Olympiad Mathematics - Algebra Problem | Vietnamese Junior Math Olympiad

Olympiad Mathematics - Algebra Problem | Vietnamese Junior Math Olympiad

Topics Covered :-
1. Math Olympiad Problem
2. Algebra Problem
3. Olympiad Mathematics
4. Algebra for Competitive Exams

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[זאבגלברד]

Another way to see that x^3 = 1 is to find the complex roots of x^2 +x +1 =0 and see that they are on the unit circle in such a way that x^3 = 1 . cis120 and cis(-120).

[ABHISHEKKUMAR-]

Senior students can use the fact that

x2 + x + 1 = 0 => x = w or x = w2, where w is the cube

root of unity.

[paulortega]

f(n+1) = f(1)*f(n) - f(n-1) where f(n) = x^n + (1/x)^n
f(1)=-1 and f(2)=-1
it is a repeating series of -1, -1, 2
Since 25 = 1 modulo 3 then f(25) = f(1) = -1

[owlsmath]

Nice problem!! Love this reciprocal type equation

[theokatcha]

Solve x^2+x+1=0 with complex numbers
--> 2 solutions x= exp (+/- i* 2*Pi/3)
--> x^25 = exp (+/- i* 50 Pi/3) = exp (+/-i* (16Pi + 2*Pi/3))= exp (+/-i *2*Pi/3)
and in the same way 1/x^25 = exp (-/+i*2*Pi/3)
-->x^25+x^-25= 2 cos(+/- 2 * Pi/3)= 2 *(-1/2)=-1

[varky]

Another way
×+1/x = -1
×^25 + 1/×^25 = (-1)^25, = -1
(×^5)^5 = -1/x^25
×^5 = 5 th root -1/×^25, when negative is in root, there is no real distinct solution .

[kaliprasadguru]

This simultaneous writing on a black strip is obstructing the screen ..( I don't know what is its technical term) Please refer to ' PreMath ' channel and design your videos in that pattern .. Regards.

[glasssmirror]

Hi friends
Can somebody pls explain as to where x cube is coming from

[martinbennett]

X = 1 cannot be the solution, so what are the possible cube roots of 1?

[benabdabdallahyassine]

wait how is x!=1 an x to the power of three is equal to 1

[oloyt]

I solved this a much easier way, and in the shower. x +1/x=-1 (x+1/x)^2=1, x^2+1/x^2=-1 and then kept building on that till i got x^25 +1/x^25

[jeremirusin]

x + 1/x = -1 doesn't have answer in reality numbers:
x, 1/x > 0 or x, 1/x < 0
x + 1/x < 0 => x, 1/x < 0
-x, - 1/x > 0
(- x + (- 1/x))/2 >= sqrt(-x(- 1/x))
- x - 1/x >= 2
x + 1/x =< -2 => x + 1/x can't be equal to -1

[NguyenNguyen-mreq]

The problem has NO real solutions! How can x=1?

[skoczeeek]

nie rozumiem prosze mi wytlumaycz jeszcze raz po polsku