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Brevet des collèges Greece

June 18, 2019

**EXERCISE 1 : 12 POINTS**

Mathilde spins two lottery wheels A and B, each with four numbered sectors

as shown in the diagram below:

The probability of obtaining each sector of a wheel is the same. The arrows indicate the two

areas obtained.

Mathilde’s experiment is as follows: she spins the two wheels to obtain a number at

two digits. The number obtained with the wheel A is the tens number and the one with the wheel B is the

number of units.

In the example above, she gets the number 27 (Wheel A: 2 and Wheel B: 7).

1. Write down all the possible numbers from this experiment.

2. Prove that the probability of obtaining a number greater than 40 is 0.25.

3. What is the probability that Mathilde gets a number divisible by 3?

**EXERCISE 2: 20 POINTS**

1. Show that the measure of the angle TSR is 60°.

2. Prove that the triangles SRT and SUP are similar

3. Determine the reduction coefficient linking the SRT and SUP triangles.

4. Calculate the length SU.

5. What is the nature of the SKL triangle? To be justified.

**EXERCISE 3 : 15 POINTS**

Marc and Jim, two running enthusiasts, are training on a track that is

of the tour measures 400 m.

Marc averages 2 minutes per lap.

Marc begins his training with a one-kilometer warm-up.

1. How long will Mark’s warm-up take?

2. What is Mark’s average running speed in km/h?

At the end of the warm-up, Marc and Jim decide to start their race from the same starting point

A and will perform a certain number of rounds.

Jim has an average time of 1 minute and 40 seconds per lap.

The diagram below represents Mark and Jim’s track and field, which consists of two segments

[AB] and [CD] and two semicircles of diameter [AD] and [BC).

*(The diagram is not to scale and the lengths shown are rounded to the nearest whole number*).

3. Calculate how long it will take for them to be together, at the same time, and to

the first time at point A.

Then determine how many laps it will take for each of them.

*All traces of research, even if not completed, must be included on the copy. It will be taken in*

*into account in the evaluation.*

**EXERCISE 4 : 16 POINTS**

To keep her little brother busy, Lucie, who likes computers, decides to make rosettes with

coloring. She decided to start with a diamond-shaped pattern.

With the help of an assisted programming software (scratch type), she represented the following pattern:

It is a rhombus whose sides are 50 pixels long and whose acute angles measure

30° and obtuse angles 150°.

In order to represent this diamond, she wrote the following program:

1. Complete the above program by replacing the dotted lines with the

correct values so that the diamond is drawn as defined.

2. Using the above rhombus, she obtains the following rosette which is not full size:

What geometric transformation, starting from the first rhombus ABCD and repeated 12 times , has been

used to obtain this figure? Define this transformation as best you can.

3. Finally, Lucie wants to complete this rose window in three different ways. For this purpose, three programs were carried out.

Copy on your copy the number of the three programs, and for each one, the letter of the figure associated with it.

**EXERCISE 5 : 15 POINTS**

The following calculation program is given:

1. Show that when we choose the number 2 at the beginning, we obtain the number 5 at the end.

2. What result do we get when we choose the number -3 at the beginning?

3. We define a function f which, to any number x chosen at the input of the program, associates the result

obtained at the end of this program.

Thus, for all x, we obtain f (x) = (x +1)² -x²

Show that f (x) = 2x +1.

4. This question is a multiple choice questionnaire (MCQ).

In each case, only one answer is correct. For each question, write on the

copy the question number and the correct answer.

No justification is required.

**EXERCISE 6 : 22 POINTS**

In the village of Jean, a flea market is organized every year during the first weekend of July.

Jean has committed to take care of the French fries stand. For this, he makes paper cones

which will be used to sell them.

In the bottom of each cone, Jean will pour some sauce: either mayonnaise or tomato sauce.

He decides to make 400 paper cones and has to estimate how many bottles of mayonnaise and tomato sauce to buy so that he doesn’t run out.

Here is the information John has to make his calculations:

Buyers:

80% of the buyers take tomato sauce and all the others take mayonnaise.

Sauces:

The mayonnaise bottle is assimilated to a cylinder of revolution whose base diameter is

5 cm and the height is 15 cm.

The tomato sauce bottle has a capacity of 500 mL.

1. Show that the radius [EF] of the sauce cone is 1.5 cm.

2. Show that the volume of sauce for a cone of fries is approximately 11.78 cm3

3. Determine how many bottles of each sauce John will need to buy.

All traces of research, even if not completed, must be included in the copy.

Find every week new **math lessons** adapted to your level!

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