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2019 Taiwan APMO Preliminary P7

Source: 2019 Taiwan APMO Preliminary

August 23, 2020

combinatoricsSequencealgebra

Problem Statement

Let positive integer

k

satisfies

1<k<100

. For the permutation of

1,2,...,100

be

a_1,a_2,...,a_{100}

, take the minimum

m>k

such that

a_m

is at least less than

(k-1)

numbers of

a_1,a_2,...,a_k

. We know that the number of sequences satisfies

a_m=1

is

\frac{100!}{4}

. Find the all possible values of

k

.

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