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Omni-MATH / Denote by Z² the set of all points (x,y) in the plane with integer coordinates. For each integer n≥0, let P_(n) be the subset of Z² consisting of the point (0,0) together with all …

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problem

Denote by Z2\mathbb{Z}^2 the set of all points (x,y)(x,y) in the plane with integer coordinates. For each integer n≥0n \geq 0, let PnP_n be the subset of Z2\mathbb{Z}^2 consisting of the point (0,0)(0,0) together with all points (x,y)(x,y) such that x2+y2=2kx^2 + y^2 = 2^k for some integer k≤nk \leq n. Determine, as a function of nn, the number of four-point subsets of PnP_n whose elements are the vertices of a square.
Plain-text mathematical notation (without MathML)
Denote by Z² the set of all points (x,y) in the plane with integer coordinates. For each integer n≥0, let P_(n) be the subset of Z² consisting of the point (0,0) together with all points (x,y) such that x²+y²=2^(k) for some integer k≤n. Determine, as a function of n, the number of four-point subsets of P_(n) whose elements are the vertices of a square.
Original LaTeX notation
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.

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Source and history

task_id
d2ee24cd-b353-5b3e-b4e0-382f1845faec
task_key
test--d2ee24cd-b353-5b3e-b4e0-382f1845faec
task_revision_id
1
config / split
/ test

Official source

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