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. Jack uses a balance scale to measure four types of objects and gets the result as follows. The total weight is 24 kg. What is the weight of the lightest object?
A. 0.5 kg
B. 0.8 kg
C. 1 kg
D. 1.2 kg
E. 2 kg
Answer : \(B\)
2. Ashley arranges five puzzle pieces below to make one mathematical expression. Find the value of that expression.
A. 17
B. 52
C. 71
D. 202.5
E. 648
Answer : \(C\)
3. Jack has a square piece of paper. He folds it twice along the dashed lines and cuts out some shapes as instructed below. What will he get after opening it?



Answer : \(D\)
4. There is a dart game in an amusement park where the goal is to hit the black portions of the regular hexagon in order to win a prize. From the choices below, which dart board will give the highest chance of winning?
Answer : \(B\)
5. Three small triangles are cut off from a bigger regular triangle to form a hexagon. Given that one of them is a right triangle, another is isosceles and the last one has an angle of 50 degrees. Find the size of the largest angle of the hexagon.
A. \(160^0\)
B. \(150^0\)
C. \(110^0\)
D. \(130^0\)
E. \(120^0\)
Answer :\(B\)
B. Warm-up
(4 points per question / No points deducted for wrong answers)
6. Each letter is a distinct digit smaller than 7. Find the greatest value of \(F\ times M\ times O\).
A. 15
B. 20
C. 30
D. 12
E. 24

Answer :\(C\)
7. Amy draws a wall consisting of 14 identical rectangles and a triangle fitted inside that wall. Find the area of the shaded region in \(cm^2\).
A. 756
B. 540
C. 234
D. 120
E. 216
Answer :\(E\)
8. Grandma has some candy jars. Each jar contains 2 apple candies and 4 banana candies OR 3 apple candies and 3 banana candies. Given that she has 25 banana candies. At least how many apple candies does she have?
A. 16
B. 17
C. 18
D. 19
E. 23
Answer :\(B\)
9. The average math score of 5 friends are 64 points. Their scores are recorded in the chart with equally-spaced lines below. Find the score of Charles.
A. 32
B. 40
C. 48
D. 50
E. 52
Answer : \(C\)
10. The numbers from 1 to 7 are to be placed in the seven circles in the diagram. The sum of the numbers lying in each triangles is the same. What is the product of X and Y?
A. 18
B. 15
C. 10
D. 12
E. 6
Answer : \(A\)
C. Challenge
(8 points per question / No points deducted for wrong answers)
11. Henry rides a bike to school. When he covers the first half distance, he looks at the clock and see that if he continues with his speed, he will be late by 5 minutes. He then decides to double his speed on the remaining path. So, he is at school 7 minutes sooner. How many minutes does it take Henry to go from home to school?
A. 12
B. 14
C. 24
D. 36
E. 48
Answer : \(D\)
12. The graph below shows Clover’s distance from home. She first goes to a grocery store to buy some food. Then she goes to the park. But it starts raining so she rushes back to her grandparent’s house. Finally, she rides bicycle to the bookstore and rides back home. Given the average speed of Clover during the whole trip is 8km/hour. How long does she stay at her grandparent’s house?
A. 9 min
B. 10 min
C. 18 min
D. 20 min
E None of the above
Answer :\(C\)
13. Dad wants to put some identical bricks into a cubic tank of side length 15 cm. At most how many bricks can he put inside the water so that the water does not spill out?
A. 17
B. 16
C. 15
D. 18
E. 19
Answer :\(A\)
14. The table below has 2021 rows with the given pattern. Anna adds all numbers in the middle column to get the result A. Find the unit digit of A.
A. 0
B. 1
C. 3
D. 7
E. 9
Answer : \(E\)
15. A rectangular paper with dimension 6cm x 18cm is folded so that the two opposite vertex coincide as the figure below. Find the area of the paper after being folded.

A. \(54cm^2\)
B. \(80cm^2\)
C. \(78cm^2\)
D. \(84cm^2\)
E. None of the above
Answer : \(C\)
D. Star of hope
(10 points per question/ 10 points deducted for wrong answer)
16. Given the map with 12 intersections below. A person standing at one intersection can see all other intersections lying on the same street. For example, a person standing at I8 can see all people standing at I2, I4, I5, I7, I9 and I11.
Six people Amy, Ben, Clover, Dan, Elsa and Fred are standing at 6 different intersections in the map. Given that:
* Dan, Amy, and Fred are standing at 3 corners of a triangle formed by 3 street segments.
* Dan can see only Amy and Fred.
* Elsa can see only Amy and Clover.
* Amy sees Ben standing in the next intersection behind Fred.
* Clover cannot see Ben or Fred.
* No one among the six is standing at I10.
Find the position of Clover.

Answer : \(I12\)

FERMAT Mathematical Olympiad Summer 2021
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