Hard

Brainteasers

$7$ men labeled $A-G$ play a coin flip game that starts with player $A$. At each turn, the player will flip a coin. If it appears heads, then the player must double the stack of each other player that plays the game, taking the money from their own stack. In other words, the payout to each of the other $6$ players is the value of their current stack. Otherwise, nothing happens. The sequence $HHHHHHH$ is observed and each of the $7$ players ends up with $\$1.28$ as their final stack. If $x_1,\dots,x_7$ represent the amount that each player started with, find the value of $10000(x_1^2 + \dots + x_7^2)$. For example, if player $A$ starts with $\$5.31$, then $x_1 = 5.31$.

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Hint