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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2003 Bosnia and Herzegovina Team Selection Test
2003 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 2003 Day 2 Problem 3
Let
a
a
a
,
b
b
b
and
c
c
c
be real numbers such that
∣
a
∣
>
2
\mid a \mid >2
∣
a
∣>
2
and
a
2
+
b
2
+
c
2
=
a
b
c
+
4
a^2+b^2+c^2=abc+4
a
2
+
b
2
+
c
2
=
ab
c
+
4
. Prove that numbers
x
x
x
and
y
y
y
exist such that
a
=
x
+
1
x
a=x+\frac{1}{x}
a
=
x
+
x
1
,
b
=
y
+
1
y
b=y+\frac{1}{y}
b
=
y
+
y
1
and
c
=
x
y
+
1
x
y
c=xy+\frac{1}{xy}
c
=
x
y
+
x
y
1
.
5
1
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Bosnia and Herzegovina TST 2003 Day 2 Problem 2
It is given regular polygon with
2
n
2n
2
n
sides and center
S
S
S
. Consider every quadrilateral with vertices as vertices of polygon. Let
u
u
u
be number of such quadrilaterals which contain point
S
S
S
inside and
v
v
v
number of remaining quadrilaterals. Find
u
−
v
u-v
u
−
v
4
1
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Bosnia and Herzegovina TST 2003 Day 2 Problem 1
In triangle
A
B
C
ABC
A
BC
A
D
AD
A
D
and
B
E
BE
BE
are altitudes. Let
L
L
L
be a point on
E
D
ED
E
D
such that
E
D
ED
E
D
is orthogonal to
B
L
BL
B
L
. If
L
B
2
=
L
D
⋅
L
E
LB^2=LD\cdot LE
L
B
2
=
L
D
⋅
L
E
prove that triangle
A
B
C
ABC
A
BC
is isosceles
3
1
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Bosnia and Herzegovina TST 2003 Day 1 Problem 3
Prove that for every positive integer
n
n
n
holds:
(
n
−
1
)
n
+
2
n
n
≤
(
n
+
1
)
n
≤
2
(
n
−
1
)
n
+
2
n
n
(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}
(
n
−
1
)
n
+
2
n
n
≤
(
n
+
1
)
n
≤
2
(
n
−
1
)
n
+
2
n
n
2
1
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Bosnia and Herzegovina TST 2003 Day 1 Problem 2
Upon sides
A
B
AB
A
B
and
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
are constructed squares
A
B
B
1
A
1
ABB_{1}A_{1}
A
B
B
1
A
1
and
B
C
C
1
B
2
BCC_{1}B_{2}
BC
C
1
B
2
. Prove that lines
A
C
1
AC_{1}
A
C
1
,
C
A
1
CA_{1}
C
A
1
and altitude from
B
B
B
to side
A
C
AC
A
C
are concurrent.
1
1
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Bosnia and Herzegovina TST 2003 Day 1 Problem 1
Board has written numbers:
5
5
5
,
7
7
7
and
9
9
9
. In every step we do the following: for every pair
(
a
,
b
)
(a,b)
(
a
,
b
)
,
a
>
b
a>b
a
>
b
numbers from the board, we also write the number
5
a
−
4
b
5a-4b
5
a
−
4
b
. Is it possible that after some iterations,
2003
2003
2003
occurs at the board ?