MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1997 Yugoslav Team Selection Test
1997 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(2)
Problem 3
1
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reversing row/column of number table
Numbers
1
,
2
,
…
,
199
7
2
1,2,\ldots,1997^2
1
,
2
,
…
,
199
7
2
are written in the cells of a
1997
×
1997
1997\times1997
1997
×
1997
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?
Problem 1
2
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binomial expression is integer, parameter
Given a natural number
k
k
k
, find the smallest natural number
C
C
C
such that
C
n
+
k
+
1
(
2
n
n
+
k
)
\frac C{n+k+1}\binom{2n}{n+k}
n
+
k
+
1
C
(
n
+
k
2
n
)
is an integer for every integer
n
≥
k
n\ge k
n
≥
k
.
volume of tetrahedron as a function
Consider a regular
n
n
n
-gon
A
1
A
2
…
A
n
A_1A_2\ldots A_n
A
1
A
2
…
A
n
with area
S
S
S
. Let us draw the lines
l
1
,
l
2
,
…
,
l
n
l_1,l_2,\ldots,l_n
l
1
,
l
2
,
…
,
l
n
perpendicular to the plane of the
n
n
n
-gon at
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots,A_n
A
1
,
A
2
,
…
,
A
n
respectively. Points
B
1
,
B
2
,
…
,
B
n
B_1,B_2,\ldots,B_n
B
1
,
B
2
,
…
,
B
n
are selected on lines
l
1
,
l
2
,
…
,
l
n
l_1,l_2,\ldots,l_n
l
1
,
l
2
,
…
,
l
n
respectively so that:(i)
B
1
,
B
2
,
…
,
B
n
B_1,B_2,\ldots,B_n
B
1
,
B
2
,
…
,
B
n
are all on the same side of the plane of the
n
n
n
-gon; (ii) Points
B
1
,
B
2
,
…
,
B
n
B_1,B_2,\ldots,B_n
B
1
,
B
2
,
…
,
B
n
lie on a single plane; (iii)
A
1
B
1
=
h
1
,
A
2
B
2
=
h
2
,
…
,
A
n
B
n
=
h
n
A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n
A
1
B
1
=
h
1
,
A
2
B
2
=
h
2
,
…
,
A
n
B
n
=
h
n
.Express the volume of polyhedron
A
1
A
2
…
A
n
B
1
B
2
…
B
n
A_1A_2\ldots A_nB_1B_2\ldots B_n
A
1
A
2
…
A
n
B
1
B
2
…
B
n
as a function in
S
,
h
1
,
…
,
h
n
S,h_1,\ldots,h_n
S
,
h
1
,
…
,
h
n
.