$(a)$ Ana chooses $17,19,23,22,11$ and puts these numbers to $(1,1),(2,2),(3,3),(4,4),(5,5)$ ,respectively and Enrique can't reach his/her goal obviously. $(b)$ $4$ is very small number to avoid making perfect square. Since there will be at least $1$ column which will be filled completely by Enrique, he/she can find $5$ numbers whose product is perfect square. So doing bash gives the result. Using some things like "at least $1$ of the numbers that Ana choose should be perfect square" and so on, can make the bash shorter,but anyway you should do bash ( at least on $v_p$s)

For (b) same reasoning for at least $1$ column which will be filled completely by Enrique. Then consider the sets with a square product and doing some bashing... : $1,4,9,16,25$ at least one must be picked at least one free... then from $2,10,12,15$ at least one must be picked so there are at least two from the squares... WLOG $1,4$ now consider the other sets $3,7,21$ $11,18,22$ $5,20$ $6,24$ and at most two picks from ana so ...