At first I thought the following problem looked simple, but I've had serious problems pinning it down:
Suppose that $\prod_{i=1}^k x_i$ is fixed. Then find the minimum value of $$\sum_{i=1}^k (1 - x_i)^k$$ in the range $0 \leq x_1, \dots, x_k \leq 1$.
A first conjecture is that it should be minimised when $x_1 = \dots = x_k$. This turns out to be false when the product is small, but I still suspect it's true when the product is reasonably large, say at least $1/2^k$. Does anyone see a simple or reasonable way to analyze this? You can get somewhere with Lagrange multipliers, but it doesn't seem to be enough. I also thought there might be an elementary argument I'm missing.
Perhaps I should flesh this out a little more. Using Lagrange multipliers, one can show that $x_i(1-x_i)^{k-1}$ is the same for each $i$. Since the function $x(1-x)^{k-1}$ has a unique maximum (at $x = 1/k$), this shows that the $x_i$ take at most $2$ values. But I've had trouble reducing the problem any further, partly because, as I said, there are cases where the minimum is not in the expected place.