We first note that there is a useful relation between $a$, $b$, $d$, and $v$. The total number of integers on the two pieces of paper is $a + b$, the number of integers on Alicia's piece of paper plus the number of integers on Britt's piece of paper. This, however, also equals $v + d$: the total number of distinct integers, plus the total number of integers that have been written down twice. Hence, we get that $a + b = v + d$.

a. We choose $a = b = 2022$ and look for a solution to $a \cdot b = d \cdot (v + d)$. We use the fact that $a + b = v + d$. This means that we are looking for solutions to $a \cdot b = d \cdot (a + b)$. If we substitute $a = b = 2022$, then we find that $2022 \cdot 2022 = d(2022 + 2022) = d \cdot 2 \cdot 2022$, so $d = 1011$. Together with $a + b = v + d$, we find that $2022 + 2022 = v + 1011$, so $v = 3033$. This situation happens for example if Alicia writes down the numbers $1$ to $2022$, and Britt writes down the numbers $1012$ to $3033$.

b. With a little bit of trying, and by choosing $d$ not too large, we find that $a = b = 3$, $d = 1$, and $v = 5$ is a solution: $3 \cdot 3 = 1 \cdot (5 + 4)$. The numbers also satisfy the equation $a + b = v + d$. This situation can occur if Alicia writes down the numbers $1$, $2$, and $3$, and Britt writes down the numbers $3$, $4$, and $5$, for example.

c. Suppose that there are numbers such that $a \cdot b = v \cdot d$. We already deduced that $a + b = v + d$, or $v = a + b - d$. Substituting this yields
$$ab = vd = (a + b - d)d = ad + bd - d^2.$$
If we now subtract $ad$ from both sides of this equation, we find $ab - ad = bd - d^2$, so $a(b - d) = d(b - d)$. Because Britt wrote down at least one number that Alicia did not write down, we have $b > d$. Therefore, we can divide the equation $a(b - d) = d(b - d)$ by the positive number $b - d$, and we find that $a = d$. On the other hand, Alicia wrote down at least one number that Britt did not write down, hence $a > d$. This gives a contradiction and hence there cannot exist numbers such that $a \cdot b = v \cdot d$.