4

Part of 2002 India Regional Mathematical Olympiad

Problems(1)

Source: Indian RMO 2002 Problem 4

Suppose the integers

1,2,\ldots 10

are split into two disjoint collections

a_1,a_2, \ldots a_5

b_1 , \ldots b_5

a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5

(i) Show that the larger number in any pair

\{ a_j, b_j \}

1 \leq j \leq 5

6

. (ii) Show that

\sum_{i=1} ^{5} | a_i - b_i|

= 25 for every such partition.