A Nice Algebra Challenge | Vietnamese Math Olympiad Problem

In this video, learn the method to solve
A Nice Algebra Challenge | Vietnamese Math Olympiad Problem

[yavrusahin]

This doesn't take much to solve. You can easily see that if one of the numbers is 2022 and the other two are just negative of each other, both equations are satisfied. Since the odd power of a number maintains the polarity, they cancel each other in the final equation also.

[멋쟁이프로도]

2:38 b+c ≠0 Without loss of generality.
So a(a+b+c) = -bc
(a+b)(a+c) = 0
a = -c
And then, go to 6:36

[MottaYT]

Today I learned the term "Nice"as in nice algebra problem is not a universal term. I am sure I wouldnt classify this as nice.

[guyhoghton]

By inspection, a = 1, b = -1, c = 2022 satisfies both equations.
Using those values the expression evaluates to 1/2022²⁰²³.

[nr]

It's immediately clear that (n, -n, 2022) works for any n

[Total_Syntheses]

Solution of equation x^3 - 2022x^2 + kx - 2022k = 0 is (A, B, C)
x^3 - 2022x^2 + kx - 2022k = (x-2022)(k+x^2) = 0
so, A = 2022, B = i × sqrt(k), C = -i×sqrt(k)
if k<0 then this eqn can have real sol (A, B, C)

[ranshen]

This can be solved by Vieta's formulas. Let p = abc. Then Vieta's formulas imply that a, b, & c are the three roots of
x³ - 2022x² + p/2022 x - p = 0.
This can be factorized into
(x² + p/2022) (x−2022) = 0.
Thus, either x=2022 or p = -2022x².

There are 4 potential cases:
(1) a=b=c=2022. This violates a+b+c=2022.
(2) a=b=2022, but c≠2022: Then -2022c²=abc implies c=−2022.
(3) a=2022 but b≠2022 & c≠2022: Then -2022b²=abc implies b=−c.
(4) a≠2022, b≠2022 & c≠2022: Then p = -2022x² implies c²=b²=a² and abc=-2022³. The only solution under this case is a=b=c=−2022, but this violates a+b+c=2022.

Combining the feasible cases (2) and (3), we conclude that one of them has to be 2022 and the other two are negative of each other.

[KM-omhm]

I think we just have to relate those first two equations
There isn't any more possible way to do that
Just have to find the values of a, b, c

[anuradhaabhayawardhana]

But this is contradict at one point you got a realtion a = -b, b = -c and a = -c...
That implies a = b. But you solve this imagining a = -b...

[shafikbara]

8:36 if b =0 its dividing on zero...
There is something wrong

[hanswust]

At 6:05

b + c = 0
a + b = 0
a + c = 0

Summimg up all these e equations:

2 (a + b + c) = 0 =>

a + b + c = 0

Which contradicts the data:

a + b + c = 2022

[prakashmadaksirashamrao]

This method is too lengthy..Multiply the LHS terms and RHS terms. We geta^2(b+c)+ b^2(c+a)+c^2(a+b))+2abc=0.Now by rearranging we get ab(a+b) write the term c(a^2+b^2) =c(a+b)^2 -2abc..And the expression factorises to be(a+b)(b+c)(c+a)=0 and the result follows.

[DilipkumarBasu-sxmk]

Ifx+ 1/x =3 find x30+x 24+x18+x12+ x6 .how to solve.all are to the power.

[matematikadanfisikaterapan]

Woww...you are a deep may I upload video with some problem as yours with different solutions ( I mean shortcut solution not algebraic solution ) ??...

[anneshamondal]

2:44 I can't understand where did the minus sign before the bracket of the previous step in the first fraction went. Can anybody explain

[salihoguz]

6:57 if c=2022 a=-c means a=-2022 and b=-c means b=-2022 but you say a=-b so -2022=2022 how is that possible?

[quakers]

Ok so yougo over the steps, the mechanics of solving it but you never explain why you are doing the things you do and that is the hard part is it not. Think of a recipe. You can just go through the steps and make a cake but if you explain things it will make more sense. You need baking soda to make it rise, eggs for a binder, it is easier to blend the sugar, eggs and butter alone then add the flour. So say something like when i have such a problem i have to eliminate one variable or make the equasion equalto zero something like that.

[pauliron]

I guess that the answer should be December 2022😃

[khalilbenzina]

Or just raise all of the equation to the power of 2023

[anonymous]

Yeah I solve this it was also one of AMC questions