1997 Yugoslav Team Selection Test

Part of Serbia Team Selection Test

Subcontests

(2)

1

reversing row/column of number table

1,2,\ldots,1997^2

are written in the cells of a

1997\times1997

table. It is allowed to apply the following transformations: exchange places of any two rows or any two columns, or reverse a row or column. (When a row or column is reversed, the first and last entry exchange their positions, so do the second and second last, etc.) Is it possible that, after finitely many such transformations, arbitrary two numbers exchange their positions and no other number changes its position?

2

binomial expression is integer, parameter

Given a natural number

k

, find the smallest natural number

C

\frac C{n+k+1}\binom{2n}{n+k}

is an integer for every integer

n\ge k

volume of tetrahedron as a function

Consider a regular

n

A_1A_2\ldots A_n

S

. Let us draw the lines

l_1,l_2,\ldots,l_n

perpendicular to the plane of the

n

A_1,A_2,\ldots,A_n

respectively. Points

B_1,B_2,\ldots,B_n

are selected on lines

l_1,l_2,\ldots,l_n

respectively so that:
(i)

B_1,B_2,\ldots,B_n

are all on the same side of the plane of the

n

-gon; (ii) Points

B_1,B_2,\ldots,B_n

lie on a single plane; (iii)

A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n

.
Express the volume of polyhedron

A_1A_2\ldots A_nB_1B_2\ldots B_n

as a function in

S,h_1,\ldots,h_n