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Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2007 Bosnia Herzegovina Team Selection Test
2007 Bosnia Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
Hide problems
Ann chooses subsets
The set
A
A
A
has exactly
n
>
4
n>4
n
>
4
elements. Ann chooses
n
+
1
n+1
n
+
1
distinct subsets of
A
A
A
, such that every subset has exactly
3
3
3
elements. Prove that there exist two subsets chosen by Ann which have exactly one common element.
5
1
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Find value of angle in right angled triangle
Triangle
A
B
C
ABC
A
BC
is right angled such that
∠
A
C
B
=
9
0
∘
\angle ACB=90^{\circ}
∠
A
CB
=
9
0
∘
and
A
C
B
C
=
2
\frac {AC}{BC} = 2
BC
A
C
=
2
. Let the line parallel to side
A
C
AC
A
C
intersects line segments
A
B
AB
A
B
and
B
C
BC
BC
in
M
M
M
and
N
N
N
such that
C
N
B
N
=
2
\frac {CN}{BN} = 2
BN
CN
=
2
. Let
O
O
O
be the intersection point of lines
C
M
CM
CM
and
A
N
AN
A
N
. On segment
O
N
ON
ON
lies point
K
K
K
such that
O
M
+
O
K
=
K
N
OM+OK=KN
OM
+
O
K
=
K
N
. Let
T
T
T
be the intersection point of angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
and line from
K
K
K
perpendicular to
A
N
AN
A
N
. Determine value of
∠
M
T
B
\angle MTB
∠
MTB
.
4
1
Hide problems
Polynomial inequality
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial such that
P
(
x
)
=
x
3
−
2
x
2
+
b
x
+
c
P(x)=x^3-2x^2+bx+c
P
(
x
)
=
x
3
−
2
x
2
+
b
x
+
c
. Roots of
P
(
x
)
P(x)
P
(
x
)
belong to interval
(
0
,
1
)
(0,1)
(
0
,
1
)
. Prove that
8
b
+
9
c
≤
8
8b+9c \leq 8
8
b
+
9
c
≤
8
. When does equality hold?
3
1
Hide problems
Square root equation
Find all
x
∈
Z
x\in \mathbb{Z}
x
∈
Z
and
a
∈
R
a\in \mathbb{R}
a
∈
R
satisfying
x
2
−
4
+
x
+
2
=
x
−
a
+
a
\sqrt{x^2-4}+\sqrt{x+2} = \sqrt{x-a}+a
x
2
−
4
+
x
+
2
=
x
−
a
+
a
2
1
Hide problems
Diophantine equation
Find all pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
x
(
x
+
2
)
=
y
2
(
y
2
+
1
)
x(x+2)=y^2(y^2+1)
x
(
x
+
2
)
=
y
2
(
y
2
+
1
)
1
1
Hide problems
Find area of triangle
Let
A
B
C
ABC
A
BC
be a triangle such that length of internal angle bisector from
B
B
B
is equal to
s
s
s
. Also, length of external angle bisector from
B
B
B
is equal to
s
1
s_1
s
1
. Find area of triangle
A
B
C
ABC
A
BC
if
A
B
B
C
=
k
\frac{AB}{BC} = k
BC
A
B
=
k