Sweden Junior Math Olympiad Problem - Algebra

Sweden Junior Math Olympiad Problem - Algebra

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[GillesF]

I followed your reasoning until (x - y)(x + y - 1) = 0 (at 01:50) and got stuck, unable to see how to progress. From this step, your reasoning is very interesting and subtle. Math is exciting and enjoyable! GREAT VIDEO !

[ARN]

Sir please solve, 3^x + 4^x = 5^x,
To find the value of x without using trial method.. (Answer is 2.. Everyone knows but how to solve)

[PS-mhts]

These are the given equations:

x²=5+y -- (i)
y²=5+x -- (ii)

After establishing that x+y=-1 -- (iii), we can also proceed like so:

Equations (i) and (iii) give:

x²=5+(-1-x)=4-x

or, x²+x-4=0 -- (iv)

Equations (ii) and (iii) give:

y²=5+(-1-y)=4-y

or y²+y-4=0 -- (v)

(iv) and (v) together imply that x and y are distinct roots (because it's given that x≠y) of the general quadratic t²+t-4=0. Thus, xy=product of roots=c/a=-4

[teobrian]

Can you guys explain me this
After x+y=-1, because (i) and (ii), i got x²y²=(5+x)(5+y)
Then I regarded xy as a
I got a²=a+20 or a²-a-20=0
The 2 zeros i got were -4 and 5
So is 5 a solution and if not, why