2024-241

Consider a particle moving in a one dimensional space. Its acceleration is proportional to the square of its velocity. Also, at \( t=0 \), it has a velocity 2 m/s and at \( t=1 \), it runs at 1 m/s. Consider that the particle starts at \( x=0 \) m.

With that in mind, choose the alternative that correctly describes the particle's position, \( x(t) \), with respect to time.

1. \( x(t) = (\ln(1 + t))/2 \)
2. \( x(t) = 1 + \ln(2 + t^2) \)
3. \( x(t) = 2 + \ln(1 + t^2) \)
4. \( x(t) = 2 \ln( 1 + t ) \)
5. None of the above.

Original idea by: Vitor Antônio Pimenta Silva

2024-240

In search for fortune, a prospector named Jack had heard tales of a legendary gold mine hidden deep in the mountains. Inspired by stories of riches, he set out to explore three promising locations: Point A, Point B, and Point C. 

Using his metal detector, he discovered at point A, a signal modeled by the function \(f(t) = 3 \sin(2t) + 5t\) ; at Point B, \(g(t) = 2 \sin(3t) + 4t\) ; and at Point C, the function was \(h(t) = 4 \sin(t) + 2t\).

Jack knew that the greater the difference between the maximum value of the derivative of the signal and a threshold of 8, the more likely he was to stumble upon gold. Select the correct alternative.

a) He is more likely to find gold at Point A.

b) He is more likely to find gold at Point B.

c) He is more likely to find gold at Point C.

d) He is equally likely to find gold at Points A, B, and C.

e) None of the above.

Original idea by: João Vitor Baptista Moreira

2024-239

In a scale-free network with 1296 nodes, the maximum degree is 144, and the degree exponent is \( \gamma = 3 \). What is the minimum degree in the network?

A) 10

B) 12

C) 8

D) 3

E) None of the above.

Original idea by: Caroline Nakazato.

 

2024-238

Consider the following statements:

I. Given a tree T, it is possible to find the diameter of T using two BFS executions.

II. If an undirected graph is bipartite, it can be detected using BFS.

III. BFS was run on a connected simple graph, starting from a node r. Let u and v be two nodes in the graph. If the distance from r to u is 2, and the distance from r to v is 1, then the distance between u and v in the graph must necessarily be 1.

Which of the statements are correct?

a) Only I

b) Only II

c) I and II

d) II and III

e) None of the above

Original idea by: Sergio Sanchez

2024-237

John is a tourist from Minas Gerais who is on vacation and wants to explore the cities in UaiSo county. There are 3 main cities that he would like to visit: CheeseBreadLand, DulceDeLecheTown and TorresmoCity, that are painted in blue in the map. But John only moves following a BFS order, and he has time to visit 5 cities only. Help John optimize his trip so he can visit the most possible interesting cities. Consider that the BFS always chooses the neighbors in ascending order and the following map is an undirected graph. 

 

To visit the most cities he would like, what should John do?

a) John should start his trip on city 1, so he can visit 2 cities of interest.

b) John should start his trip on cities 4 or 9, so he can visit 2 cities of interest.

c) John should start his trip on city 3, so he can visit 3 cities of interest.

d) John should start his trip on city 2, so he can visit 2 cities of interest.

e) None of the above

Original idea by: João Vitor Baptista Moreira