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Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2006 Bosnia and Herzegovina Team Selection Test
2006 Bosnia and Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
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Function with some periodic injectivity
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
,...,
a
n
a_n
a
n
be constant real numbers and
x
x
x
be variable real number
x
x
x
. Let
f
(
x
)
=
c
o
s
(
a
1
+
x
)
+
c
o
s
(
a
2
+
x
)
2
+
c
o
s
(
a
3
+
x
)
2
2
+
.
.
.
+
c
o
s
(
a
n
+
x
)
2
n
−
1
f(x)=cos(a_1+x)+\frac{cos(a_2+x)}{2}+\frac{cos(a_3+x)}{2^2}+...+\frac{cos(a_n+x)}{2^{n-1}}
f
(
x
)
=
cos
(
a
1
+
x
)
+
2
cos
(
a
2
+
x
)
+
2
2
cos
(
a
3
+
x
)
+
...
+
2
n
−
1
cos
(
a
n
+
x
)
. If
f
(
x
1
)
=
f
(
x
2
)
=
0
f(x_1)=f(x_2)=0
f
(
x
1
)
=
f
(
x
2
)
=
0
, prove that
x
1
−
x
2
=
m
π
x_1-x_2=m\pi
x
1
−
x
2
=
mπ
, where
m
m
m
is integer.
5
1
Hide problems
Isosceles triangle in simple configuration
Triangle
A
B
C
ABC
A
BC
is inscribed in circle with center
O
O
O
. Let
P
P
P
be a point on arc
A
B
AB
A
B
which does not contain point
C
C
C
. Perpendicular from point
P
P
P
on line
B
O
BO
BO
intersects side
A
B
AB
A
B
in point
S
S
S
, and side
B
C
BC
BC
in
T
T
T
. Perpendicular from point
P
P
P
on line
A
O
AO
A
O
intersects side
A
B
AB
A
B
in point
Q
Q
Q
, and side
A
C
AC
A
C
in
R
R
R
. (i) Prove that triangle
P
Q
S
PQS
PQS
is isosceles (ii) Prove that
P
Q
Q
R
=
S
T
P
Q
\frac{PQ}{QR}=\frac{ST}{PQ}
QR
PQ
=
PQ
ST
4
1
Hide problems
Geometric progression in arithmetic sequence
Prove that every infinite arithmetic progression
a
a
a
,
a
+
d
a+d
a
+
d
,
a
+
2
d
a+2d
a
+
2
d
,... where
a
a
a
and
d
d
d
are positive integers, contains infinte geometric progression
b
b
b
,
b
q
bq
b
q
,
b
q
2
bq^2
b
q
2
,... where
b
b
b
and
q
q
q
are also positive integers
3
1
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Fractional part inequality
Prove that for every positive integer
n
n
n
holds inequality
{
n
7
}
>
3
7
14
n
\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}
{
n
7
}
>
14
n
3
7
, where
{
x
}
\{x\}
{
x
}
is fractional part of
x
x
x
.
2
1
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Inscribed rectangles in triangle
It is given a triangle
△
A
B
C
\triangle ABC
△
A
BC
. Determine the locus of center of rectangle inscribed in triangle
A
B
C
ABC
A
BC
such that one side of rectangle lies on side
A
B
AB
A
B
.
1
1
Hide problems
Tetrislike Z shape
Let
Z
Z
Z
shape be a shape such that it covers
(
i
,
j
)
(i,j)
(
i
,
j
)
,
(
i
,
j
+
1
)
(i,j+1)
(
i
,
j
+
1
)
,
(
i
+
1
,
j
+
1
)
(i+1,j+1)
(
i
+
1
,
j
+
1
)
,
(
i
+
2
,
j
+
1
)
(i+2,j+1)
(
i
+
2
,
j
+
1
)
and
(
i
+
2
,
j
+
2
)
(i+2,j+2)
(
i
+
2
,
j
+
2
)
where
(
i
,
j
)
(i,j)
(
i
,
j
)
stands for cell in
i
i
i
-th row and
j
j
j
-th column on an arbitrary table. At least how many
Z
Z
Z
shapes is necessary to cover one
8
×
8
8 \times 8
8
×
8
table if every cell of a
Z
Z
Z
shape is either cell of a table or it is outside the table (two
Z
Z
Z
shapes can overlap and
Z
Z
Z
shapes can rotate)?