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n arithmetic progressions of integers each of k terms

Source: India tst 2006 p9

June 27, 2012
number theory unsolvednumber theory

Problem Statement

Let A1,A2,⋯ ,AnA_1,A_2,\cdots , A_n be arithmetic progressions of integers, each of kk terms, such that any two of these arithmetic progressions have at least two common elements. Suppose bb of these arithmetic progressions have common difference d1d_1 and the remaining arithmetic progressions have common difference d2d_2 where 0<b<n0<b<n. Prove that b≤2(k−d2gcd(d1,d2))−1.b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.
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