2000 APMO

Part of APMO

Subcontests

(5)

1

Two integers

n,k

be given positive integers with

n>k

\frac{1}{n+1} \cdot \frac{n^n}{k^k (n-k)^{n-k}} < \frac{n!}{k! (n-k)!} < \frac{n^n}{k^k(n-k)^{n-k}}

1

Prove perpendicular

ABC

be a triangle. Let

M

N

be the points in which the median and the angle bisector, respectively, at

A

BC

Q

P

be the points in which the perpendicular at

N

NA

MA

BA

, respectively. And

O

the point in which the perpendicular at

P

BA

AN

produced. Prove that

QO

is perpendicular to

BC

1

Permutations

Given a permutation (

a_0, a_1, \ldots, a_n

) of the sequence

0, 1,\ldots, n

. A transportation of

a_i

a_j

is called legal if

a_i=0

i>0

a_{i-1}+1=a_j

. The permutation (

a_0, a_1, \ldots, a_n

) is called regular if after a number of legal transportations it becomes (

1,2, \ldots, n,0

). For which numbers

n

is the permutation (

1, n, n-1, \ldots, 3, 2, 0

1

Equal sums

Find all permutations

a_1, a_2, \ldots, a_9

1, 2, \ldots, 9

a_1+a_2+a_3+a_4=a_4+a_5+a_6+a_7= a_7+a_8+a_9+a_1

a_1^2+a_2^2+a_3^2+a_4^2=a_4^2+a_5^2+a_6^2+a_7^2= a_7^2+a_8^2+a_9^2+a_1^2

1

Compute a sum

Compute the sum:

\sum_{i=0}^{101} \frac{x_i^3}{1-3x_i+3x_i^2}

x_i=\frac{i}{101}