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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2012 Bosnia Herzegovina Team Selection Test
2012 Bosnia Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
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Polygons inscribed in a square
A unit square is divided into polygons, so that all sides of a polygon are parallel to sides of the given square. If the total length of the segments inside the square (without the square) is
2
n
2n
2
n
(where
n
n
n
is a positive real number), prove that there exists a polygon whose area is greater than
1
(
n
+
1
)
2
\frac{1}{(n+1)^2}
(
n
+
1
)
2
1
.
5
1
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Bosnia and Herzegovina TST 2012 Problem 5
Given is a triangle
△
A
B
C
\triangle ABC
△
A
BC
and points
M
M
M
and
K
K
K
on lines
A
B
AB
A
B
and
C
B
CB
CB
such that
A
M
=
A
C
=
C
K
AM=AC=CK
A
M
=
A
C
=
C
K
. Prove that the length of the radius of the circumcircle of triangle
△
B
K
M
\triangle BKM
△
B
K
M
is equal to the lenght
O
I
OI
O
I
, where
O
O
O
and
I
I
I
are centers of the circumcircle and the incircle of
△
A
B
C
\triangle ABC
△
A
BC
, respectively. Also prove that
O
I
⊥
M
K
OI\perp MK
O
I
⊥
M
K
.
4
1
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Bosnia and Herzegovina TST 2012 Problem 4
Define a function
f
:
N
→
N
f:\mathbb{N}\rightarrow\mathbb{N}
f
:
N
→
N
,
f
(
1
)
=
p
+
1
,
f(1)=p+1,
f
(
1
)
=
p
+
1
,
f
(
n
+
1
)
=
f
(
1
)
⋅
f
(
2
)
⋯
f
(
n
)
+
p
,
f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,
f
(
n
+
1
)
=
f
(
1
)
⋅
f
(
2
)
⋯
f
(
n
)
+
p
,
where
p
p
p
is a prime number. Find all
p
p
p
such that there exists a natural number
k
k
k
such that
f
(
k
)
f(k)
f
(
k
)
is a perfect square.
2
1
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Bosnia and Herzegovina TST 2012 Problem 2
Prove for all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
, such that
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
:
a
3
b
2
+
c
+
b
3
c
2
+
a
+
c
3
a
2
+
b
≥
3
1
+
3
.
\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.
b
2
+
c
a
3
+
c
2
+
a
b
3
+
a
2
+
b
c
3
≥
1
+
3
3
.
3
1
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Bosnia and Herzegovina TST 2012 Problem 3
Prove that for all odd prime numbers
p
p
p
there exist a natural number
m
<
p
m<p
m
<
p
and integers
x
1
,
x
2
,
x
3
x_1, x_2, x_3
x
1
,
x
2
,
x
3
such that:
m
p
=
x
1
2
+
x
2
2
+
x
3
2
.
mp=x_1^2+x_2^2+x_3^2.
m
p
=
x
1
2
+
x
2
2
+
x
3
2
.
1
1
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Bosnia and Herzegovina TST 2012 Problem 1
Let
D
D
D
be the midpoint of the arc
B
−
A
−
C
B-A-C
B
−
A
−
C
of the circumcircle of
△
A
B
C
(
A
B
<
A
C
)
\triangle ABC (AB<AC)
△
A
BC
(
A
B
<
A
C
)
. Let
E
E
E
be the foot of perpendicular from
D
D
D
to
A
C
AC
A
C
. Prove that
∣
C
E
∣
=
∣
B
A
∣
+
∣
A
C
∣
2
|CE|=\frac{|BA|+|AC|}{2}
∣
CE
∣
=
2
∣
B
A
∣
+
∣
A
C
∣
.