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. Lucy collects dolls. Each month, the number of her dolls equals the sum of the numbers of her dolls in the previous two months. In April, she had 8 dolls and in June she has 21 dolls. How many dolls did she have in January that year?
A) 1
B) 2
C) 3
D) 4
E) 5
2. A rectangle sheet of paper is folded twice and cut along the red lines as shown. How many pieces of paper can we get after cutting?

A) 16
B) 9
C) 12
D) 10
E) 8
3. The caterpillar and the butterfly below are both natural numbers. Find the largest value of the caterpillar given that the butterfly is a 2-digit number.
A) 130
B) 140
C) 150
D) 160
E) 170

4. 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 covers a total distance of 21 km. If he cycles for 45 minutes, how long does it take him to finish the whole race?

A) 20 min
B) 30 min
C) 45 min
D) 60 min
E) 75 min

5. A star is given below. Find the size of the interior angle at vertex A.

A) $$10^o$$
B) $$20^o$$
C) $$30^o$$
D) $$35^o$$
E) $$40^o$$

B. Speed-up
(6 points per question / No points deducted for wrong answers)

6. When Peter adds two 2-digit numbers, he fills each of the cells below with one of the digits 0, 1, 2, 3, 4, 5, 6 so that every cell is different. What is the units digit of the sum?

A) 2
B) 3
C) 4
D) 5
E) 6

7. At the weekend, Anna drives to her grandparents’ house. She goes 30km straight, then turns right and kept going for 25km. Then she turns right again and goes 10km straight. Finally, she turns right and goes another 10km. Find the shortest distance between Anna’s house and the grandparent’s house.

A) 25km
B) 75km
C) 35km
D) 15km
E) 55km

8. Based on the pattern below, how many small squares are there in the 50th figure counting from the left?

A) 302
B) 308
C) 312
D) 316
E) None of the above

9. Given three balances below. Find the weight of the heaviest ball.

A) 1.8kg
B) 0.4kg
C) 2.0kg
D) 1.2kg
E) 1.6kg

10. Candace stacked 6 boxes of volume 8cm^3 each to build the figure below. She painted all surfaces of the stair (except the bottom). Find the total unpainted area in cm^2.

A) 72
B) 56
C) 144
D) 76
E) 112

C. Challenge
(8 points per question / No points deducted for wrong answers)

11. Two identical equilateral triangles are overlapped to form a star as in the figure below. Find the area of the star given that each triangle has an area of 36cm^2?

A) 60cm^2
B) 72cm^2
C) 54cm^2
D) 48cm^2
E) None of the above

12. A grocery store sold 9 packets with 3, 7, 8, 10, 11, 12, 14, 15, and, 20 eggs respectively. Sofia and Nana bought 4 packets for each person. Given that the number of eggs Nana bought is 2 times the number of Sofia’s eggs. How many eggs are there in the remaining packet?

A) 7
B) 11
C) 10
D) 14
E) 20

13. Straight lines are drawn on 3 faces of a cuboid as in the figure below. An ant is standing at one corner of the cuboid. It wants to reach the sugar cubes but it can only go along the drawn lines or the edges. Given that it always moves in such a way that it gets closer to sugar and farther away from the original spot. How many different ways can it do that?

A) 20
B) 26
C) 24
D) 16
E) 18

14. Given a square paper with a perimeter of 8m. It is placed on a round table such that all vertices lie on the circle. Can you find the area of the table NOT covered by the paper?

A) 8π – 8
B) 32π – 64
C) 4π – 8
D) 2π – 4
E) 4π – 4

15. Father goes from home to the park, then goes to Lucy’s summer camp to take her home. Given that the diagram below represents the father’s distance from home and the average speed of the whole trip is 24km/h. When does father get to Lucy’s camp?

A) 12:30
B) 11:30
C) 12:15
D) 12:15
E) None of the above