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**Exercise 1: 20 points.**

**Part 1**

We’re interested in a race conducted in early 2018. There are 80 participants, including 32 women

and 48 men.

Women wear red bibs numbered from 1 to 32. Men wear green bibs

numbered from 1 to 48.

There is therefore a red number 1 for a woman, and a green number 1 for a man, and so on.

right away …

1. What is the percentage of women participating in the race?

2. A facilitator randomly draws a participant’s bib to award a consolation prize.

a. Let the event V be: “The bib is green”. What is the probability of the event V?

b. Let the event M be: “The number of the bib is a multiple of 10”. What is the probability

of the event M?

c. The announcer announces that the bib number is a multiple of 10. What is then the

probability that it belongs to a woman?

**Part 2**

At the end of the race, the standings are posted below.

We are interested in the birth years of the first 20

runners.

1. The years of birth of the runners were arranged

in ascending order :

Give the median of the series.

2. The average of the series was calculated in cell B23.

What formula was entered in cell B23?

3. Astrid notes that the mean and median of this series are equal.

Is this the case for any other statistical series?

Explain your answer.

**Exercise 2: 11 points.**

1. The number 588 can be decomposed into the form 588 = 22×3×72.

What are its prime divisors, i.e. numbers that are both prime numbers

and divisors of 588?

2. a. Determine the prime factor decomposition of 27,000,000.

b. What are its prime divisors?

3. Determine the smallest odd positive integer that has three different prime divisors.

Explain your reasoning.

**Exercise 3: 13 points.**

After one of his running workouts, Bob

receives a summary of his or her performance from his or her coach

race, reproduced below.

The average pace of the runner is the quotient of the duration of the race by the distance covered and is expressed in min/km.

Example: if Bob takes 18 min to walk 3 km, his pace is 6 min/km.

**Exercise 4: 17 points.**

Worker bees go back and forth between the flowers and the hive to transport nectar and

the pollen of the flowers that they store in the hive.

1. A bee has an average mass of 100 mg and brings in an average of 80 mg of filler (nectar,

pollen) on each trip.

A man has a mass of 75 kg. If it were to be charged in proportion to its mass, like a

bee, what mass would this man carry?

2. When they return to the hive, the bees deposit the collected nectar in cells.

We consider that these cells have the shape of a prism of 1.15 cm height and whose base

is a hexagon with an area of about 23 mm², see the figure below.

a. Check that the volume of a hive cell is equal to 264.5 mm².

b. The bee stores nectar in its crop. The crop is a small pocket under the abdomen

with a volume of liter. How many times does the bee have to go out to

fill a cell?

(reminder: 1 = 1 liter)

3. The graph below shows the French honey production in 2015 and 2016.

a. Calculate the total amount of honey (in tons) harvested in 2016.

b. Knowing that the total amount of honey harvested in 2015 is 24,224 tons, calculate the

percentage decrease in honey harvest between 2015 and 2016.

**Exercise 5: 15 points.**

Sam has written the following program to draw a rectangle

as shown on the right.

This program has two variables (Length) and (Width) which

represent the dimensions of the rectangle.

It is recalled that the instruction means that we are moving to the right.

1. Complete the rectangle block above with numbers and variables to make the script work.

Only the repeat loop will be copied and completed on its copy.

2. When running the program, what are the coordinates of the end point and in

Which direction are we facing?

3. Sam modified his script to also draw the image of the rectangle by the homothety of center

the point with coordinates (0; 0) and ratio 1.3.

a. Complete the new Sam script given opposite to obtain the figure below.

We will copy and complete on his copy the lines 9 and 10 as well as the missing instruction

online 11.

b. Sam runs his script. What are the new values of the variables Length and Width

at the end of the script execution ?

**Exercise 6: 12 points.**

The figure below gives a schematic of a calculation program.

1. If the starting number is 1, show that the result is -15.

2. If we choose any number as a starting number, among the following expressions,

which one gives the result obtained by the calculation program? Justify.

3. Lily claims that the expression gives the same results as expression B for all values of x.

Is Lily’s statement true? Justify.

**Exercise 7: 12 points.**

For mountain running, some athletes measure their performance by speed

ascending, noted Va.

Va is the quotient of the difference in altitude of the race, expressed in meters, by the duration, expressed in

time.

For example: for a difference in altitude of 4,500 meters, the journey takes 3 hours: Va = 1,500 m/h.

**Reminder:** the difference in elevation of the race is the difference between the altitude at the finish and the altitude at the start.

A high level runner wants to reach a climb speed of at least 1400 m/h when

of his next race.

The course is broken down into two stages (see Figure 2):

– First stage of 3 800 m for a horizontal displacement of 3 790 m.

– Second stage of 4.1 km with a slope angle of about 12°.

1. Check that the first step is about 275.5 m.

2. What is the elevation gain of the second stage?

3. From the start, the runner takes 48 minutes to reach the summit.

Does the runner reach his goal?

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