Ukraine | A Nice Algebra Problem | Math Olympiad
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- Опубліковано 28 лис 2024
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Find the value of x and y?
How to solve √8^ √x - √5^√y = 39
In this video, we'll show you How to Solve Math Olympiad Question A Nice Radical Equation √8^ √x - √5^√y = 39 in a clear , fast and easy way. Whether you are a student learning basics or a professtional looking to improve your skills, this video is for you. By the end of this video, you'll have a solid understanding of how to solve math olympiad exponential equations and be able to apply these skills to a variety of problems.
#matholympiad #maths #math #algebra
Without constraints, the first solution is obvious x = y = 16. A second less obvious lower bound solution when y = 0 is x =(2*log(39)/log(8))^2. With a family of other non-integer solutions, where y>0, x = (2*lg(39+(sqrt(5))^Sqrt(y))/lg(8))^2
much easier when you recognize that 39 = 64 - 25 = 8^2 - 5^2. and that's the first thing i checked to see if a simple solution exists.
Very nice! ❤
Yes. Another way to see this is that, since 1 and 3 are the only factors of 39 less than its square root, the only possible gnomon sequences (ie differences between perfect squares) adding to 39 are:
a) 39 = (19 + 20) = 20^2 - 19^2
b) 39 = 11 + 13 + 15 = (5+6) + (6 +7) + (7+8) = 8^2 - 5^2
The second one quickly yields the desired answer.
Вапще-то должно было быть так:
x = Украина, y = Крым
Задача: найти (x - y)
(x,y)=( (16/9)(log(2)20)^2,16(log(5)19)^2 )
Very nice! ❤
Why did you limit a and b to integers? It was not stated in the problem
In genral, a and b are integers. ❤
In general, to find value of n variables, you need n pieces of information. Thus, for 2 variables, you usually need 2 pieces of information, such as two equations. In this case, we have 2 variables and only 1 equation. Thus, there is a missing piece of information, which is CASUALLY assumed during the solution. This assumption is that both x and y are real / integers. Ideally, this assumption should have been stated in the very beginning, because we have ignored non-real as well as non-integral solutions (which may or may not exist, but we never checked for them anyway).
First view : 64 - 25 = 39
You are right! ❤
This took about 5 seconds to solve. Math Olympiad? Hahahaha……
Very Nice! ❤