MathDB
c_1 + c_2 + ... + c_k = n^2 - 1, when c_1=a_i if a_i = a_{i-1} etc

Source: Dutch IMO TST 2015 day 1 p3

August 30, 2019
algebraSequenceSum

Problem Statement

Let nn be a positive integer. Consider sequences a0,a1,...,aka_0, a_1, ..., a_k and b0,b1,,..,bkb_0, b_1,,..,b_k such that a0=b0=1a_0 = b_0 = 1 and ak=bk=na_k = b_k = n and such that for all ii such that 1≤i≤k1 \le i \le k , we have that (ai,bi)(a_i, b_i) is either equal to (1+ai−1,bi−1)(1 + a_{i-1}, b_{i-1}) or (ai−1;1+bi−1)(a_{i-1}; 1 + b_{i-1}). Consider for 1≤i≤k1 \le i \le k the number ci={ai   if   ai=ai−1bi   if   bi=bi−1c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\ b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases} Show that c1+c2+...+ck=n2−1c_1 + c_2 + ... + c_k = n^2 - 1.
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