Let's eat cake

[BMAD]

At a wedding reception n guests have assembled into m groups to converse. (the groups are not necessarily equal sized.) The host is preparing m square cakes, each with an ornate ribbon adorning its perimeter, to serve to the m groups. No guest is allowed to have more than 25 cm^{2 of cake. Prove that no more than 20*sqrt(mn) cm of ribbon is needed to embellish the m cakes.}

[Barcallica]

Let a1, a2, ....am be the sides of the cakes. we need to prove 4*(a1+a2+......+am)<=20*sqrt(mn) ------> (a1+a2+......+am)<=5*sqrt(mn) -------> (a1+a2+......+am)2<=25mn We know 25n>=a12+a22+.........+am2 so we need to prove m*(a12+a22+.........+am2)>=(a1+a2+......+am)2 which can be easily proven by writing equation a2+b2>=2ab; (m-1)*(m-2)/2 times.

[Brainy Binary]

I guess that is rigth