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1 0 1 x 3 − 2 0 1 9 x y + 1 0 1 y 3 = 1 0 0
If all the positive integer solutions of the equation above are ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) , ⋯ ( x n , y n ) , find the value of x 1 + y 1 + x 2 + y 2 + x 3 + y 3 + ⋯ + x n + y n .
The answer is 20.
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2 solutions
What, if x 3 − 2 0 x y + y 3 < 0 ?
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Happens when (x+y)(x^2-xy+y^2)<20xy. Let x+y=a and xy=b. So it happens when
a^3-3ab<20b.By AM GM, a^2 is greater than or equal to 4b. So it happens when a^2(a-20)<0. So a<20. Simple bruteforcing gives this is impossible
that bruteforcing is not simple. But still it can be done in a few minutes
Oh yes i didnt notice it!!!
Here is an alternate solution. We are given that x, y are positive integers such that. 101x^3-2019xy+101y^3=100. The key trick of this problem is to note that 2019=-1(mod101). So the given equation can be written as
101(x^3-20xy+y^3)=100-xy. So 100-xy is divisible by 101. So 100-xy=0 as 100-xy<101 because xy>0. So xy=100. Hence, x^3-20xy+y^3=0 so.
x^3+y^3=2000. Let x+y=z. Then with a little factorising, z(z^2-300)=2000. So z^3-300z-2000=0. So ((z+10)^2)(z-20)=0. Hence, z=20 because z>0. Simple solving gives x=y=10
Since x and y are positive, x 3 + y 3 ≥ 2 ( x y ) 1 . 5 . So 1 0 0 = 1 0 1 ( x 3 + y 3 ) − 2 0 1 9 x y ≥ 2 0 2 ( x y ) 1 . 5 − 2 0 1 9 x y . Simplifying we get ( x y − 1 0 0 ) ( 4 0 8 0 4 x 2 y 2 + 4 0 3 9 x y + 1 0 0 ) ≤ 0 . Since 4 0 8 0 4 x 2 y 2 + 4 0 3 9 x y + 1 0 0 > 0 , therefore x y − 1 0 0 ≤ 0 , or x y ≤ 1 0 0 . So it suffices to check only ten values of any one of x and y , say x = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 . A quick check shows that the only values of x and y satisfying the given equation are x = 1 0 , y = 1 0 , and so x + y = 2 0
Superb solution
The equation can be rewritten as: 1 0 1 ( x 3 − 2 0 x y + y 3 ) + x y = 1 0 0 since x,y are positive integers, it forces : ( x 3 − 2 0 x y + y 3 ) = 0 because if ( x 3 − 2 0 x y + y 3 ) = 1 then L.H.S>R.H.S hence it has to be 0. So the equation reduces to x y = 1 0 0 . A little bit of bruteforcing gives x=y=10 as the only solution. And hence the answer is 10+10=20.