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. Jake is watching a flock of healthy sheep. He sees that the number of sheep’s noses is 120 fewer than the number of sheep’s legs. How many sheep are there in total?

A) 17
B) 21
C) 40
D) 12
E) 14
2. Kellie and Liam cycle on a track that is 300 m around. It takes Kellie 100 seconds to cycle 3 times around the track and it takes Liam 200 seconds to cycle 5 times around the track. Who is the slower cyclist and at what speed does he/she run?
A) Kellie, 9m/s
B) Kellie, 7.5m/s
C) Liam, 3.3m/s
D) Liam, 7.5m/s
E) Liam, 9m/s
3. The first group has 1 square. The second group has 5 squares. The third group has 14 squares. Refer to the pattern below, how many squares are there in the 4th group?

A) 16
B) 20
C) 25
D) 28
E) 30
4. Each letter in the operation below represents a distinct digit. Can you find the sum of A, B, and C?

A) 9
B) 10
C) 11
D) 12
E) 13
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 2022 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. Leo folds the square paper in half twice and cuts out a rectangular hole as in the figure below. How many pieces does he receive?

A) 13
B) 16
C) 12
D) 15
E) 20
7. Greg has 100 identical rectangles with dimensions 4cm x 2cm. He arranges them as the pattern below so that the top-left vertex of each rectangle is placed at the midpoint of the bottom edge of the rectangle above it. Find the total perimeter of the final figure.

A) 800cm
B) 404cm
C) 804cm
D) 402cm
E) 808cm
8. Uncle John has 3 nieces named Abby, Bella, and Cindy. Abby is two years older than Bella and two years younger than Cindy. The age of Uncle John is the sum of 3 nieces’ ages. Which number below can be Uncle’s John age?

A) 35
B) 36
C) 37
D) 38
E) 40
9. Three sacks of rice are weighed two at a time in all possible ways. The weights of the pairs of rice sacks are 127 kg, 128 kg, and 131 kg. How much does the lightest sack weigh?

A) 63kg
B) 62kg
C) 64kg
D) 65kg
E) 66kg
10. A restaurant represents their sale of 4 dishes by the pie chart below. The sale of the cookies is $270 more than the sale of the sandwiches. Find the sale of the fries. A)$630
B) $595 C)$665
D) $700 E)$735
C. Challenge (8 points per question / No points deducted for wrong answers)
11. Five different locks have passwords as the figure below and each letter corresponds to the same digit. All the keys fell on the floor and the label of one key was lost. Find the missing label.

A) 312
B) 134
C) 314
D) 132
E) 413
12. Thirteen identical cubes are combined to get the figure below with a total surface area of 1656cm^2. Find the total volume of the figure.

A) 351cm^3
B) 832 cm^3
C) 1625cm^3
D) 2808cm^3
E) None of the above
13. In the art lesson, each student is asked to make a figure by combining 6 identical rectangles. The two figures below are the final product of Harry and Ron. If the perimeter of Harry’s figure is 32cm less than Ron’s figure, find the perimeter of one small rectangle.

A) 24cm
B) 48cm
C) 54cm
D) 30cm
E) 42cm
14. Given a box with 100 balls numbered consecutively from 1 to 100. At least how many balls should be picked out to ensure that you get two balls with the sum of numbers being 100?

A) 49
B) 50
C) 51
D) 52
E) 53
15. A car left a gas station and traveled towards the airport at a constant speed of 60 km/h. A truck left the same gas station 10 minutes after the car and traveled in the same direction at a constant speed. It took the truck 1 hour and 40 minutes to overtake the car. At what speed did the truck travel?
A) 62km/h
B) 64km/h
C) 66km/h
D) 68km/h
E) None of the above