= Solution
A <state-price density> is a strictly positive <random variable> $Z$ with
$$
\boxed{\mathbb EZ=1/R,\qquad\mathbb E(ZS_1)=1.}
$$
On the stock-state space this means a positive integrable function $z(s)$ satisfying $\int_0^Cz(s)ds=C/R$ and $\int_0^Csz(s)ds=C$. Equivalently $dQ/dP=RZ$ is an equivalent probability measure with $\mathbb E_QS_1=R$. Such measures exist because $0<R<C$, for example exponential tilts of the uniform law with mean tuned to $R$.
By the <fundamental theorem of asset pricing>, any arbitrage-free claim price $p$ is represented by such a density in the enlarged market. For payout $S_1^2$,
$$
p=\frac1R\mathbb E_QS_1^2>\frac1R(\mathbb E_QS_1)^2=R,
$$
with strictness because equivalence preserves the stock's nondegenerate distribution. Also $S_1^2<CS_1$ almost surely, so $p<C$. Hence \b[$R<p<C$].
When $C=3R$, nonnegative wealth for all stock states requires $-1/2\le\pi\le1$. At $\pi=1$ wealth is $S_1$. The tangent inequality for the strictly concave square root gives, for $s>0$,
$$
\sqrt{R+\pi(s-R)}\le\sqrt s+\frac{(\pi-1)(s-R)}{2\sqrt s}.
$$
Uniform integration gives $\mathbb E\sqrt{S_1}=2\sqrt C/3$ and $\mathbb ES_1^{-1/2}=2/\sqrt C$, so the expected linear term vanishes exactly when $C=3R$. <Strict concavity> makes equality possible only for $\pi=1$. Thus \b[the unique optimal holding is $\pi^*=1$], including this boundary optimum without assuming an interior maximizer.
The <marginal utility price> density is
$$
Z_* =\frac{U'(S_1)}{R\mathbb EU'(S_1)}
=\frac{\sqrt C}{2R\sqrt{S_1}}.
$$
It is integrable and satisfies both pricing constraints: $\mathbb EZ_*=1/R$ and $\mathbb E(Z_*S_1)=C/(3R)=1$. Therefore
$$
\boxed{p_* =\mathbb E(Z_*S_1^2)
=\frac{\mathbb ES_1^{3/2}}{R\mathbb ES_1^{-1/2}}
=\frac{C^2}{5R}=\frac95R.}
$$
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