6

Part of 1995 Putnam

Problems(2)

Source:

Suppose that each of

n

people writes down the numbers

1, 2, 3

in random order in one column of a

3\times n

matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums

a, b, c

of the resulting matrix be rearranged (if necessary) so that

a \le b \le c

. Show that for some

n \ge 1995

,it is at least four times as likely that both

b = a+1

c = a+2

a = b = c

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Source: Putnam

a>0

\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}

. Show that there are no three positive reals

a,b,c

\mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset

\mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N}

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