MathDB

Let

M

be a positive integer. At a party with 120 people, 30 wear red hats, 40 wear blue hats, and 50 wear green hats. Before the party begins,

M

pairs of people are friends. (Friendship is mutual.) Suppose also that no two friends wear the same colored hat to the party.
During the party,

X

and

Y

can become friends if and only if the following two conditions hold: [*] There exists a person

Z

such that

X

and

Y

are both friends with

Z

. (The friendship(s) between

Z,X

and

Z,Y

could have been formed during the party.) [*]

X

and

Y

are not wearing the same colored hat.
Suppose the party lasts long enough so that all possible friendships are formed. Let

M_1

be the largest value of

M

such that regardless of which

M

pairs of people are friends before the party, there will always be at least one pair of people

X

and

Y

with different colored hats who are not friends after the party. Let

M_2

be the smallest value of

M

such that regardless of which

M

pairs of people are friends before the party, every pair of people

X

and

Y

with different colored hats are friends after the party. Find

M_1+M_2

. [hide="Clarifications"]
[*] The definition of

M_2

should read, ``Let

M_2

be the smallest value of

M

such that...''. An earlier version of the test read ``largest value of

M

''.
Victor Wang

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