Germany | Can you solve ? | Math Olympiad Simplification without Calculators
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- Опубликовано: 8 сен 2024
- Germany | Can you solve ? | Math Olympiad Simplification | Calculators NOT allowed
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This isn't as rigorous as your solution but here's an alternative.
The problem suggests a solution of the form x = a + b√2.
Then x^3 = a^3 + 3a^2 * b√2 + 6a * b^2 + 2√2b^3 = 55 + 63√2
Regrouping x^3 = a^3 + 6a*b^2 + √2*(3a^2*b + 2b^3)
Alternately x^3 = a(a^2 + 6b^2) + √2*b(3a^2 + 2b^2)
If the proposed solution is valid, the first term should equal 55 and the second should equal 63√2.
Or 55 = a(a^2 + 6b^2) and 63 = b(3a^2 + 2b^2)
The first equation above indicates that a > 0 since the terms inside the parentheses are positive. Similarly the second equation indicates that b>0.
So the first equation factors into:
a a^2 + 6b^2
1 55
5 11
11 5
55 1
b can be determined for each pair. It equals 3 for the first and is imaginary for the others, i.e. only the first pair gives a valid solution of the first equation.
Substituting a = 1 and b = 3 into the second equation gives 63, which satisfies it.
So the solution is 1 + 3√2
Wow! Nice solution from awesome approach. Thanks 👍💯 for your input and support
If this is indeed a "math Olympiad" then it HAS to have an elegant solution of the form S = a + b*sqrt(2) where a and b are integers. Raise left and right to power 3 and determine that a = 1 and b = 3.
Diffecult to under stand why you but y from where you bring
He introduced the conjugate!
I like your explanation. Thanks 👍💯 for your input
@@superacademy247 I hope you to ignore any negative comments on this. For similar problems, your way is somewhat new. I like your way, too!
I will dieeeee by solving this complex solution...
15mins long explnation for a single question..... !!!!!!!!!!! (
A whole Lot of arkward manipulations, senseless!
Show us your method. Senseless how? It's produced the solution. What do you mean awkward? Why use insultive language with no apparent reason? Why do you want me to think like you while I'm an individual?
Be careful to choose your word for other's hard works. I think you have no right to judge one's work simply by "senseless". You obviously awe him an apology!