P7

Part of 2019 Taiwan APMO Preliminary Test

Problems(1)

2019 Taiwan APMO Preliminary P7

Source: 2019 Taiwan APMO Preliminary

Let positive integer

k

1<k<100

. For the permutation of

1,2,...,100

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

, take the minimum

m>k

a_m

is at least less than

(k-1)

a_1,a_2,...,a_k

. We know that the number of sequences satisfies

a_m=1

\frac{100!}{4}

. Find the all possible values of

k

combinatoricsSequencealgebra