# When is the Probability One-Half?

Number Theory Level pending

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the probability that the two marbles are of opposite color is $$\tfrac12$$. Let $$k_1<k_2<\cdots<k_{100}$$ be the $$100$$ smallest possible values for the total number of marbles in the urn. Compute the remainder when $k_1+k_2+k_3+\cdots+k_{100}$ is divided by $$1000$$.

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