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2019 Azerbaijan Senior NMO, 4

Is it possible to construct a equilateral triangle such that: $\text{a)}$ Coordinates of this triangle are integers in two dimensional plane? $\text{b)}$ Coordinates of this triangle are integers in three dimensional plane?

Revenge EL(S)MO 2024, 3

Tags: dimension , sequence , algebra

Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and \[ a_k = (n-k+1) \cdot c_{k-1}, \quad b_k = \binom nk - c_k - a_k, \quad \text{and} \quad c_k = \frac{b_{k-1}}{k} \] for each integer $1 \leq k \leq n$. $ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$. Proposed by [i]Jonathan Du[/i]

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