The constant product is used in decentralized finance. It is used to calculate how much someone gets for currency $Y$ when giving funds of currency $X$. $X$ and $Y$ indicate the size of funds in the corresponding currencies, $z$ stands for the amount of funds of currency $X$ given into the pool, $a$ stands for the amount of funds received in currency $Y$.
$k = X * Y$
$a = Y - k / (X + z)$
I want to split $Y$ into $Y_1$ and $Y_2$:
$k = X * (Y_1 + Y_2)$
$a = (Y_1 + Y_2) - k / (X + z)$
Now my question is: How much does someone have to give to exactly receive $Y_1$? What the size of $z$ so that $a = Y_1$?
$X = 100, Y_1 = 50, Y_2 = 50, z = 10$
$k = 100 *(50+50)$
$a = (50+50) - 10000 / (100+10) = 9.09...$
So here the question is how to calculate $z$ so that $a=Y_1=50$
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