# Brevet des Collèges 2017 North America June 7, 2017

North American math patent 2017 subject and answer key.

Exercise 1. 4.5 points
Copy the correct answer (no justification is expected).

Exercise 2. 9.5 points

With a geometry software, we execute the program below.

Construction Program:
– Construct a square ABCD;
– Draw the circle with center A and radius [AC];
– Place the point E at the intersection of the circle and the
half line [AB) ;
– Construct a DEFG square.

1. On the copy, make the construction with AB = 3 cm.

2. In this question, AB = 10 cm.
2. a. Show that AC =p200 cm.
2. b. Explain why AE =p200 cm.
2. c. Show that the area of square DEFG is three times the area of square ABCD.
3. For this question, we assume that for any length of the side [AB], the area of the squareDEFG is always three times
of the area of the square ABCD.
By executing this construction program, we wish to obtain a DEFG square with an area of 48 cm2.
What length AB should be chosen at the beginning?

Exercise 3. 6 points
There are 12 balls in an urn, indistinguishable by touch, numbered from 1 to 12. We want to draw a ball at random.
1. Is it more likely to get an even number or a multiple of 3?
2. What is the probability of getting a number less than 20?
3. Remove from the urn all the balls whose number is a divisor of 6. We want to draw a ball at random again.
Explain why the probability of getting a number that is a prime number is then 0.375.

Exercise 4. 10 points
The data and questions in this exercise are for metropolitan France.

Part 1:

1. Determine an estimate of the number of people, to the nearest 100,000, who suffered from food allergies in France
in 2010.
2. Is it true that in 2015 there were about 6 times as many people affected as in 1970?

Part 2:
In 2015, in a middle school with 681 students, 32 students had food allergies.
The following table shows the types of foods they responded to.

1. Is the proportion of students in this school with food allergies higher than that of the French population?

2. Jawad is surprised: “I added up all the numbers in the table and I got 39 instead of 32.
Explain this difference.
3. Lucas andMargot each started a chart to represent the allergies of the 32 students in their middle school:

3. a. Who of Lucas or Margot made the best choice for the situation? Justify the answer.

3. b. Reproduce and complete the diagram chosen in question a.

Exercise 5. 5 points
The image below shows the position obtained when the start block of a game program is triggered.

The background consists of dots spaced 40 units apart.

In this position, the cat has the coordinates (-120; -80).
The goal of the game is to position the cat on the ball.

1. What are the coordinates of the center of the ball shown in this position?
2. In this question, the cat is in the position obtained at the start of the start block.
Here is the script of the “cat” sprite that moves.

3. What happens when the cat reaches the ball?

Exercise 6. 10 points

The diagram below represents Leïla’s garden. It is not to scale.
[OB] and [OF] are walls, OB = 6and OF = 4m.
The dotted line BCDEF represents the fence that Leila wants to install to delimit a rectangular OECD enclosure.
She has a 50m roll of wire mesh that she wants to use entirely.

Leila considers several possibilities for placing point C.

1. Placing C so that BC = 5 m, she gets that FE = 15 m.
1. a. Check that it uses the 50m of wire mesh.
1. b. Justify that the area A of the OECD enclosure is 209 m².
2. To have a maximum area, Leïla calls on her math teacher neighbor who, a bit in a hurry, writes on a piece of paper:
“By noting BC = x, we have A(x)= -x² +18x +144 “
Check that the neighbor’s formula is consistent with the result of question 1.
3. In this section, the questions a. and b. do not require justification.
3. a. Leila entered a formula in B2 and then stretched it to cell 12.

What formula is then written in cell F2?

3. b. Which of the values in the table will Leila choose for BC to obtain an enclosure of maximum area?
3. c. Give the dimensions of the resulting enclosure.

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