This open problem appeared on the bulletins of Evans Hall at Berkeley this week.
I hope this doesn't violate StackExchange policy (the solution carries a $500 prize), but I thought why not re-post it here!
Definition 1 Let $x$ and $y$ be i.i.d. random variables distributed according to $F$ with continuous support over $R^{+}$. Let continuously differentiable function $g(x, y)$ exhibit:
1. Symmetry: $\forall x, y \in [0, \infty), g(x,y) = g(y, x).$
2 Synergy: $\forall x, y \in [0, \infty), g(x, y) > x+y.$
3 Increasing Differences: $\forall x, y \in [0, \infty), \frac{d}{dx}g(x,y) > 1.$
4 Initial Coincidence: $\forall x \in [0, \infty), g(x,0) = x.$
Conjecture 2 (Fixed Point)
$\exists \gamma \in (\frac{1}{2} , 1) : \gamma = E[\frac{x}{x+y} : g(x, y)\gamma > x$ AND $g(x,y)(1-\gamma) > y]. $
Remark 3
Conditions 1 and 2 on $g$ are required. Optional conditions 3 and 4 reduce difficulty but a solution that does not require them would be superior.
Remark 4
Contest ends on 5/31/13. If more than one correct solution is submitted, the first will be selected. If a counterexample is discovered, a prize will also be awarded.
Contact 5
Justin Tumlinson (Ifo Institute for Econonmic Research at Munich) [email protected] John Morgan (U.C. Berkeley) [email protected]