as translators for the opening ceremony of the "Belt and Road" summit, with 2 people serving as English translators and 2 people serving as Russian translators. There are a total of \_\_\_\_\_\_ different ways to select them.
|
60
| 12.5 |
11,463 |
Find the maximum of $x^{2} y^{2} z$ under the condition that $x, y, z \geq 0$ and $2 x + 3 x y^{2} + 2 z = 36$.
|
144
| 35.15625 |
11,464 |
The military kitchen needs 1000 jin of rice and 200 jin of millet for dinner. Upon arriving at the rice store, the quartermaster finds a promotion: "Rice is 1 yuan per jin, with 1 jin of millet given for every 10 jin purchased (fractions of 10 jins do not count); Millet is 2 yuan per jin, with 2 jins of rice given for every 5 jin purchased (fractions of 5 jins do not count)." How much money does the quartermaster need to spend to buy enough rice and millet for dinner?
|
1200
| 19.53125 |
11,465 |
A triangle has two medians of lengths 9 and 12. Find the largest possible area of the triangle. (Note: A median is a line segment joining a vertex of the triangle to the midpoint of the opposite side.)
|
72
| 28.125 |
11,466 |
On a plane with 100 seats, there are 100 passengers, each with an assigned seat. The first passenger ignores the assigned seat and randomly sits in one of the 100 seats. After that, each subsequent passenger either sits in their assigned seat if it is available or chooses a random seat if their assigned seat is taken. What is the probability that the 100th passenger ends up sitting in their assigned seat?
|
\frac{1}{2}
| 92.96875 |
11,467 |
At 8:00 AM, Xiao Cheng and Xiao Chen set off from locations A and B respectively, heading towards each other. They meet on the way at 9:40 AM. Xiao Cheng says: "If I had walked 10 km more per hour, we would have met 10 minutes earlier." Xiao Chen says: "If I had set off half an hour earlier, we would have met 20 minutes earlier." If both of their statements are correct, how far apart are locations A and B? (Answer in kilometers).
|
150
| 32.8125 |
11,468 |
Find the maximum value of \( x + y \), given that \( x^2 + y^2 - 3y - 1 = 0 \).
|
\frac{\sqrt{26}+3}{2}
| 19.53125 |
11,469 |
From the 10 numbers $0, 1, 2, \cdots, 9$, select 3 such that their sum is an even number not less than 10. How many different ways are there to make such a selection?
|
51
| 92.96875 |
11,470 |
During the 2013 National Day, a city organized a large-scale group calisthenics performance involving 2013 participants, all of whom were students from the third, fourth, and fifth grades. The students wore entirely red, white, or blue sports uniforms. It was known that the fourth grade had 600 students, the fifth grade had 800 students, and there were a total of 800 students wearing white sports uniforms across all three grades. There were 200 students each wearing red or blue sports uniforms in the third grade, red sports uniforms in the fourth grade, and white sports uniforms in the fifth grade. How many students in the fourth grade wore blue sports uniforms?
|
213
| 9.375 |
11,471 |
If the acute angle \(\alpha\) satisfies \(\frac{1}{\sqrt{\tan \frac{\alpha}{2}}}=\sqrt{2 \sqrt{3}} \sqrt{\tan 10^{\circ}}+\sqrt{\tan \frac{\alpha}{2}}\), then the measure of the angle \(\alpha\) in degrees is \(\qquad\)
|
50
| 64.84375 |
11,472 |
The instructor of a summer math camp brought several shirts, several pairs of trousers, several pairs of shoes, and two jackets for the entire summer. In each lesson, he wore trousers, a shirt, and shoes, and he wore a jacket only on some lessons. On any two lessons, at least one piece of his clothing or shoes was different. It is known that if he had brought one more shirt, he could have conducted 18 more lessons; if he had brought one more pair of trousers, he could have conducted 63 more lessons; if he had brought one more pair of shoes, he could have conducted 42 more lessons. What is the maximum number of lessons he could conduct under these conditions?
|
126
| 53.90625 |
11,473 |
Given that $a, b \in R^{+}$ and $a + b = 1$, find the supremum of $- \frac{1}{2a} - \frac{2}{b}$.
|
-\frac{9}{2}
| 68.75 |
11,474 |
The number of integer points inside the triangle $OAB$ (where $O$ is the origin) formed by the line $y=2x$, the line $x=100$, and the x-axis is $\qquad$.
|
9801
| 61.71875 |
11,475 |
In $\triangle ABC, AB = 10, BC = 9, CA = 8$ and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$.
|
16
| 71.09375 |
11,476 |
There are 1235 numbers written on a board. One of them appears more frequently than the others - 10 times. What is the smallest possible number of different numbers that can be written on the board?
|
138
| 41.40625 |
11,477 |
In a certain place, four people, $A$, $B$, $C$, and $D$, successively contracted the novel coronavirus, with only $A$ having visited an epidemic area.
1. If the probabilities of $B$, $C$, and $D$ being infected by $A$ are $\frac{1}{2}$ each, what is the probability that exactly one of $B$, $C$, and $D$ is infected with the novel coronavirus?
2. If $B$ is definitely infected by $A$, for $C$, since it is difficult to determine whether they were infected by $A$ or $B$, assume that the probabilities of $C$ being infected by $A$ and $B$ are both $\frac{1}{2}$. Similarly, assume that the probabilities of $D$ being infected by $A$, $B$, and $C$ are all $\frac{1}{3}$. Under these assumptions, the number of people directly infected by $A$ among $B$, $C$, and $D$, denoted as random variable $X$, is to be determined. Find the probability distribution and the mean (expected value) of the random variable $X$.
|
\frac{11}{6}
| 13.28125 |
11,478 |
Determine the number of ways to select 4 representatives from a group of 5 male students and 4 female students to participate in an activity, ensuring that there are at least two males and at least one female among the representatives.
|
100
| 98.4375 |
11,479 |
In the final of the giraffe beauty contest, two giraffes, Tall and Spotted, reached the finals. There are 135 voters divided into 5 districts, with each district divided into 9 precincts, and each precinct having 3 voters. The voters in each precinct choose the winner by majority vote; in a district, the giraffe that wins in the majority of precincts wins the district; finally, the giraffe that wins in the majority of the districts is declared the winner of the final. The giraffe Tall won. What is the minimum number of voters who could have voted for Tall?
|
30
| 59.375 |
11,480 |
Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\{a_n\}$. Given that $a_3 = 8a_6$, find the value of $\frac{S_4}{S_2}$.
|
\frac{5}{4}
| 89.0625 |
11,481 |
There are two hourglasses - one for 7 minutes and one for 11 minutes. An egg needs to be boiled for 15 minutes. How can you measure this amount of time using the hourglasses?
|
15
| 56.25 |
11,482 |
How many unique five-digit numbers greater than 20000, using the digits 1, 2, 3, 4, and 5 without repetition, can be formed such that the hundreds place is not the digit 3?
|
78
| 34.375 |
11,483 |
Consider the set $\{2, 7, 12, 17, 22, 27, 32\}$. Calculate the number of different integers that can be expressed as the sum of three distinct members of this set.
|
13
| 75.78125 |
11,484 |
Let the set \( P = \{1, 2, \ldots, 2014\} \) and \( A \subseteq P \). If the difference between any two numbers in the set \( A \) is not a multiple of 99, and the sum of any two numbers in the set \( A \) is also not a multiple of 99, then the set \( A \) can contain at most how many elements?
|
50
| 39.84375 |
11,485 |
Snow White entered a room with a round table surrounded by 30 chairs. Some of the chairs were occupied by dwarfs. It turned out that Snow White couldn't sit in a way such that no one was sitting next to her. What is the minimum number of dwarfs that could have been at the table? Explain how the dwarfs must have been seated.
|
10
| 42.96875 |
11,486 |
In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $a=2$ and $\cos C=-\frac{1}{4}$.
1. If $b=3$, find the value of $c$.
2. If $c=2\sqrt{6}$, find the value of $\sin B$.
|
\frac{\sqrt{10}}{4}
| 50 |
11,487 |
Divide a square into 25 smaller squares, where 24 of these smaller squares are unit squares, and the remaining piece can also be divided into squares with a side length of 1. Find the area of the original square.
|
25
| 77.34375 |
11,488 |
In each cell of a $15 \times 15$ table, the number $-1, 0,$ or $+1$ is written such that the sum of the numbers in any row is nonpositive and the sum of the numbers in any column is nonnegative. What is the minimum number of zeros that can be written in the cells of the table?
|
15
| 48.4375 |
11,489 |
Evaluate the expression $\frac{3^2 + 3^0 + 3^{-1} + 3^{-2}}{3^{-1} + 3^{-2} + 3^{-3} + 3^{-4}}$.
|
\frac{3807}{180}
| 0 |
11,490 |
Given real numbers \( x, y \in (1,+\infty) \) such that \( xy - 2x - y + 1 = 0 \), find the minimum value of \( \frac{3}{2} x^{2} + y^{2} \).
|
15
| 53.90625 |
11,491 |
A certain model of hybrid car travels from point $A$ to point $B$ with a fuel cost of $76$ yuan, and with an electricity cost of $26 yuan. It is known that for every kilometer traveled, the fuel cost is $0.5$ yuan more than the electricity cost.
$(1)$ Find the cost of traveling one kilometer using electricity only.
$(2)$ If the total cost of fuel and electricity for a hybrid trip from point $A$ to point $B$ does not exceed $39$ yuan, how many kilometers at least must be traveled using electricity?
|
74
| 48.4375 |
11,492 |
In a shooting competition, nine clay targets are set up in three hanging columns. The first column contains four targets, the second column contains three targets, and the third column contains two targets. A sharpshooter must break all the targets under the following conditions:
1) The sharpshooter first selects any one of the columns from which a target is to be broken.
2) The sharpshooter must then break the lowest remaining target in the chosen column.
What are the different possible orders in which the sharpshooter can break all nine targets?
|
1260
| 93.75 |
11,493 |
In $\triangle ABC$, given that $\sin A = 10 \sin B \sin C$ and $\cos A = 10 \cos B \cos C$, what is the value of $\tan A$?
|
11
| 22.65625 |
11,494 |
In how many ways can 8 people be seated in a row of chairs if two of the people, Alice and Bob, must not sit next to each other, and Charlie has to sit at one end of the row?
|
7200
| 50 |
11,495 |
30 students from five courses created 40 problems for the olympiad, with students from the same course creating the same number of problems, and students from different courses creating different numbers of problems. How many students created exactly one problem?
|
26
| 4.6875 |
11,496 |
Squirrels $A$, $B$, and $C$ have several pine cones in total. Initially, squirrel $A$ has 26 pine cones, and it takes 10 pine cones to evenly divide between $B$ and $C$. Then, squirrel $B$ takes 18 pine cones and evenly divides them between $A$ and $C$. Finally, squirrel $C$ divides half of its current pine cones evenly between $A$ and $B$. At this point, all three squirrels have the same number of pine cones. How many pine cones did squirrel $C$ originally have?
|
86
| 20.3125 |
11,497 |
Calculate: $-{1^{2022}}+{({\frac{1}{3}})^{-2}}+|{\sqrt{3}-2}|$.
|
10-\sqrt{3}
| 88.28125 |
11,498 |
If \( y+4=(x-2)^{2} \) and \( x+4=(y-2)^{2} \), and \( x \neq y \), then the value of \( x^{2}+y^{2} \) is:
|
15
| 12.5 |
11,499 |
Given the function $f(x)= e^{-2x}+1$, find the area of the triangle formed by the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$ and the lines $y=0$ and $y=x$.
|
\frac{1}{3}
| 17.1875 |
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