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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2004 Bosnia and Herzegovina Team Selection Test
2004 Bosnia and Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
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Bosnia and Herzegovina TST 2004 Day 2 Problem 3
It is given triangle
A
B
C
ABC
A
BC
and parallelogram
A
S
C
R
ASCR
A
SCR
with diagonal
A
C
AC
A
C
. Let line constructed through point
B
B
B
parallel with
C
S
CS
CS
intersects line
A
S
AS
A
S
and
C
R
CR
CR
in
M
M
M
and
P
P
P
, respectively. Let line constructed through point
B
B
B
parallel with
A
S
AS
A
S
intersects line
A
R
AR
A
R
and
C
S
CS
CS
in
N
N
N
and
Q
Q
Q
, respectively. Prove that lines
R
S
RS
RS
,
M
N
MN
MN
and
P
Q
PQ
PQ
are concurrent
5
1
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Bosnia and Herzegovina TST 2004 Day 2 Problem 2
For
0
≤
x
<
π
2
0 \leq x < \frac{\pi}{2}
0
≤
x
<
2
π
prove the inequality:
a
2
tan
(
x
)
⋅
(
cos
(
x
)
)
1
3
+
b
2
sin
x
≥
2
x
a
b
a^2\tan(x)\cdot(\cos(x))^{\frac{1}{3}}+b^2\sin{x}\geq 2xab
a
2
tan
(
x
)
⋅
(
cos
(
x
)
)
3
1
+
b
2
sin
x
≥
2
x
ab
where
a
a
a
and
b
b
b
are real numbers.
4
1
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Bosnia and Herzegovina TST 2004 Day 2 Problem 1
On competition which has
16
16
16
teams, it is played
55
55
55
games. Prove that among them exists
3
3
3
teams such that they have not played any matches between themselves.
3
1
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Bosnia and Herzegovina TST 2004 Day 1 Problem 3
Let
a
a
a
,
b
b
b
and
c
c
c
be positive real numbers such that
a
b
c
=
1
abc=1
ab
c
=
1
. Prove the inequality:
a
b
a
5
+
b
5
+
a
b
+
b
c
b
5
+
c
5
+
b
c
+
a
c
c
5
+
a
5
+
a
c
≤
1
\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1
a
5
+
b
5
+
ab
ab
+
b
5
+
c
5
+
b
c
b
c
+
c
5
+
a
5
+
a
c
a
c
≤
1
2
1
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Bosnia and Herzegovina TST 2004 Day 1 Problem 2
Determine whether does exists a triangle with area
2004
2004
2004
with his sides positive integers.
1
1
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Bosnia and Herzegovina TST 2004 Day 1 Problem 1
Circle
k
k
k
with center
O
O
O
is touched from inside by two circles in points
S
S
S
and
T
,
T,
T
,
respectively. Let those two circles intersect at points
M
M
M
and
N
N
N
, such that
N
N
N
is closer to line
S
T
ST
ST
. Prove that
O
M
OM
OM
and
M
N
MN
MN
are perpendicular iff
S
S
S
,
N
N
N
and
T
T
T
are collinear