Fermat Mathematical Olympiad FMO is an annual competition developed and held by Fermat Education – an authorized Vietnamese organization board of many international Olympiad competitions. FMO is a mathematical playground for students from Kindergarten to high schoolers. Unlike other Mathematical Olympiad competitions, which have many difficult problems requiring complicated calculations or various formulae to be solved, FMO 2022 focuses on how students read, understand, and analyze the problems. As a friendly, accessible, and suitable competition for the majority of students, the examination questions of FMO 2022 will be arranged in increasing difficulty and composed relatively close to the curriculum framework. The tedious knowledge in textbooks is now vividly illustrated, thought-provoking, and in specific real-life settings. Students participating in FMO 2022 will be provided an opportunity to review the prior knowledge in a new approach, cultivate their passion for Mathematics and challenge themselves with real-world problems. Any student can participate in FMO 2022. as long as they have an interest in Mathematics and adequate basic mathematical knowledge. Ultimately, the purpose of this competition is to help every student answer the age-old question: “Why study Mathematics?”.
In 2022, Fermat Mathematical Olympiad FMO takes place for the third time. In the previous seasons, the competition was a great success by having attracted the attention of many students from all over the world such as Thailand, Philippines, Bulgaria, Indonesia, Turkey, India, …. Following that success, with some alterations in the examination structure and the appearance of real-life problems, together with many valuable cash prizes, FMO 2022 promises to be an educational, interesting but also challenging mathematical playground for students. This is also a great opportunity for the participants to compete on a global scale and have memorable experiences. (sc: Facebook Fermat Mathematical Olympiad)
A. Warm-up (4 points per question / No points deducted for wrong answers)
1. Given a long strip of paper with cells numbered from 1 as below. The castle can move 4 cells or 7 cells to the right each time. Which number CANNOT be stepped on by the castle?

A) 21
B) 19
C) 15
D) 22
E) 17
2. The caterpillar and the butterfly below are both natural numbers. Find the value of the caterpillar given that the butterfly is the largest 2-digit even number.

A) 130
B) 150
C) 140
D) 170
E) 160
3. The black ants are marching. Suddenly, two red ants join the line. Two red ants see that the number of black ants in front of them is 3 times the number of black ants behind them. Which answer can be the total number of ants of the new line?

A) 20
B) 21
C) 22
D) 23
E) 24
4. Peter draws and paints a pineapple on a paper grid with each cell of side length 2cm. Can you help Peter find the total painted area?

A) 64cm^2
B) 36cm^2
C) 80cm^2
D) 72cm^2
E) 144cm^2
5. Each time, Jane rotates the square clockwise as in the figure below (from Figure 1 to Figure 2). If she rotates Figure 2 another 22 times, which answer can she get?

A)

B)

C)

D)

E)
B. Speed-up (6 points per question / No points deducted for wrong answers)
6. Henry folds the square paper in half twice and cuts it as the figure below. How many pieces does he receive?

A) 12
B) 8
C) 6
D) 4
E) 10
7. In an eating competition, each person needs to finish 10 burgers. Their amounts of time have been recorded by the bar graph below. The person who finishes second is 12 minutes faster than the person who finishes last. How many minutes does it take the winner of the competition to finish all the burgers?

A) 3
B) 2
C) 4
D) 6
E) 8
8. Fred and Peter play a game in which the winner gains 5 points, the loser loses 2 points, and there are no ties. If Fred won exactly 18 games and Peter had a final score of 134 points, how many games did they play in total?
A) 54
B) 50
C) 34
D) 52
E) 32
9. Given 4 identical small squares. Kellie arranges them in a row to get a rectangle. Then Kyle rearranges them to get a big square. Kyle sees that the perimeter of his figure has become 10cm less than Kellie’s. Find the perimeter of 1 small square.

A) 10cm
B) 20cm
C) 80cm
D) 40cm
E) 100cm
10. Andrea has some blank cards. On one side of the card, she writes consecutive odd numbers starting from 11. On the other side of the card, she writes consecutive even numbers starting from 26. If Andrea has 10 cards, find the sum of all numbers on those cards.

A) 450
B) 400
C) 460
D) 550
E) 540
C. Challenge (8 points per question / No points deducted for wrong answers)
11. Lucy glued 14 white cubes together with the side length of 3cm and painted all the surfaces in yellow (except for the bottom). Find the total painted area.

A) 378cm^2
B) 297cm^2
C) 279cm^2
D) 387cm^2
E) 287cm^2
12. There are three types of suitcases which are yellow, red, and green as the figure below. Find the weight of the heaviest type of suitcase.

A) 4kg
B) 6kg
C) 8kg
D) 10kg
E) None of the above
13. Six squares of many sizes are combined to form a big rectangle with a perimeter of 156cm as follows. Find the perimeter of the greatest square.

A) 60cm
B) 100cm
C) 124cm
D) 120cm
E) 96cm
14. Given a box with 121 balls numbered consecutively from 25 to 145. At least how many balls should be picked out to ensure that you get two balls with the difference of numbers being 18?

A) 64
B) 65
C) 66
D) 67
E) 68
15. James writes nine consecutive odd numbers in the “Magic square” below so that the sum of each column, row, or diagonal is equal. Some numbers have been replaced by letters. Find the sum of A and E.

A) 32
B) 28
C) 26
D) 30
E) 24