# Bizarre Bag of Balls

Two bags of balls each contain balls numbered $$1, 2, \ldots , n$$. For how many values of $$n$$ such that $$1 \leq n \leq 500$$, does there exist a $$k$$ such that the probability of getting a sum of $$n+1$$ when a ball is selected from each bag is equal to the probability that the sum is $$\leq k$$ when a ball is selected from each bag?

Details and assumptions

Since a ball is selected from each bag, there are 2 balls selected, and we are looking at the sum of these values.

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