MathDB

3

Part of 1999 China Team Selection Test

Problems(2)

China TST 1999 any 3-element subset

Source: China TST 1999, problem 3

5/22/2005
Let S={1,2,…,15}S = \lbrace 1, 2, \ldots, 15 \rbrace. Let A1,A2,…,AnA_1, A_2, \ldots, A_n be nn subsets of SS which satisfy the following conditions: I. ∣Ai∣=7,i=1,2,…,n|A_i| = 7, i = 1, 2, \ldots, n; II. ∣Ai∩Aj∣≤3,1≤i<j≤n|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n III. For any 3-element subset MM of SS, there exists AkA_k such that M⊂AkM \subset A_k. Find the smallest possible value of nn.
combinatorics unsolvedcombinatorics
Permutation Minimum and Maximum

Source: China TST 1999, problem 6

5/22/2005
For every permutation τ \tau of 1,2,…,10 1, 2, \ldots, 10, \tau \equal{} (x_1, x_2, \ldots, x_{10}), define S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|. Let x_{11} \equal{} x_1. Find I. The maximum and minimum values of S(τ) S(\tau). II. The number of τ \tau which lets S(τ) S(\tau) attain its maximum. III. The number of τ \tau which lets S(τ) S(\tau) attain its minimum.
combinatorics unsolvedcombinatorics