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. A packet has 12 orange candies, 10 apple candies, 14 strawberry candies. Erick picks some candies from the packet without looking. What is the smallest number of candies he must pick to be sure that he has all flavors?
A) 23
B) 24
C) 25
D) 26
E) 27
2. They need 9 paint jars to paint the triangle ABE. Given that AB // CD, AB = 15 cm and CD = 35 cm. How many paint jars do they need to paint the triangle CDE?

A) 21
B) 28
C) 35
D) 49
E) 63
3. Given that the same objects have the same weights, find the heaviest object.

A)

B)

C)
D)

E)
4. One morning, John bought a new clock and set it correctly at 8:30. However, John’s clock moves 3 minutes faster every hour than a normal clock. In the afternoon of the same day, John looks at the clock and it shows 15:30. What is the correct time now?
A) 14:40
B) 15:00
C) 15:10
D) 17:10
E) 15:50
5. Jimmy 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) 17
B) 19
C) 20
D) 21
E) 22
B. Speed-up
(6 points per question / No points deducted for wrong answers)
6. Annie wants to fill each cell with a number so that the sums of the numbers in each row, each column, and each diagonal line are the same. Which number should be filled into the question mark?

A) 19
B) 18
C) 20
D) 24
E) None of the above
7. A factory has 37kg of sugar and 81kg of flour. They use all of them to make 3 types of cakes. The amounts of sugar and flour to make a box of each type are shown below. How many boxes of cakes do they make in total? (The number of boxes is an integer.)

A) 12
B) 11
C) 10
D) 8
E) 7
8. Given 3 circles externally tangent to each other with center A, B, and C. Given that triangle ABC is right at A, its perimeter is 30cm and AB = 5cm. Find the sum of the areas of 3 circles in cm^2.

A) 104π
B) 109π
C) 113π
D) 208π
E) 218π
9. Derek, Eden, and Fred competed in a 100-meter running race and each person always ran at a constant speed. Two participants competed against each other in 3 rounds. In the first round, when Fred reached the finish line, Eden was 20m behind. In the second round, when Eden reached the finish line, Derek was 10m behind. In the final round, when Fred reached the finish line, how many meters was Derek behind?

A) 20
B) 25
C) 28
D) 32
E) 40
10. To build a grid with 1 small square, we only need 4 matches. To build a grid with 4 small squares, we need 12 matches. To build a grid with 9 small squares, we need 24 matches. If we use 180 matches to build a grid with the same pattern, how many small squares does that grid have?

A) 49
B) 121
C) 64
D) 100
E) 81
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. Can you find its total surface area in cm^2?

A) 1696
B) 1856
C) 1664
D) 1792
E) None of the above
12. If the death rate of the bulls on a farm is 3 times of the death rate of the cows, there will be 160 bulls left by the time all the cows die. However, if the death rate of cows is 4 times of the death rate of bulls there will be 600 bulls by the time all the cows die. Find the total number of bulls and cows on the farm?
A) 800
B) 860
C) 900
D) 960
E) 720
13. Oliver bought some apples, books, cakes, desks, and eggs with a total number of 42 objects. The number and total price of each type are shown in the diagram below by points A, B, C, D, and E respectively. Given that the same types have the same prices. How many dollars does a cake cost, given that Oliver must pay 85 dollars in total?

A) 0.8
B) 0.95
C) 1.25
D) 1.75
E) 1.8
14. A flag is split into 4 triangles as in the figure below. Lucy wants to paint each triangle with one of the following colors: blue, yellow, pink, red, grey. If every pair of triangles sharing the same side must have different colors, in how many different ways can we paint the flag?

A) 200
B) 220
C) 180
D) 240
E) 260
15. Rick had two candles, one of which was 32 cm shorter than the other. He lit the longer one at 5 p.m. After 4 hours, he lit the shorter one. At 11 p.m., they were both the same length. The longer one was completely burned out at midnight, and the shorter one was completely burned at 2 a.m. The two candles burned at different but constant rates. What is the original length of the longer candle?
A) 48cm
B) 35cm
C) 42cm
D) 49cm
E) None of the above