France - Math Olympiad Challenge | Best Trick
Vložit
- čas přidán 14. 06. 2024
- You should know this approach. Many goes WRONG! Solution
What do you think about this problem? Don't forget to like and subscribe to my channel for more helpful math tricks. Thanks For Watching!
Check out my latest video (Math Olympiad Challenge): • Germany - Math Olympia...
If you're reading this ❤️. Thank You!
Hello My Friend ! Welcome to my channel. I really appreciate it!
@higher_mathematics
#maths #math
since x≠0 we have
(1-x)/x=ξ with ξ one of the 6th roots of the unit. Then
1/x = ξ+1
x= 1/(ξ+1)
We can find all the solutions knowing
ξ=exp(2πin/6)
=cos(πn/3) + i sin(πn/3)
There is a substitution that turns this and several similar problems many are posting about into something of the form (a + b)^(2 * n) = (a - b)^(2 * n). (b will be a constant upon performing this substitution.)
For example, with a 6th power, using the Binomial Coefficients Formula:
a^6 + 6 a^5 b + 15 a^4 b^2 + 20 a^3 b^3 + 15 a^2 b^4 + 6 a b^5 + b^6 = a^6 - 6 a^5 b + 15 a^4 b^2 - 20 a^3 b^3 + 15 a^2 b^4 - 6 a b^5 + b^6.
Much cancels out.
0 = 12 a^5 b + 40 a^3 b^3 + 12 a b^5 = 4ab(3a^4 + 10a^2 b^2 + 3b^4) =4ab(3 a^2 + b^2) (a^2 + 3 b^2).
1/2 is the obvious real solution. There will be 5 complex solutions.
❤❤
It can be done much simpler: the absolute value of (x) is equal to the absolute value of (x-1)....having three sets of values: (minus infinity ; 0) , (0;1) ang (1; infinity) you preper simplest solution
Solution of X[2,3] must include ±, not just +.
1/2
VERY longwinded!
I find it incredibly annoying that the obvious and sufficient solution x=1/2 is deemed insufficient or "wrong".
The way this is presented here, there is no mention of requiring ALL the roots including complex roots. It may well be assumed by the questioner and those whose (singular) "Solution" only finds the one real root may well get it "wrong" in the examiners mind, but the fact remains that the question is deficient because it does not clearly state what constitutes the correct "solution".
This omission of relevant detail by assumption in questions to make the obtusely simple far more complicated than the question actually requires is a technique exploited to the hilt in elitist testing particularly where poor English grammar is also evident. So x=1/2 is not a wrong answer if that is offered as THE solution because the question as it is presented here did not ask for anything more than a "Solution".
If you ask an incomplete question, you'll get a incomplete answer. A correct answer to an incomplete question is not by any means "wrong", at all. Quite the contrary. The questioner is "wrong" by marking "wrong" a correct and complete answer to an incomplete question.