2021/083) show that for no positive integers a, b,c can all the 3 expresssion $a^2+b+c, b^2+a + c , c^2 + a +b$ be perfect squares

 Proof:

The samllest square above $a^2$ is $a^2+2a+1$. so we must have for $a^2+b+c$ to be a prefect square

$a^2 + b+ c \ge (a+1)^2$

Or $b+c \ge 2a + 1\cdots(1)$

Similarly $c+a \ge 2b + 1\cdots(2)$

And $a + b  \ge 2c + 1\cdots(3)$

Adding above 3 equations we must have $2(a+b+c) >= 2(a + b+ c) + 3$ or $0 \ge  3$ which is contradiction

So above is impossible  or No solution exists