Combinatorics Problems

69. (China TST 2000, Problem 2) Given positive integers k, m, n such that
1 ≤ k ≤ m ≤ n. Evaluate
n∑
i=0
1
n + k + i · (m + n + i)!
i!(n − i)!(m + i)! .
70. (China Team Selection Test 2002, Day 1, Problem 3) Seventeen
football fans were planning to go to Korea to watch the World Cup football
match. They selected 17 matches. The conditions of the admission tickets they
booked were such that
• One person should book at most one admission ticket for one match;
• At most one match was same in the tickets booked by every two persons;
• There was one person who booked six tickets.
How many tickets did those football fans book at most?
71. (China Team Selection Test 2003, Day 1, Problem 2) Suppose
A ⊆ {0, 1, . . ., 29}. It satisfies that for any integer k and any two members
a, b ∈ A(a, b is allowed to be same), a+b+30k is always not the product of two
consecutive integers. Please find A with largest possible cardinality.
72. (China Team Selection Test 2003, Day 2, Problem 2) Suppose
A = {1, 2, . . ., 2002} and M = {1001, 2003, 3005}. B is an non-empty subset of
A. B is called a M-free set if the sum of any two numbers in B does not belong
to M. If A = A 1 ∪ A 2 , A 1 ∩ A 2 = ∅ and A 1 , A 2 are M-free sets, we call the
ordered pair (A 1 , A 2 ) a M-partition of A. Find the number of M-partitions of
A.
1.3.2 Vietnam IMO Team Selection Test Problems
73. (Vietnam TST 1994, Problem 6) Calculate
T = ∑ 1
n 1 ! · n 2 ! · · · ·n 1994 ! · (n 2 + 2 · n 3 + 3 · n 4 + . . . + 1993 · n 1994 )!
where the sum is taken over all 1994-tuples of the numbers n 1 , n 2 , . . .,n 1994 ∈
N ∪ {0} satisfying n 1 + 2 · n 2 + 3 · n 3 + . . . + 1994 · n 1994 = 1994.
74. (Vietnam TST 1996, Problem 5) There are some people in a meeting;
each doesn’t know at least 56 others, and for any pair, there exist a third one
who knows both of them. Can the number of people be 65?
75. (Vietnam TST 1996, Problem 1) In the plane we are given 3 · n points
(n >1) no three collinear, and the distance between any two of them is ≤ 1.
Prove that we can construct n pairwise disjoint triangles such that: The vertex
set of these triangles are exactly the given 3n points and the sum of the area of
these triangles < 1/2.
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