Using Quantifiers to express "At least two"

Question

I have been stumped trying to solve problem e. I am confused as how to state "At least two students..." using quantifiers!

I know if it were ONE student it would be: ∃x,y(Q(x,Jeopardy)).
And all students would be:∀x,y(Q(x,Jeopardy))

So how can I expand on this?

Answer 1

Suppose $P(x)$ denotes the predicate that $x$ has been on Jeopardy.

One can assert $\exists x,y$ s.t. $P(x) \land P(y) \land x\neq y$.

Answer 2

As another way to put this, this is the negation of the "there exists at most one" quantifier, which I usually denote as $\;\langle !x :: P(x) \rangle\;$. Looking at another answer of mine and doing some basic manipulations, that negation would be $$ \langle \forall z :: \langle \exists x :: x \not= z \land P(x) \rangle \rangle $$ where of course $\;P(x)\;$ stands for "$\;x\;$ has been on Jeopardy", and $\;x,z\;$ range over the school students.

Perhaps that looks counterintuitive, but it is correct: you could read it as, "Whichever $\;z\;$ you take away (even if $\;P\;$ holds for it), you will still have at least one $\;x\;$ left that satisfies $\;P\;$."

Note how this is the only formula which contains $\;P\;$ only once: the usual formulations have two occurrences of $\;P\;$.