5

Part of 2021 Korea Winter Program Practice Test

Problems(2)

Source: 2021 Korea Winter Program Practice Test

E,F

AB,AC

(B,E,F,C)

D

is the intersection of

BC

and the perpendicular bisecter of

EF

B',C'

are the reflections of

B,C

AD

X

is a point on the circumcircle of

\triangle{BEC'}

AB

is perpendicular to

BX

Y

is a point on the circumcircle of

\triangle{CFB'}

AC

is perpendicular to

CY

DX=DY

geometryperpendicular bisectorcircumcircle

Permutation consisting of A&B

Source: 2021 Korea Winter Program Test2 Day2 #5

For positive integers

k

n

, express the number of permutation

P=x_1x_2...x_{2n}

A

B

that satisfies all three of the following conditions, using

k

n

.

(i)

A, B

n

times respectively in

P

.

(ii)

1\le i\le n

, if we denote the number of

A

x_1,x_2,...,x_i

a_i

,

\mid 2a_i -i\mid \le 1

.

(iii)

AB

appears exactly

k

P

. (For example,

AB

appears 3 times in

ABBABAAB

combinatoricspermutation