CAT 2006: LRDI - Erdos Number скачать в хорошем качестве

CAT 2006: LRDI - Erdos Number

Mathematicians are assigned a number called Erdös number (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös number is illustrated below: Suppose a mathematician X has co-authored papers with several other mathematicians. From among them, mathematician Y has the smallest Erdös number. Let the Erdös number of Y be y. Then X has an Erdös number of y+1. Hence any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity. In a seven day long mini conference organized in memory of Paul Erdös, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some research problems. At the beginning of the conference, A was the only participant who had an infinite Erdös number. Nobody had an Erdös number less than that of F. On the third day of the conference F co-authored a paper jointly with A and C. This reduced the average Erdös number of the group of eight mathematicians to 3. The Erdös numbers of B, D, E, G and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös number of the group of eight to as low as 3. At the end of the third day, five members of this group had identical Erdös numbers while the other three had Erdös numbers distinct from each other. On the fifth day, E co-authored a paper with F which reduced the group's average Erdös number by 0.5.The Erdös numbers of the remaining six were unchanged with the writing of this paper. No other paper was written during the conference. Q1. How many participants in the conference did not change their Erdös number during the conference? (1) 2 (2) 3 (3) 4 (4) 5 (5) cannot be determined Q2. The person having the largest Erdös number at the end of the conference must have had Erdös number (at that time): (1) 5 (2) 7 (3) 9 (4) 14 (5) 15 Q3. How many participants had the same Erdös number at the beginning of the conference? (1) 2 (2) 3 (3) 4 (4) 5 (5) cannot be determined Q4. The Erdös number of C at the end of the conference was: (1) 1 (2) 2 (3) 3 (4) 4 (5) 5 Q5. The Erdös number of E at the beginning of the conference was: (1) 2 (2) 5 (3) 6 (4) 7 (5) 8