Indifference between 2 risky asset
It can be shown that under the assumption of strict concavity of utility function the investor will choose to split her wealth equally in the two assets. This happens because the variance on returns decreases by investing in two assets (assuming that the returns on two assets are independent).
We can consider two scenarios: first when the individual invests only in $X$ and the second where she splits her investment between $X$ and $Y$. It can be shown that in second case the variance of return goes down.
Let $r \equiv E(R_x) = E(R_y)$ and $\sigma^2 = Var(R_x) = Var(R_y)$
Assume that in second case, she invests $\alpha$ fraction of wealth in $X$ and $(1-\alpha) $ fraction in asset $Y$. Clearly $E(R) = r$. However:
\begin{align}
Var(R) &amp;= Var(\alpha R_x + (1-\alpha)R_y)\\
&amp;= (\alpha^2+(1-\alpha^2))\sigma^2 \tag{from independence} \\
&amp;=(2\alpha^2+1-2\alpha) \sigma^2
\end{align}
The investor will choose $\alpha$ to minimize the term in braces which happens at $\alpha=0.5$ and so $Var(R) = \sigma^2/2$
Now it is a well know result that a portfolio with mean preserving spread is less preferred by risk-averse (concave utility) individuals. So the individual will choose to split her wealth equally among the two assets.