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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
1998 Bosnia and Herzegovina Team Selection Test
1998 Bosnia and Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
Hide problems
Bosnia and Herzegovina TST 1998 Day 2 Problem 3
Sequence of integers
{
u
n
}
n
∈
N
0
\{u_n\}_{n \in \mathbb{N}_0}
{
u
n
}
n
∈
N
0
is given as:
u
0
=
0
u_0=0
u
0
=
0
,
u
2
n
=
u
n
u_{2n}=u_n
u
2
n
=
u
n
,
u
2
n
+
1
=
1
−
u
n
u_{2n+1}=1-u_n
u
2
n
+
1
=
1
−
u
n
for all
n
∈
N
0
n \in \mathbb{N}_0
n
∈
N
0
a
)
a)
a
)
Find
u
1998
u_{1998}
u
1998
b
)
b)
b
)
If
p
p
p
is a positive integer and
m
=
(
2
p
−
1
)
2
m=(2^p-1)^2
m
=
(
2
p
−
1
)
2
, find
u
m
u_m
u
m
5
1
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Bosnia and Herzegovina TST 1998 Day 2 Problem 2
Let
a
a
a
,
b
b
b
and
c
c
c
be integers such that
b
c
+
a
d
=
1
bc+ad=1
b
c
+
a
d
=
1
a
c
+
2
b
d
=
1
ac+2bd=1
a
c
+
2
b
d
=
1
Prove that
a
2
+
c
2
=
2
b
2
+
2
d
2
a^2+c^2=2b^2+2d^2
a
2
+
c
2
=
2
b
2
+
2
d
2
4
1
Hide problems
Bosnia and Herzegovina TST 1998 Day 2 Problem 1
Circle
k
k
k
with radius
r
r
r
touches the line
p
p
p
in point
A
A
A
. Let
A
B
AB
A
B
be a dimeter of circle and
C
C
C
an arbitrary point of circle distinct from points
A
A
A
and
B
B
B
. Let
D
D
D
be a foot of perpendicular from point
C
C
C
to line
A
B
AB
A
B
. Let
E
E
E
be a point on extension of line
C
D
CD
C
D
, over point
D
D
D
, such that
E
D
=
B
C
ED=BC
E
D
=
BC
. Let tangents on circle from point
E
E
E
intersect line
p
p
p
in points
K
K
K
and
N
N
N
. Prove that length of
K
N
KN
K
N
does not depend from
C
C
C
3
1
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Bosnia and Herzegovina TST 1998 Day 1 Problem 3
Angle bisectors of angles by vertices
A
A
A
,
B
B
B
and
C
C
C
in triangle
A
B
C
ABC
A
BC
intersect opposing sides in points
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
, respectively. Let
M
M
M
be an arbitrary point on one of the lines
A
1
B
1
A_1B_1
A
1
B
1
,
B
1
C
1
B_1C_1
B
1
C
1
and
C
1
A
1
C_1A_1
C
1
A
1
. Let
M
1
M_1
M
1
,
M
2
M_2
M
2
and
M
3
M_3
M
3
be orthogonal projections of point
M
M
M
on lines
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively. Prove that one of the lines
M
M
1
MM_1
M
M
1
,
M
M
2
MM_2
M
M
2
and
M
M
3
MM_3
M
M
3
is equal to sum of other two
2
1
Hide problems
Bosnia and Herzegovina TST 1998 Day 1 Problem 2
For positive real numbers
x
x
x
,
y
y
y
and
z
z
z
holds
x
2
+
y
2
+
z
2
=
1
x^2+y^2+z^2=1
x
2
+
y
2
+
z
2
=
1
. Prove that
x
1
+
x
2
+
y
1
+
y
2
+
z
1
+
z
2
≤
3
3
4
\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}
1
+
x
2
x
+
1
+
y
2
y
+
1
+
z
2
z
≤
4
3
3
1
1
Hide problems
Bosnia and Herzegovina TST 1998 Day 1 Problem 1
Let
P
1
P_1
P
1
,
P
2
P_2
P
2
,
P
3
P_3
P
3
,
P
4
P_4
P
4
and
P
5
P_5
P
5
be five different points which are inside
D
D
D
or on the border of figure
D
D
D
. Let
M
=
m
i
n
{
P
i
P
j
∣
i
≠
j
}
M=min\left\{P_iP_j \mid i \neq j\right\}
M
=
min
{
P
i
P
j
∣
i
=
j
}
be minimal distance between different points
P
i
P_i
P
i
. For which configuration of points
P
i
P_i
P
i
, value
M
M
M
is at maximum, if :
a
)
a)
a
)
D
D
D
is unit square
b
)
b)
b
)
D
D
D
is equilateral triangle with side equal
1
1
1
c
)
c)
c
)
D
D
D
is unit circle, circle with radius
1
1
1