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Fix $m\geq1$, put $d=2m+1$, and read subscripts modulo $d$.

Statement

Integers $z_0,\ldots,z_{d-1}$ satisfying

$$|2z_i+z_{i-1}|<2\qquad(0\leq i<d)$$

are all zero.

Proof

Let $z_i\ne0$. If $z_{i-1}$ were zero or of the same sign as $z_i$, then $|2z_i+z_{i-1}|=2|z_i|+|z_{i-1}|\geq2$, against the hypothesis; so $z_{i-1}$ is nonzero, of the sign opposite to that of $z_i$. Walking back from index $i$ through all $d$ indices to $i$ again reverses the sign $d$ times, and as $d$ is odd, $z_i$ would have the sign opposite to its own. Hence no coordinate is nonzero.

Source and dependencies

An explanation of the proof of Erdős Problem 1, preliminary exposition with no named author (erdosproblems.com, 2026), §2, Lemma 2.1, p. 2. The edition read is named on the source card. Only integrality, the triangle order on $\mathbb Z$, and oddness of $d$ are used.

Bears on. #1.