Solve in integers | 2021 Junior Singapore Math Olympiad SMO 1 Mathematical Questions Solutions 2022

There are around 50 ideas in each topic (algebra, number theory, geometry, combinatorics, ...) If you want to know them feel free to send message to my Whatsapp number: 00989122125462

You can use this idea to solve many problems of Harvard, MIT, Stanford, International, America, British, Singapore, Canada, Hong Kong, Philippine, ... Mathematical Olympiad and by working on the useful ideas you can prepare for 2022 and 2023 Mathematical Olympiads

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Mobile number: 00989122125462
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Farshid Bateni's CV(Resume):

Teaching Experience:
-Iranian team members of 2018 International Mathematical Olympiad
-Allame Helli High School(Won 181 International Olympiad medals)

Some of My Honors:
-Winner of gold medal for mathematics in the Iranian National Olympiad
-Winner of silver medal for informatics(computer science) in the Iranian National Olympiad
-Ranked 8th in National University Entrance Exam for M.Sc. degree in MBA

Authoring and Proposing:
-Author of two international papers:
1."Diffusion-Based Nanonetworking : A New Modulation Technique and Performance Analysisi," IEEE Communication letter soon, 2013
2."Deteccting Matrices for random CDMA systems", 20th International Conference onTelecommunications,ICT 2013

-Author:
-All the unsolved answers of Iranian Informatics(computer science) National Olympiad second round for the first time
-for two publishers

Propose:
- First, second and third round of Iranian Mathematical Olympiad exam
-Second round of Iranian informatics (computer science) National Olympiad exam

[mathgoldmedalist]

There are around 50 important ideas in each topic(Algebra, Geometry, Number Theory, Combinatorics, ...)
For solving geometry there are around 50 important formulas that you should learn and memorize
If you want to know them and wanna improve your problem solving feel free to send message to my Whatsapp number: 00989122125462

[prathikkannan]

I have seen math olympiad problems of the same structure, so I knew immediately how to proceed! Thank you for the video 👍

[sy]

I think it takes a little long time to solve. So, I want to show my method for this problem.
At first, if a=b=c, (1/a)+(1/b)+(1/c) = 3×(1/a) [=1] . ∴ a=b=c=3
And (next step), if "a is not b" or "b is not c", we can say (at least, ) a<3 and c>3 . (∵ a ≦ b ≦ c and 1/a ≧ 1/b ≧ 1/c )
From the problem, a, b, c cannot be 1. This means a<3 and "a is not 1". ∴ a=2 .
Therefore, (except a=b=c=3, ) a=2 for any other solutions.
Where a=2, (1/a)+(1/b)+(1/c) = (1/2)+(1/b)+(1/c) [=1] ∴ (1/b)+(1/c) = 1/2
(Next, the same way can be used.) If b=c, the solution is b=c=4 . (So, one of other solutions is a=2, b=4, c=4.)
If b is not c, "b<4 (and c>4)". From "a(=2) ≦ b", b=3. (It is the last candidate.) Then, 1/3+1/c=1/2. ∴ c=6 (So, the last solution is a=2, b=3, c=6)
∴ (a, b, c) = (3, 3, 3), (2, 4, 4), (2, 3, 6)
Anyway, as for the problem, I think this method is easier and faster.

Appendix: According to this video information, it is "2021 Junior Singapore Math Olympiad" problem. But I remember there was the same problem twice in the past.
One is a problem of "pre-first grade Suken test" nearly 30 years ago. When I took the qualification test, there was the same problem, and I solved by the other method.
(The method was different from the video's one and also different from my new method that I wrote above.)
The other is a problem of "a girl's private high school entrance examination test" in my country. I noticed it nearly 40 years ago.
[ ・・・ Judging from this, I guess it was easy for Singapore junior students. ]