Express the statement “Every one has exactly one best friend “, as a logical expression involving predicates, quantifiers.. 
Express the statement “Every one has exactly one best friend “, as a logical expression involving predicates, quantifiers with a universe of discourse.

So far, I have tried the following:
$p(x)$: $X$ has exactly one best friend , where domain of $X$ is people 'for all $x$ $p(x)$'
 A: Writing $p(x,y)$ to mean that $y$ is a best friend of $x$, you can express that everyone has a best friend by the sentence
$$
\forall x\ \exists y\ p(x,y)
$$
To express unicity, it suffices to add that everyone has at most one best friend, using the sentence
$$
\forall x\ \forall y\ \forall z\ \bigl(p(x,y) \wedge p(x,z) \to y = z\bigr)
$$
A: Your suggestion is a very inaccurate formalization, because it just absorbs almost the entire sentence into a single predicate; but the point of formalizing a sentence is to reflect as accurately as possible the logical structure of the proposition using quantification and predication.
Instead, you should explicitly express the two quantifiers in the sentence ("everyone", "exactly one") and the fact that there are two persons involved in a friendship: There should be a quantifier over every person and a quantifier over their best friend, and "best friend" should be a two-place predicate of which one argument is the person and the other is their best friend.
"Exactly one $P$" can be expressed as "There exists someone such that the people who are $P$ are exactly that person" (i.o.w., there is a person who is $P$ and there is noone who is also $P$ but different from that person):
$$\text{exactly one $P$} \equiv \exists z \forall y (P(y) \leftrightarrow y = z)$$
Now for the placeholder $P(y)$ insert a two-place predicate $P(x,y)$ expressing that $y$ is the best friend of $x$, and combine it with the universal quantifier over people to express that everyone has such a friend.
A: Let $F(p,q)$ denote the degree of friendship between individuals $p$ and $q$, such that the better friends they are, the higher the value. The friendship with oneself is minus infinity.
$$\forall p:\exists b:F(p,b)=\max_q F(p,q)\implies \forall q\ne b:F(p,q)<F(p,b)$$
or simply
$$\forall p:\exists! b:F(p,b)=\max_q F(p,q).$$
A: What you've done is correct. But if you want to write it in a fancier way, you could use  the $\exists !$ which reads as: 'there exists a unique'.
Let $x\in U$ and $y\in U$, where $U=\{\text{All people} \}$. And let $P(x,y):$ $y$ is the best friend of $x$.

 \begin{equation}\forall x \exists ! y \ (x\neq y \wedge P(x,y))\end{equation}

This would require that the best friend of a person can't be the person itself. If you would like to allow for a person to be it's own best friend, it would look like

 \begin{equation}\forall x \exists ! y \ P(x,y)\end{equation}

Hope this makes it more clear to you.
