MathDB

1

Part of 1995 China Team Selection Test

Problems(2)

p that cannot be represented in the form |3^{a} - 2^{b}

Source: China TST 1995, problem 1

5/17/2005
Find the smallest prime number pp that cannot be represented in the form ∣3a−2b∣|3^{a} - 2^{b}|, where aa and bb are non-negative integers.
modular arithmeticnumber theory
the distance between any 2 sets is at least 5

Source: China TST 1995, problem 4

5/17/2005
Let S={A=(a1,…,as)∣ai=0S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0 or 1,i=1,…,8}1, i = 1, \ldots, 8 \rbrace. For any 2 elements of SS, A={a1,…,a8}A = \lbrace a_1, \ldots, a_8\rbrace and B={b1,…,b8}B = \lbrace b_1, \ldots, b_8\rbrace. Let d(A,B)=∑i=18∣ai−bi∣d(A,B) = \sum_{i=1}{8} |a_i - b_i|. Call d(A,B)d(A,B) the distance between AA and BB. At most how many elements can SS have such that the distance between any 2 sets is at least 5?
combinatorics unsolvedcombinatorics