benchmarks.wiki

benchmarks.wiki / Public workspace

OlympiadBench / 2012 / Let Z_(≥0) be the set of non-negative integers, and let …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Let Z≥0\mathbb{Z}_{\geqslant 0} be the set of non-negative integers, and let f:Z≥0×Z≥0→Z≥0f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0} be a bijection such that whenever f(x1,y1)>f(x2,y2)f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right), we have f(x1+1,y1)>f(x2+1,y2)f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right) and f(x1,y1+1)>f(x2,y2+1)f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}, y_{2}+1\right). Let NN be the number of pairs of integers (x,y)(x, y), with 0≤x,y<1000 \leqslant x, y<100, such that f(x,y)f(x, y) is odd. Find the smallest and largest possible value of NN.
Plain-text mathematical notation (without MathML)
Let Z_(≥0) be the set of non-negative integers, and let f:Z_(≥0)×Z_(≥0)→Z_(≥0) be a bijection such that whenever f(x₁,y₁)>f(x₂,y₂), we have f(x₁+1,y₁)>f(x₂+1,y₂) and f(x₁,y₁+1)>f(x₂,y₂+1).

Let N be the number of pairs of integers (x,y), with 0≤x,y<100, such that f(x,y) is odd. Find the smallest and largest possible value of N.
Original LaTeX notation
Let $\mathbb{Z}_{\geqslant 0}$ be the set of non-negative integers, and let $f: \mathbb{Z}_{\geqslant 0} \times \mathbb{Z}_{\geqslant 0} \rightarrow \mathbb{Z}_{\geqslant 0}$ be a bijection such that whenever $f\left(x_{1}, y_{1}\right)>f\left(x_{2}, y_{2}\right)$, we have $f\left(x_{1}+1, y_{1}\right)>f\left(x_{2}+1, y_{2}\right)$ and $f\left(x_{1}, y_{1}+1\right)>f\left(x_{2}, y_{2}+1\right)$.

Let $N$ be the number of pairs of integers $(x, y)$, with $0 \leqslant x, y<100$, such that $f(x, y)$ is odd. Find the smallest and largest possible value of $N$.

answer type

Numerical

is multiple answer

true

language

English

question type

Open-ended

subject

Math

Discussion

Discussion

No discussion posts on this page yet. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

Source and history

upstream_id / Original problem ID
2012
task_id
51e0c1a9-47cd-5601-aab6-11ffd4a60d3a
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--2012
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import