设m为正整数，数列a1,a2,···,a_{4m+2}是公差不为0的等差数列，若从中删去两项ai和aj(i＜j)后剩余的4m项可被平均分为m组，且每组的4个数都能构成等差数列，则称数列a1,a2,···,a_{4m+2}是(i,j)-可分数列。 （1）写出所有的(i,j)，1≤i,j≤6，使数列a1,a2,···,a6是(i,j)-可分数列; （2）当m≥3时，证明:数列a1,a2,···,a_{4m+2}是(2,13)-可分数列; （3）从1,2,3,···,4m+2中一次任取两个数ī,j(i＜j)，记数列a1,a2,···,a_{4m+2}是(i,j)-可分数列的概率为Pm，证明:Pm＞1/8。

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We have an arithmetic sequence problem. Given a sequence with 4m+2 terms, we remove two specific terms and check if the remaining 4m terms can be divided into m groups, where each group of 4 terms forms an arithmetic sequence. This is called an (i,j)-divisible sequence. Let's explore this concept step by step. For part one, we need to find all pairs (i,j) where the sequence a₁ through a₆ is (i,j)-divisible. Since m equals 1, we have 6 terms total, and after removing 2 terms, the remaining 4 terms must form a single arithmetic sequence. We systematically check all possible pairs. For example, if we remove a₂ and a₅, we need a₁, a₃, a₄, a₆ to form an arithmetic sequence. After checking all cases, we find that only (2,5) and (3,4) work. For part two, we prove that when m is at least 3, the sequence is (2,13)-divisible. We remove the second and thirteenth terms from our arithmetic sequence. The key insight is to group the remaining terms systematically. We form groups where group k contains terms a₄ₖ₋₃, a₄ₖ₋₂, a₄ₖ₋₁, and a₄ₖ. Since the original sequence has common difference d, each group maintains this arithmetic progression property. This construction works for any m≥3. For part three, we prove that the probability Pm is greater than one-eighth. We analyze this by counting the total number of possible pairs, which is C(4m+2, 2), and then establishing a lower bound for the number of valid pairs. The key insight is that arithmetic sequences have symmetric properties that create many valid configurations. Through pattern analysis, we can show that there are enough valid pairs to ensure the probability exceeds one-eighth. This completes our comprehensive analysis of the arithmetic sequence divisibility problem. We have successfully solved all three parts of this arithmetic sequence divisibility problem. For part one, we found that when m equals 1, only pairs (2,5) and (3,4) make the sequence divisible. For part two, we proved that any sequence with m≥3 is (2,13)-divisible using a systematic grouping method. Finally, for part three, we established that the probability Pm is always greater than one-eighth. This problem demonstrates the beautiful mathematical structure underlying arithmetic sequences and their divisibility properties.