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**MATHEMATICS PATENT**

**NORTH AMERICA**

**Session: JUNE 2022**

Duration of the test: 2 hours

Exercise 1: 22 points.

The figure below is not to scale.

- the points M, A and S are aligned
- the points M, T and H are aligned
- MH = S cm
- MS = 13 cm
- MT = 7 cm

- Show that the length HS is equal to 12 cm.
- Calculate the length AT.
- Calculate the measure of the angle . The result will be rounded to the nearest degree.
- Among the following transformations which one allows to obtain the triangle MAT from

of the MHS triangle?*In this question, no justification is expected*.*Copy the answer on**the copy*.

5. Knowing that the length MT is times longer than the length HM, a student states:

“The area of the triangle MA T is 1.4 times larger than the area of the triangle MHS ”

Is this statement true? Remember that the answer must be justified.

Exercise 2: 15 points

In this exercise, no justification is expected.

This exercise is a multiple choice questionnaire. For each question, only one of the four

answers is correct.

On the copy, write the number of the question and the chosen answer.

Exercise 3: 20 points

1. Of the 1.6 million adolescents aged 11 to 17 surveyed, how many do not comply with this

recommendation?

After reading this release, a teenager sets a goal.

**Goal: “Get at least one hour of physical practice per day on average.”**

For 14 consecutive days, he notes in the following calendar, the daily time he spends

its physical practice:

2.a. What is the range of the 14 daily durations noted in the calendar?

b. Give a median of these 14 daily durations.

3.a. Show that, over the first 14 days, this teenager did not reach his goal.

b. For the next 7 days, this teenager decided to devote more time to sports

to reach its goal over the 21 days.

Over the last 7 days, what is the total amount of physical activity he/she must do at least

plan to achieve its goal?

Exercise 4: 21 points

In this exercise, no justification is expected.

1. Using a scale of 1 cm per 20 steps, represent the resulting pattern as a “rectangle” block.

2. Here is an example of the display obtained by running the main program:

What is the distance d between the two rectangles on the display, expressed in steps?

3. What is the probability that the first pattern drawn by the sprite is a cross?

4. Draw freehand the 8 different displays that could be obtained with the program

principal.

5. We will admit that the 8 displays have the same probability to appear.

What is the probability that the player will win?

6. It is now desired that, for each reason, there should be twice as many chances of obtaining a

rectangle than a cross. To do this, you need to change the instruction in line 5.

On the copy, copy the following instruction, filling in the boxes:

Exercise 5 : 22 points

Consider the following calculation program applied to integers:

PART A

1. Verify that if the starting number is 15, then the ending number is 240.

2. Here is a table of values made with a spreadsheet:

It gives the results obtained by the calculation program according to some values of the number chosen at the beginning.

What formula could have been entered in cell B2 before being stretched down?

*No justification is expected.*

3. We note x the starting number.

Write, as a function of x, an expression of the result obtained with this calculation program.

PART B

We consider the following statement:

“To obtain the result of the calculation program, simply multiply the starting number

by the next whole number.”

4. Verify that this statement is true when the integer chosen at the beginning is 9.

5. Show that this statement is true whatever the integer chosen at the beginning.

6. Prove that the number obtained by the calculation program is an even number

regardless of the integer chosen at the beginning.

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