MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1995 Yugoslav Team Selection Test
1995 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
Hide problems
find pyramid side ratios
Let
S
A
B
C
D
SABCD
S
A
BC
D
be a pyramid with the vertex
S
S
S
whose all edges are equal. Points
M
M
M
and
N
N
N
on the edges
S
A
SA
S
A
and
B
C
BC
BC
respectively are such that
M
N
MN
MN
is perpendicular to both
S
A
SA
S
A
and
B
C
BC
BC
. Find the ratios
S
M
:
M
A
SM:MA
SM
:
M
A
and
B
N
:
N
C
BN:NC
BN
:
NC
.
Problem 2
1
Hide problems
# of 1s in binary representation
A natural number
n
n
n
has exactly
1995
1995
1995
units in its binary representation. Show that
n
!
n!
n
!
is divisible by
2
n
−
1995
2^{n-1995}
2
n
−
1995
.
Problem 1
1
Hide problems
rational number theory
Determine all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of positive rational numbers with
x
≤
y
≤
z
x\le y\le z
x
≤
y
≤
z
such that
x
+
y
+
z
,
1
x
+
1
y
+
1
z
x+y+z,\frac1x+\frac1y+\frac1z
x
+
y
+
z
,
x
1
+
y
1
+
z
1
, and xyz are natural numbers.