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. Tony has a square piece of paper. He folds it twice along the dashed lines and cuts the paper as in the picture below. How many pieces will he get after opening it?

A) 8

B) 9

C) 10

D) 11

E) 12

Answer : B

2. Tom can cycle at the speed of 20km/h and run at the speed of 12km/h. He takes part in a combined cycling and running race which lasts 56 minutes in total. If the cycling path and the running path have equal length, in how long does he cycle?

A) 21 min

B) 35 min

C) 28 min

D) 14 min

E) 42 min

Answer : A

3. Linda use a compass to draw a circle. One leg of the compass is 9cm and always perpendicular to the surface of the paper. The other leg with pencil has a length of 12cm in total. Find the area of the circle.

A) 15π

B) 21π

C) 30π

D) 54π

E) 63π

Answer : E

4. Given that the same objects have the same weights, what is the heaviest object?

A)

B)

C)

D)

E)

Answer : E

5. Three siblings named Fermat (F), Euclid (E), and Pythagoras (P) have different heights. One day, they talk to each other:

Pythagoras: “I am not the tallest.”

Euclid: “I am the tallest.”

Fermat: “I am not the shortest.”

Only one of them says the truth. Can you list their height in order from the tallest to the shortest?

A) F, E, P

B) P, F, E

C) E, P, F

D) P, F, E

E) E, F, P

Pythagoras: “I am not the tallest.”

Euclid: “I am the tallest.”

Fermat: “I am not the shortest.”

Only one of them says the truth. Can you list their height in order from the tallest to the shortest?

A) F, E, P

B) P, F, E

C) E, P, F

D) P, F, E

E) E, F, P

Answer : B

**B. Speed-up**

**(6 points per question / No points deducted for wrong answers)**

6. When practicing adding numbers, Jimmy didn’t write the last digit of a term. So, his result was 3698 instead of 33285. Let’s help him to find the digit that he didn’t write.

A) 3

B) 4

C) 5

D) 6

E) 7

Answer : B

7. Given 3 circles tangent to each other with center A, B, and C. If the perimeter of triangle ABC is 12, what is the area of the circle with center A?

A) 12π

B) 24π

C) 36π

D) 72π

E) 144π

Answer : C

8. From the pattern below, how many matchsticks do we need to build Figure 20?

A) 232

B) 244

C) 228

D) 224

E) 248

Answer : A

9. Andy and Alan start riding a bike at the same time and at the same speed from A to D, but they follow 2 different paths as the figure below. Andy is at D 15 minutes earlier than Alan. How many minutes does it take Alan to ride from A to D?

A) 90

B) 100

C) 105

D) 95

E) 110

Answer : C

10. Thomas has 133 Math questions. He does 2 questions more each day than the preceding day. In exactly a week, he finishes all questions. How many pages does he do on the second day?

A) 15

B) 13

C) 17

D) 14

E) 12

A) 15

B) 13

C) 17

D) 14

E) 12

Answer : A

**C. Challenge**

**(8 points per question / No points deducted for wrong answers)**

11. Given a triangular prism whose base is an isosceles triangle. It is placed on a cuboid to get a combined figure as below. What is its total volume in cm3? (Note that the volume of a triangular prism is equal to the product of the area of the base and its height.)

A) 4736

B) 4032

C) 5376

D) 4608

E) None of the above

Answer : D

12. Father goes from home to the park, takes some rest then goes to Lucy’s summer camp to get her home. Given that the diagram below represents the father’s distance from home and the time he gets to each place. The average speed of the whole trip is 12km/h. How far is it from the home to the park?

A) 15km

B) 24km

C) 12km

D) 30km

E) None of the above

Answer : A

13. A confectionery factory has 141kg of sugar and 207kg of flour. Each box of candies needs 5kg of sugar, 7kg of flour and costs 12 dollars. Each box of cakes needs 8kg of sugar, 12kg of flour and costs 15 dollars. If they use all amounts of sugar and flour, how many dollars do they get after selling candies and cakes?

A) 144

B) 156

C) 245

D) 288

E) 360

Answer : D

14. Jack wants to create a sequence by writing even consecutive numbers starting from 0. To make it more challenging, he splits all numbers into digits as below. Can you find the 2022nd digit in the sequence counting from the left?

**0, 2, 4, 6, 8, 1, 0, 1, 2, 1, 4, 1, 6, …**

A) 2

B) 6

C) 5

D) 8

E) 1

Answer : E

15. In the figure, ABC is a quarter of a circle with a radius of 8. Two semi-circles with diameters AB and BC are drawn inside. Can you find the area of the shaded region and choose the closest answer below? (Take π =3.14)

A) 22

B) 16

C) 25

D) 18

E) 20

Answer : D

**D. Star of hope**

**(10 points per question/ 10 points deducted for the wrong answer)**

16. In a football tournament at Fermat Schoool, there are four teams named Gauss, Pascal, Euclid, and Pythagoras. After each team has played with every other team exactly 4 times, the total points are given in the table below:

If each team gets 3 points for a win, 1 point for a tie, and no points for a loss, how many games ended in a tie?

Answer : 5