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Omni-MATH / Lucy starts by writing s integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples v=(v₁,…,v₂₀₂₂) and w=(w₁,…,w₂₀₂₂) that…

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problem

Lucy starts by writing ss integer-valued 20222022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples v=(v1,…,v2022)\mathbf{v}=(v_1,\ldots,v_{2022}) and w=(w1,…,w2022)\mathbf{w}=(w_1,\ldots,w_{2022}) that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued 20222022-tuple on the blackboard after finitely many steps. What is the smallest possible number ss of tuples that she initially wrote?
Plain-text mathematical notation (without MathML)
Lucy starts by writing s integer-valued 2022-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples v=(v₁,…,v₂₀₂₂) and w=(w₁,…,w₂₀₂₂) that she has already written, and apply one of the following operations to obtain a new tuple:
\begin{align*}
\mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\
\mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022}))
\end{align*}
and then write this tuple on the blackboard.

It turns out that, in this way, Lucy can write any integer-valued 2022-tuple on the blackboard after finitely many steps. What is the smallest possible number s of tuples that she initially wrote?
Original LaTeX notation
Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple:
\begin{align*}
\mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\
\mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022}))
\end{align*}
and then write this tuple on the blackboard.

It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

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Source and history

task_id
f5f860eb-2f0f-5e5e-a2a3-acfffa23b1c0
task_key
test--f5f860eb-2f0f-5e5e-a2a3-acfffa23b1c0
task_revision_id
1
config / split
/ test

Official source

initial import