Combinatorics · Invariants · Parity · Pigeonhole · Prefix sums · Extremal example
A finite sequence of \(n\) numbers is given, each of them equal to \(0\) or \(1\), where \(n\) is a positive integer. One move consists of choosing two adjacent terms \(x\) and \(y\), deleting both of them, and writing in their place the single number \(x + y \pmod 2\), so that the sequence becomes one term shorter.
Determine the largest \(k\), as a function of \(n\), for which we can guarantee the following: whatever the initial sequence is, after finitely many moves we can obtain a sequence containing at least \(k\) consecutive mutually equal terms.
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Serbian National Competition (Drzavno takmicenje) 2026, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source