MathDB

4

Part of 2008 South East Mathematical Olympiad

Problems(2)

derived sequence

Source: China south east mathematical Olympiad 2008 day1 problem 4

7/15/2013
Let m,nm, n be positive integers (m,n>=2)(m, n>=2). Given an nn-element set AA of integers (A={a1,a2,⋯ ,an})(A=\{a_1,a_2,\cdots ,a_n\}), for each pair of elements ai,aj(j>i)a_i, a_j(j>i), we make a difference by aj−aia_j-a_i. All these Cn2C^2_n differences form an ascending sequence called “derived sequence” of set AA. Let Aˉ\bar{A} denote the derived sequence of set AA. Let Aˉ(m)\bar{A}(m) denote the number of terms divisible by mm in Aˉ\bar{A} . Prove that Aˉ(m)≥Bˉ(m)\bar{A}(m)\ge \bar{B}(m) where A={a1,a2,⋯ ,an}A=\{a_1,a_2,\cdots ,a_n\} and B={1,2,⋯ ,n}B=\{1,2,\cdots ,n\}.
combinatorics unsolvedcombinatorics
wave numbers

Source: China south east mathematical Olympiad 2008 day2 problem 8

7/15/2013
Let nn be a positive integer. f(n)f(n) denotes the number of nn-digit numbers a1a2⋯an‾\overline{a_1a_2\cdots a_n}(wave numbers) satisfying the following conditions : (i) for each ai∈{1,2,3,4}a_i \in\{1,2,3,4\}, ai≠ai+1a_i \not= a_{i+1}, i=1,2,⋯i=1,2,\cdots; (ii) for n≥3n\ge 3, (ai−ai+1)(ai+1−ai+2)(a_i-a_{i+1})(a_{i+1}-a_{i+2}) is negative, i=1,2,⋯i=1,2,\cdots. (1) Find the value of f(10)f(10); (2) Determine the remainder of f(2008)f(2008) upon division by 1313.
number theory unsolvednumber theory