a and b are real numbers such that
a 4 + a 2 b 2 + b 4 = 9 0 0
a 2 + a b + b 2 = 4 5
What is the value of 2 a b ?
This problem is from the OMO-2012.
This problem is from the set "Olympiads and Contests Around the World -1". You can see the rest of the problems here .
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I got it!!
damn freaking hard. I almost forgot how to factorize this tricky polynomials.
a 4 + a 2 b 2 + b 4 = ( a 2 + a b + b 2 ) ( a 2 − a b + b 2 ) = 4 5 ( a 2 − a b + b 2 ) = 9 0 0 ⟹ a 2 − a b + b 2 = 2 0
∴ 2 a b = ( a 2 + a b + b 2 ) − ( a 2 − a b + b 2 ) = 4 5 − 2 0 = 2 5
Even I Did it this way
Problem:
a and b are real numbers such that
a 4 + a 2 b 2 + b 4 = 9 0 0
a 2 + a b + b 2 = 4 5
What is the value of 2 a b ?
Solution:
Okay, seeing all square terms in the first equauon, we are motivated to find some factorization using difference of squares. Notice that we can complete a square by adding an extra a 2 b 2 to the LHS: a 4 + a 2 b 2 + b 4 + a 2 b 2 = ( a 2 + b 2 ) 2 . So to return back to our original equation, by subtracting the extra term that we added: 9 0 0 = ( a 2 + b 2 ) 2 − ( a b ) 2 = ( a 2 + a b + b 2 ) ( a 2 − a b + b 2 ) . Amazing! We factorized our first equation, and that too contains a factor a 2 + a b + b 2 , the value of which we know from the second equation! So, simple substitution yields: 9 0 0 = 4 5 ( a 2 − a b + b 2 ) ⇒ a 2 − a b + b 2 = 2 0 . Now we are ready to fire the problem: 2 a b = a 2 + a b + b 2 − ( a 2 − a b + b 2 ) = 4 5 − 2 0 = 2 5 . Thus the problem is solved. :)
A very well-written solution! Voted up!
Hello,
given, a^4 + (ab)^2 + b^4 = 900 --------> a^4 + b^4 = 900 - (ab)^2 (1st)
a^2 + b^2 + ab = 45 ------> a^2 + b^2 = 45 - ab (2nd)
So i will take the below equation,
(a^2 + ab + b^2 )^2= (45)^2 -----> by squaring both sides,
a^4 + b^4 + 3(ab)^2 + 2ab(a^2+b^2) = 2025 (3rd),
by substituting (1st) & (2nd) into (3rd),
900 - (ab)^2 + 3(ab)^2 + 2ab(45-ab) = 2025
90ab = 2025 - 900
ab = 1125 / 90 , therefore for 2ab = 2 x (1125 / 90 ) = 25
Thanks...
a^2+b^2=45-a b so a^4+b^4+a^2b^2=900 (a^2+b^2)^2 -a^2 b^2=900 (45-a b)^2 - a^2 b^2=900 2025 - 90a b + a^2 b^2 - a^2 b^2 =900 1125 = 90a b 2a*b=25
(a^2 + b^2 )^2- a^2b^2=900
(45-ab)^2- a^2b^2=900
Solving u get ab=25/2 2ab= 25
Square the second equation. Luckily it includes the first equation inside of it. So basically just substitute and simplify and found the second equation in it as well!!! Then just divide.
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a 2 + a b + b 2 = 4 5
⇒ ( a 2 + a b + b 2 ) 2 = 4 5 2
⇒ a 4 + a 2 b 2 + b 4 + 2 a 3 b + 2 a 2 b 2 + 2 a b 3 = 2 0 2 5
⇒ ( 9 0 0 ) + 2 a b ( a 2 + a b + b 2 ) = 2 0 2 5
⇒ 2 a b × 4 5 = 1 1 2 5
⇒ 2 a b = 2 5