MathDB

Peter has a deck of

1001

cards, and with a blue pen he has written the numbers

1,2,\ldots,1001

on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card

C

, he took

500

consecutive cards following

C

(clockwise order), and denoted by

f(C)

the number of blue numbers written on those

500

cards that are greater than the blue number written on

C

itself. After all, he wrote this

f(C)

number on the top side of the card

C

with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.

1