Question 197
Alicia writes down $a$ distinct integers on a piece of paper and Britt writes down $b$ distinct integers on another piece of paper. Alicia wrote down at least one integer that Britt did not write down, and Britt wrote at least one integer down that Alicia did not write down. Vera counts the number of distinct integers on the two pieces of paper; let this number of distinct integers be $v$. Daan counts how many of the integers that have been written down by Alicia, have also been written down by Britt; let $d$ be this number. For example, if Alicia wrote down $1$, $2$, and $5$, and Britt wrote down $2$, $5$, $7$, and $8$, then we have $a = 3$ and $b = 4$ while $v = 5$ and $d = 2$.
a. Find an example for which $a = b = 2022$ and $a \cdot b = d \cdot (v + d)$.
b. Is it possible that $a \cdot b = d \cdot (v + 4)$? Give an example or prove that it is impossible.
c. Is it possible that $a \cdot b = d \cdot v$? Give an example or prove that it is impossible.
a. Find an example for which $a = b = 2022$ and $a \cdot b = d \cdot (v + d)$.
b. Is it possible that $a \cdot b = d \cdot (v + 4)$? Give an example or prove that it is impossible.
c. Is it possible that $a \cdot b = d \cdot v$? Give an example or prove that it is impossible.