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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