P(x) is a polynomial with integral coefficients such that for four distinct integers $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d},\mathrm{P}\left(\mathrm{a}\right)=\mathrm{P}\left(\mathrm{b}\right)=\mathrm{P}\left(\mathrm{c}\right)=\mathrm{P}\left(\mathrm{d}\right)=3$. If $\mathrm{P}\left(\mathrm{e}\right)=5$ (e is an integer), then

$\mathrm{P}\left(\mathrm{a}\right)=\mathrm{P}\left(\mathrm{b}\right)=\mathrm{P}\left(\mathrm{c}\right)=\mathrm{P}\left(\mathrm{d}\right)=3$

$\Rightarrow \mathrm{P}\left(\mathrm{x}\right)=3$ has a, b, c, d as its roots

$\Rightarrow \mathrm{P}\left(\mathrm{x}\right)-3=(\mathrm{x}-\mathrm{a})(\mathrm{x}-\mathrm{b})(\mathrm{x}-\mathrm{c})(\mathrm{x}-\mathrm{d})\mathrm{Q}\left(\mathrm{x}\right)$

[Q(x) has integral coefficient]

Given P(e) = 5, then

$(\mathrm{e}-\mathrm{a})(\mathrm{e}-\mathrm{b})(\mathrm{e}-\mathrm{c})(\mathrm{e}-\mathrm{d})\mathrm{Q}\left(\mathrm{e}\right)=5$

This is possible only when at least three of the five integers

$(\mathrm{e}-\mathrm{a}),(\mathrm{e}-\mathrm{b}),(\mathrm{e}-\mathrm{c}),(\mathrm{e}-\mathrm{d}),\mathrm{Q}\left(\mathrm{e}\right)$ are equal to 1 or -1. Hence,
two of them will be equal, which is not possible. Since a, b, c,
d are distinct integers , P(e) = 5 is not possible.