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Omni-MATH / On some planet, there are 2^(N) countries (N≥4). Each country has a flag N units wide and one unit high composed of N fields of size 1×1, each field being either yellow or blue. No…

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problem

On some planet, there are 2N2^N countries (N≥4).(N \geq 4). Each country has a flag NN units wide and one unit high composed of NN fields of size 1×1,1 \times 1, each field being either yellow or blue. No two countries have the same flag. We say that a set of NN flags is diverse if these flags can be arranged into an N×NN \times N square so that all NN fields on its main diagonal will have the same color. Determine the smallest positive integer MM such that among any MM distinct flags, there exist NN flags forming a diverse set. [i]
Plain-text mathematical notation (without MathML)
On some planet, there are 2^(N) countries (N≥4). Each country has a flag N units wide and one unit high composed of N fields of size 1×1, each field being either yellow or blue. No two countries have the same flag. We say that a set of N flags is diverse if these flags can be arranged into an N×N square so that all N fields on its main diagonal will have the same color. Determine the smallest positive integer M such that among any M distinct flags, there exist N flags forming a diverse set.

[i]
Original LaTeX notation
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.

[i]

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task_id
e74acae9-f45a-5512-8b63-e35d29ecfe2f
task_key
test--e74acae9-f45a-5512-8b63-e35d29ecfe2f
task_revision_id
1
config / split
/ test

Official source

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