Problem 1

Problem Description:

A group of students goes on a city tour with 6 landmarks: A, B, C, D, E, and F. Each student explores independently for a period and then observes which landmarks they cannot see. Each student records which landmarks are visible (A, B, C, D) and which ones are not (E, F). If the students notice that their sets of visible and invisible landmarks have no common landmarks, then the set is considered valid. The students can identify which landmarks are visible in pairs but cannot identify individual landmarks.

The question is to find the maximum number of students in the group that can have unique and valid sets of visible/invisible landmarks.

Options:

1. 3
2. 4
3. 6
4. 12

Solution:

For each landmark, a student can either see it or not see it. Therefore, each student has a unique combination of visible/invisible landmarks. Since there are 6 landmarks, each landmark has 2 possible states (visible or invisible).

The total number of unique combinations of visible/invisible landmarks for 6 landmarks is $2^6 = 64$.

However, the problem states that the combination of visible/invisible landmarks should not have any common landmarks. This significantly reduces the number of possible valid combinations.

Considering pairs (A, B, C, D) are visible, and (E, F) are not, we can see that:

For any specific combination of 4 landmarks being visible, there are $\binom{6}{4} = 15$ ways to choose 4 landmarks from 6, where the remaining 2 will automatically be invisible.

Therefore, the total number of unique and valid sets is 15.

Since the maximum number of unique valid combinations is 15, the maximum number of students in the group that can have unique and valid sets is 15.

Therefore, the answer is:

Option 4: 12

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Problem 2

Problem Description:

A student (Xiao Ming) plays a game with an initial score of 2 points. During the game, the score increases continuously (with a rate of 1 unit per game time unit). The game starts and at each random game time unit, there is a probability (relative to the time unit) for the game to end. If the game ends and Xiao Ming hits a specific game unit, he can earn an additional score. The game has multiple levels of difficulty, and if the game ends and Xiao Ming fails to hit the target, the game ends without earning additional points.

The probability that Xiao Ming hits the target decreases exponentially with each subsequent attempt.

Questions:

1. If Xiao Ming fails on his first attempt, what is the probability that he succeeds on the second attempt?
2. If Xiao Ming's total accumulated score increases to a maximum, at what point should he continue to play the game?

Options:

1. 1
2. 2
3. 3
4. 4

Solution:

The detailed solution to this problem requires understanding the exponential decay function for the probability of hitting the target. For a complete mathematical solution, the exponential decay function would be formulated, and the optimal strategy for maximizing the score would be identified.

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Would you like the detailed solution for Problem 2 or any other specific assistance?

Suggested Questions:

1. Can you solve a system of linear equations?
2. What are the steps to solve a quadratic equation?
3. How do you find the derivative of a function?
4. Can you explain the concept of integration?
5. What are the properties of logarithms?

Tip:

Always double-check your calculations for accuracy, especially when dealing with exponential functions or probability.