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Combinatorics with arrows

Source: 2023 Polish Junior Math Olympiad Finals

May 5, 2023

Problem Statement

Let n≥1n\geq 1 be odd integer. There are nn arrows are arranged from left to right, such that each arrow points either to the left or to the right. Prove that there exists an arrow that is pointed to by exactly as many arrows as it is pointing to.
Note: For example, for n=5n=5 and the arrangement →→←←→\rightarrow\rightarrow\leftarrow\leftarrow\rightarrow, the successive arrows (from the left) point respectively to 44, 33, 22, 33, 00 arrows.
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