In response to our policy on homework questions, I will work on the following abstract version of your question:
A single prize is to be allocated by an all-pay auction. Assume the value of the prize is 1 to both bidders, and this is common knowledge. Both bidders have a budget constraint $m$. If there is a tie, each bidder receives the prize with $\frac{1}{2}$ probability. Find a Nash Equilibrium in this game.
We shall focus on the symmetric equilibrium in this game, that is, both bidders are using the same equilibrium strategy. Before we get started, let's make two observations:
Claim 1: Whenever there is a mass point in the equilibrium strategy (i.e. there is positive probability to submit one bid), this mass should be put on the upper bound $m$. For if not, one bidder can always deviate to a slightly higher bid and obtain higher surplus.
Claim 2: Whenever a bidder randomizes, the bid should be uniformly drawn from $(0,t)$, and the probability that this happens should be $t$. This is because only in this case, a bidder is indifferent between any bid $b\in(0,t)$:
$$\text{Expected payoffs}=t\cdot \frac{b}{t}\cdot 1-b=0.$$
Therefore, we can guess that the equilibrium strategy takes the form
$$t\cdot \text{Uniform}(0,t)+(1-t)\cdot\delta_m.$$
That is, with probability $t$, a bidder submits a bid uniformly drawn from $(0,t)$, and with complementary probability she submits $m$. The only thing left is to pin down $t$. Note that if this is an equilibrium strategy, a bidder should be indifferent between any bid $b\in(0,t)\cup\{m\}$. Therefore, the payoffs from submitting $m$ should also be 0, i.e.
$$t\cdot 1+(1-t)\cdot\frac{1}{2}-m=0,$$
which implies $t=2m-1$. (Notice that for the existence of such an equilibrium, we must have $2m>1$.)
Therefore, a Nash Equilibrium in this game is that both bidders play the strategy
$$(2m-1)\cdot \text{Uniform}(0,2m-1)+(2-2m)\cdot\delta_m.$$
How does it map back to the original problem? I believe the OP can take it from here.
