Update 12/2015. This problem is now solved in the preprint Optimal Surviving Strategy for Drifted Brownian Motions with Absorption by Wenpin Tang and Li-Cheng Tsai.
First consider K particles performing independent Brownian motion on the positive half-line, with state 0 absorbing, and each particle started at state 1 at time 0. Now suppose we have at our disposal a unit quantity of positive drift, which we can distribute amongst the non-absorbed particles at time t according to any control policy we choose. Let F(K) be the maximum, over all control policies, of the expectation of the number of particles that survive forever.
It is a challenging exercise in stochastic calculus to show that F(K) grows as order K^{1/2}.
Problem. Prove F(K) = (c + o(1))K^{1/2} for some explicit constant c.
Problem. Analyze the policy of assigning drift 1 to the leftmost particle.
Here are some unpublished 2002 notes on this problem. Related work appears in One-dimensional Brownian particle systems with rank dependent drifts by Soumik Pal and Jim Pitman, and in The advantage of capitalism vs. socialism depends on the criterion by H. P. Mckean and L. A. Shepp.
History. The model was inspired by a (now discontinued) child's board game Up The River -- Race to the Harbor by Ravensburger. I gave a talk on this topic in February 2002 at a Berkeley meeting to honor David Blackwell and Lester Dubins. That talk prompted the "related work" above.