Olympiad Mathematics | Learn How to Solve the System of Equations Fast | Math Olympiad Preparation

Learn how to solve the Olympiad Question quickly with these tips and tricks for a, b, and c variables. The system of equation involves a, b, and c variables.

Today I will teach you tips and tricks to solve the given olympiad math question in a simple and easy way. Learn how to prepare for Math Olympiad fast!

Need help with solving this Math Olympiad Question? You're in the right place!

I have over 20 years of experience teaching Mathematics at American schools, colleges, and universities. Learn more about me at

Olympiad Mathematics | Learn How to Solve the System of Equations Fast | Math Olympiad Preparation

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[Feelu]

Wow Your explanation is very good thank you so much sir

[davidfromstow]

Looking at it, it was fairly obvious that 0 would be an answer, without doing any work. Knowing you, I just knew there had to be another answer, and you very cleverly showed what it was! Thank you. as always.

[montynorth]

a+b=abc.
Also b+c=abc.
Therefore a+b=b+c.
Subtract b from both sides.
Therefore a=c.
b+c=abc
Also c+a=abc.
Therefore b+c=c+a
Subtract c from both sides.
Therefore b=a.
Thus b&c both equal a
So a=b=c.
abc can now be written as a^3
Also, a+b can be written as 2a
Thus 2a=a^3
Divide both sides by a.
2=a^2
Taking square root of both sides.
Sq.root of 2=a. (also b & c of course)

[Galileosays]

Since an interchange of a<->b, b<->c, c<->a doesn't change the set of eqn. a=b=c. therefore a+a=a^3 hence a=b=c=+/- sqr(2)

[renesperb]

A somewhat faster way to solve this problem is to first observe that the equations are completely symmetric in a, b, c. The conclusion therefore is that for the solution one has a=b=c, i.e 2 a= a^3, i.e. a= +, - sqrt2. Of course there is also the trivial solution a=b=c=0.

[chrismcgowan]

Wow, you got me on this one, a quick look at it and a=b=c=0 was a solution. I could not see a possibility of another solution. A great lesson in logic, thanks for this :-).

[akramst]

I knew that a=b=c just by looking at it and using some logic however I didn't complete my thinking to give the value of each of them, so thank you for showing me how.

[themanwithaplan.]

It is easy to see a, b and c are all equal. From there you can form an equation for which there are three. solutions, zero and plus or minus sqr2.

[Awesome-]

Actually this one was quite easy for me.
I really enjoyed doing it

[SubhrodeepBandyopadhyay]

Sir directly from symmetry could we conclude a=b=c ?

[szkoclaw]

Does it count if know the answer by just looking on the equations? Just felt like bragging a little :)

[learningmath]

Please, someone tells me what app teacher use for teaching ????, i need help, thanks

[Sam_on_YouTube]

Solved this one in my head. First realized that if any of them is zero, they must all be zero. Then realized that they must all be the same regardless. So 2x=x^3. From there, its simple.

[devondevon]

a+b=abc equation 1
b+c = abc
hence a= c
a+c = abc
hence b=a
hence a=b=c
a+a=aaa substitute in equation 1
2a=a^3
2= a^2
a=sqrt 2
a=b=c= sqrt 2 answer

[hafizzuddinnur]

How about if abc is not axbxc, instead its a 3(or more) digit number ?

[Gmdtechinfo]

Wow So Really Nice Olympiad Mathematics and Really Really Beautiful! Keep it up Good Job! Stay Connected! 😍😍😍🤗🤗🤗

[asmrrelaxationtv]

Wonderful Content. Thank You for sharing. have a good day!👍❤

[DrLiangMath]

Great presentation for solving this system of equations!

[HappyFamilyOnline]

Very brilliant 👍
Thank you so much sir💖💖

[pascalfrancois]

Well, as on the board :
a+b = a×b×c (1)
b+c = a× b×c (2)
(1) and (2) <=> a+b = b+c <=> a= c.
(Directly, while regular property of +)
Idem for : a+b = abc (1)
with c+a = a+c = abc (3)
Directly b= c
So, finely, a = b = c.
(1) <=> abc = a+b
axaxa = a+a
a×(a)^2 = 2a (4)
If a=0, (1) IS True. Ok for first triple
(a, b, c) = (0, 0, 0)
If a # 0
(4) <=> (a)^2 = 2
(directly regular property of × ).
<=> a = +/- √2
So two triples more.
Very Happy.😊