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Find the number of ordered triples (a,b,c) of integers such that  0≤a,b,c≤5  and
a^2b+b^2c+c^2a-ab^2-bc^2-ca^2=0

 Feb 16, 2024
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Factoring the equation, we get a2(b−c)+b(b2−c2)+c(c2−a2)=0. This can be rewritten as (b−c)(a2−ab+c)+b(b−c)(b+c)=0.

 

Therefore, (b−c)(a−b)(a−c)+b(b−c)(b+c)=(b−c)(a−b)(a−c)(b+c)=0. So b−c or a−b, a−c, or b+c must equal 0.

 

If b−c=0, then b=c. Since 0≤b,c≤5, it follows that b=c=0,1,2,3,4,5. In each case, we can choose a to be any integer between 0 and 5, inclusive.

 

This gives us 6⋅6=36​ possibilities.

 

Therefore, the answer is 36.

 Feb 16, 2024

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