2026-336

A research data center needs to transfer data to a remote backup facility during a 4‑hour maintenance window. The data transfer process is modeled as a directed network, where nodes represent logical components (functional stages) of the data transfer pipeline, and edges represent logical communication channels, with limited capacity.

In this model, node \(A\) represents the point where data is generated in the primary data center. The intermediate nodes (\(B, C, D\)) represent functional stages of the transfer process, such as internal processing, aggregation, or interfaces to external networks. And node \(E\) represents the logical destination where data is finally stored at the backup site.

All edges have constant transmission capacity, except for one critical link, whose capacity varies over time due to shared usage with other services. The network links are as follows, with capacities measured in GB/hour:

\( A \rightarrow B : 8 \)

\( B \rightarrow C : 4 + \pi \sin(\pi t/4) \)

\( C \rightarrow E : 9 \)

\( A \rightarrow D : 3 \)

\( D \rightarrow E : 6 \)

where \(t\) is the time in hours since the start of the maintenance window.

Which of the following alternatives represents approximately the maximum total amount of data (in GB) that can be transferred from \(A\) to \(E\) during the maintenance window?

a) 40

b) 36

c) 28

d) 18

e) None

 

Original idea by: Ingrid Barbosa

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

2023-211

What is the volume generated by rotating the region bounded by the curves \( y=x^2 \) and \( y=4 \) around the \(x\)-axis?

1. 8π
2. 4π
3. 2π
4. π
5. None of the above

Original idea by: João Marcos

2023-210

Consider a random network with constant probability of connexion \( p = 0.1 \) at all times and a number of nodes that increases over time according to \( N(t) = 10t \). What would it be the function that estimates the variation of average links over time \( d \langle L \rangle/dt \)?

1. \( 10t - 1 \)
2. \( 0.1 * [10t - 1] \)
3. \( 10t - 1/2 \)
4. \( 0.1 * [10t - 1/2] \)
5. None of the above

Original idea by: Germán Darío Buitrago Salazar

2023-209

Consider a scale-free network with a fixed \( \gamma = 2.5 \) at all times and a number of nodes that increases over time according to the relation \( N(t) = e^{10^6t} \). Which alternative correctly estimates and interprets the variation of the average distance over time (\( d\langle d \rangle/dt \)):

1. \( e^{10^6t} \), as the network grows rapidly, \( d\langle d \rangle/dt \) also increases rapidly over time, causing the network to lose its ultra-small-world property.
2. \( 1/(te^{10^6} + \ln t) \), as the network grows rapidly, \( d\langle d \rangle/dt \) gets too small, a consequence of the ultra-small-world property of scale-free networks in this regime.
3. \( e^{10^6t}/(10^6t) \), as the network grows rapidly, \( d\langle d \rangle/dt \) also increases rapidly over time, but not too fast, which may cause the loss of its ultra-small-world property in some cases.
4. \( 1/(t(e^{10^6} + \ln t)) \), as the network grows rapidly, \( d\langle d \rangle/dt \) gets too small, a consequence of the ultra-small-world property of scale-free networks in this regime.
5. None of above

Original idea by: Anderson Nogueira Cotrim

2022-169

A critical point for a function \( f \) is a value \( c \) in the domain of \( f \) where \( f'(c) = 0 \) or \( f'(c) \) does not exist.  Choose the statement that presents the critical points of $$ f(x) = x^{3/5}(4 - x) $$

1. 0 and 2/3
2. 2/3 and 4/3
3. 4/3 and 3/2
4. 0 and 3/2
5. None of the above.

Original idea by: Rubens de Castro Pereira

2022-166

The function \( f(x) = x^3 - 6x^2 + 9x - 2 \) has a local maximum value at the point:

1. (0, 2)
2. (1, 2)
3. (2, 0)
4. (2, 1)
5. None of the above

Original idea by: Matheus Fernandes Sarmento

2022-165

Suppose you are analyzing a scale-free network with currently 10000 nodes, degree exponent \( \gamma = 3.0 \), minimum degree of 10 and maximum degree of 1000. However, you know this network is growing over time at a rate of 1000 nodes/year. Assuming that the minimum degree is growing at 10 nodes/year while the degree exponent remains constant, what is currently the rate of change for the largest hub in this network?

(A) 5 nodes/year

(B) 55 nodes/year

(C) 550 nodes/year

(D) 5500 nodes/year

(E) None of the above.

Original idea by: Luan Silveira

2022-164

Consider functions \( f \) and \( g \) satisfying:

\( f'' = x^{-1/3} \)
\( g = 4 x^5 + 2 x^3 - 3 x + 5 \)

Which option below is compatible with the preceding equations?

1. \( f''' = -(1/3)x^{-(3/4)} \) and \( g' = 20 x^4 + 6 x - 3 \)
2. \( f''' = -(1/3)x^{-(3/2)} \) and \( g' = 20 x^4 + 6 x^2 - 3 \)
3. \( f''' = (3/2)x^{2/3} \) and \( g' = 4 x^4 + 2 x^2 - 3 \)
4. \( f''' = (3/2)x^{2/3} \) and \( g' = 20 x^4 + 6 x^2 - x \)
5. None of the above

Original idea by: Fábio Assunção.

2022-104

Consider the following integral:

$$ \int \cos^2 x dx $$

Which of the alternatives has a correct intermediate step and the solution, using Euler's formulas below, where \( i \) represents the imaginary unit?

$$ \cos \theta = \frac{e^{\theta i} + e^{-\theta i}}{2} $$ $$ \sin \theta = \frac{e^{\theta i} - e^{-\theta i}}{2i} $$

1. $$ \frac{1}{2} \int (e^{2ix} + 2 - e^{-2ix}) dx = \frac{1}{2}(\frac{e^{2ix}}{2i} + 2x - \frac{e^{-2ix}}{2i}) + c $$
2. $$ \frac{1}{4} \int (e^{2ix} + 2 + e^{-2ix}) dx = \frac{1}{4}(\frac{e^{2ix}}{2i} + 2x - \frac{e^{-2ix}}{2i}) + c $$
3. $$ \frac{1}{4} \int (e^{2ix} + e^{-2ix}) dx = \frac{1}{4}(\frac{e^{2ix}}{2i} + \frac{e^{-2ix}}{2i}) + c $$
4. $$ \frac{1}{4} \int (e^{2x} + 2 + e^{-2x}) dx = \frac{1}{4}(\frac{e^{2x}}{2} + 2x + \frac{e^{-2x}}{2}) + c $$
5. None of the above
   

Original idea by: Diogo Souza

2022-090

What is the value of the following definite integral?

$$ \int_0^{10} e^x - 1 dx $$

1. \( x e^{x - 1} \)
2. 10
3. \( e^{10} - 11 \)
4. \( 1/e^{10} \)
5. None of the above

Original idea by: Filipe Maciel

2022-089

Select the false statement regarding derivatives:

1. The derivative at a point of the function y=f(x) represents the instantaneous rate of change of y with respect to x at that point
2. If \( f'(x) = 0 \), the point \( x \) can be a local maximum point of the function \( f \)
3. If \( f'(x) = 0 \), the point \( x \) can be a local minimum point of the function \( f \)
4. Derivatives determine the area under the function graph in the cartesian plane
5. None of the above

Original idea by: Victória Pedrazzoli

2022-088

Consider that you work at a company that produces wooden boxes. A client requests a batch of 1000 boxes. However, he has some requirements:

• All boxes must have a height of 1 meter.
• All boxes must have a volume of 10000 liters.

Using the information above choose the best base dimensions a and b of the boxes to spend the least possible amount of money on wood (round to 3 decimal places):

1. a = 3.713m, b = 3.162m
2. a = 3.713m, b = 3.713m
3. a = 3.162m, b = 3.713m
4. a = 3.162m, b = 3.162m
5. None of the above

Original idea by: Felipe Crispim da Rocha Salvagnini

2022-086

Pink Dolphins Extinction Calculation

The pink dolphin is the largest freshwater dolphin in the world. It is found in northern South America, in the basins of the Amazon and Orinoco rivers. Even with the efforts of environmentalists and researchers, it is at risk of extinction.

The function below gives the number of pink dolphins after \( t \) years since the species began to be monitored.

After 5 years, if nothing is done, what is the instantaneous rate of change of pink dolphins (round to nearest integer)?

1. -209 pink dolphins per year
2. 209 pink dolphins per year
3. -674 pink dolphins per year
4. 674 pink dolphins per year
5. None of the above

Original idea by: Márcia Jacobina

2021-013

Choose the option that contains the derivative of \( f(x) = g(x) h(x) \), where \( g(x) = x^3 \) and \( h(x) = (1 + \ln x)^2 \).

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

Original idea by: Maria Tejada Begazo

2021-012

Find the derivative of the following function:

$$ f(x) = 2\sqrt{x} - 6\sqrt[3]{x^2} + 5x^2 - 20 $$

1. \( x^{1/2} - 6x^{1/3} + 10x \)
2. \( x^{1/2} - 4x^{1/3} + 10x \)
3. \( x^{-1/2} - 4x^{-1/3} + 10x \)
4. \( x^{-1/2} - 6x^{-1/3} + 10x \)
5. none of the above

Original idea by: Angelica Oliveira

2021-010

A crack appeared in a container and the liquid in it began to leak. At instant \( t \ge 0 \), a total of \( 10t - \sqrt{t} \) units of the liquid will have leaked. What is the flow rate \( r \) of the liquid through the crack at instant \( t = 100 \) ?

1. \( r = 99/2 \)
2. \( r = 99/20 \)
3. \( r = 199/2 \)
4. \( r = 199/20 \)
5. None of the above

Original idea by: Bruno Almêda

2021-009

Find the derivative of $$ f(t) = \frac{1}{t} − \frac{1}{2t^3} + \frac{1}{2t^5} $$.

1. \( f'(t) = \dfrac{1}{t} − \dfrac{1}{2t^3} + \dfrac{1}{2t^5} \)
2. \( f'(t) = {t^2} − {2t^4} + \dfrac{1}{2t^5} \)
3. \( f'(t) = -\dfrac{1}{t^2} + \dfrac{1}{2t^4} + \dfrac{t^5}{2} \)
4. \( f'(t) = {t} - {2t^3} + {2t^5} \)
5. None of the above

Original idea by: André Regino

2021-008

The space exploration agency AES is testing a new type of submarine, called XY, which can fly out from under the ocean and reach space. In its first test, there were failures that caused the first XY prototype to have a total loss of its engines, stopping its ascent at a certain time and starting to fall afterwards. The trajectory of the prototype in the time interval \( 0 \leq t \leq 20 \) is given by \( f(t) = 56 + 5t - (t-8)^{2} \).

What is the exact time \( t \) when the XY prototype stopped going up towards space?

1. 20/2.
2. 21/2.
3. 2/21.
4. -21/2.
5. None of the above.

Original idea by: Adolfo Schneider

2021-007

Choose the incorrect alternative below:

1. If \( f(x) = 2x^{−2} \), then \( f'(x) = −4x^{−1} \)
2. If \( f(x) = xe^x \), then \( f'(x) = (x+1)e^x \)
3. If \( f(x) = \sqrt{x^2+1} \), then \( f'(x) = x / \sqrt{x^2+1} \)
4. If \( f(x) = x^{−1} \), then \( \int 𝑓 = \ln x + C \)
   
5. none of the above

Original idea by: Mauricio Schiezaro