Olympiad Question! Can you solve this? | Brain Teaser!

[dipinds]

Take the reciprocal..it will ease to get the answer fast.This solution is great also

[philipkudrna]

Very impressive! (But no chance figuring that out on my own!) 🙂

[guyhoghton]

It helps to note that (x² + x + 1)² = (x⁴ + x² + 1) + 2x(x² + x + 1).
Substituting from the 1st equation:
(x² + x + 1)² = (x⁴ + x² + 1) + 2x²/a.
Square the 1st equation, substitute for the denominator as above, then work towards the result from there.

[pwmiles]

Great problem! Without guessing the factorisation you can put a = 1/(x+1+1/x), x+1+1/x = 1/a, x+1/x = 1/a-1. Square both sides, x^2+2+1/x^2 = (1/a-1)^2. Subtract 1 from both sides, x^2+1+1/x^2 = (1/a-1)^2-1. Reciprocal of both sides is the answer

[xyz]

Another way to do it, flip the first equation it will become x +1+ 1/x = 1/a thus x+ 1/x = 1/a -1 and square both side x^2 + 1/x^2 + 2 = (1/a -1)^2 and subtract 1 from both side x^2 + 1/x^2 + 1 = (1/a -1)^2 -1 = ((1/a -1 )+ 1) ((1/a -1 )-1)= 1/a (1/a -2) = (1-2a)/a^2
Divide the 2nd equation by x^2 and bottom part becomes x^2 + 1/x^2 + 1, and final answer a^2/(1-2a)

[Gargaroolala]

I have to agree. I solved this qns using reciprocal and square both sides. Ur solution is hard to imagine. Like how are regular people suppose to know the first step is to + x and - x to the polynomial term?

[j.r.]

I solved this by squaring both sides; rearranging the terms in the resulting long polynomial; cross-multiplying to get most of the terms into the numerator; substituting "a" back into the equation when I could; factoring and rearranging some more to isolate the "a" terms; then flipping the resulting fractions on both sides to get the final answer. It was like riding a bike on a path into a maze: you don't know when you start exactly how long it will take to get out, but you can sense when you're making the correct turns. And it was all pure algebra, with no tricks.

[clementyip]

You should NOT need to factor the quartic polynomial.
We simply let b be our answer we are looking for. Then we have:
1/a = x + 1 + 1/x
1/b = x^2 + 1 + 1/x^2
Then we can easily express b in term of a
We will get b = a^2 / (1 - 2a)

The solution is much easier.

[michaelpurtell]

Wasn’t this posted on sybermath a while ago?

[afriend]

Hi there! I chose your channel randomly to ask my question after seeing a thumbnail of your Cambridge question video. I'm diving back into my math studies, and I'm wondering what you recommend to learn/understand, what I'm guessing is called equation(s) math (number minus a number divided by a number equals a number, that sort of thing). Is this an officially catalogued/classed subject, or would I just understand this type of math from refreshing my memory on basic math operations (addition, subtraction, multiplication, division), and learning how to put these all together? Thanks! =).

[prabhudasmandal]

I did it this way.x/a=(x^2+x+1). and x^2+1=(x/a_x)or (x_ax)/a.now denom of the given expression is nothing but (x^2+x+1)(x^2_x+1).now putting the above values it becomes x^2 whole divided byx/a(x_2ax)/a=x^2*a^2 whole divided by x^2(1_2a)=a^2/(1_2a). Ans.

[govindashit]

Wow, what a great solution !!!
Thanks sir 🙋🙋🙋

[gila]

Regarding factoring X^4+X^2+1, another way is just add X^2, then minus X^2, giving (X^2 + 1)^2 - (X^2)

[zplusacademy]

Above superb and technically interesting sir by taking raciprocal ❤️❤️❤️🙏🙏🙏🌄🌅🌅🙏

[chaosredefined]

x/(x^2 + x + 1) = a
Taking the reciprocal gives:
(x^2 + x + 1)/x = 1/a
Cleaning up:
x + 1 + 1/x = 1/a
Subtract 1 from both sides:
x + 1/x = 1/a - 1
Squaring both sides:
x^2 + 2 + 1/x^2 = 1/a^2 - 2/a + 1
Subtract one from both sides:
x^2 + 1 + 1/x^2 = 1/a^2 - 2/a
Cleanup:
(x^4 + x^2 + 1)/x^2 = (1 - 2a)/a^2
Take the reciprocal of each side:
x^2/(x^4 + x^2 + 1) = a^2/(1 - 2a)

[yanghwanlim]

Easiest way is to reciprocate(or inverse?) the given equation.
That is . . . X+ 1/X +1= 1/a or X+ 1/X= (1/a) -1
It follows that X^2+1/X^2={(1/a)-1}^2 -2
Like wise, instead of calculating X^2/(X^4+X^2+1), let us get the reciprocal of this one!
Which is X^2+1/X^2+1 and this is just {(1/a)-1}^2 -2 +1 = {(1-a)/a}^2 -1
So the answer is just reciprocal of {(1-a)/a}^2 -1 .

[divyanshuraghav]

This question is also easy sir, but thanks for uploading 😊

[rangaswamyks]

Oh my god you are awsome
You are brilliant
You are great mathematician
You are great teacher
You are my master
Namasthe gurujee
Such a brilliant. Beautiful.. Explanation.. Never seen such a master in my life
You are gifted by god.. Sir

[shaikparveen]

Why did you subtracted -2 on both sides

[davidpardi]

At 2:48 how come you didn’t distribute x into (x^2-x+1) since that is in the bracket with (x+1)?