MathDB

6

Part of 2010 NZMOC Camp Selection Problems

Problems(2)

each person knew exactly 22 others at a strange party

Source: New Zealand NZMOC Camp Selection Problems 2010 Juniors 6

9/18/2021
At a strange party, each person knew exactly 2222 others. For any pair of people XX and YY who knew one another, there was no other person at the party that they both knew. For any pair of people XX and YY who did not know each other, there were exactly six other people that they both knew. How many people were at the party?
combinatorics
3 of 8 integers from {1, 2, . . . , 16, 17} such that a_i - a_j = k

Source: New Zealand NZMOC Camp Selection Problems 2010 Seniors 6

9/18/2021
Suppose a1,a2,...,a8a_1, a_2, . . . , a_8 are eight distinct integers from {1,2,...,16,17}\{1, 2, . . . , 16, 17\}. Show that there is an integer k>0k > 0 such that there are at least three different (not necessarily disjoint) pairs (i,j)(i, j) such that ai−aj=ka_i - a_j = k. Also find a set of seven distinct integers from {1,2,...,16,17}\{1, 2, . . . , 16, 17\} such that there is no integer k>0k > 0 with that property.
number theory