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Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2010 Bosnia Herzegovina Team Selection Test
2010 Bosnia Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
Hide problems
Partitions of positive integer n
Prove that total number of ones which is showed in all nonrestricted partitions of natural number
n
n
n
is equal to sum of numbers of distinct elements in that partitions.
5
1
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Inequality with sides of a triangle
Let
a
a
a
,
b
b
b
and
c
c
c
be sides of a triangle such that
a
+
b
+
c
≤
2
a+b+c\le2
a
+
b
+
c
≤
2
. Prove that
−
3
<
a
3
b
+
b
3
c
+
c
3
a
−
a
3
c
−
b
3
a
−
c
3
b
<
3
-3<{\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}-\frac{a^3}{c}-\frac{b^3}{a}-\frac{c^3}{b}}<3
−
3
<
b
a
3
+
c
b
3
+
a
c
3
−
c
a
3
−
a
b
3
−
b
c
3
<
3
4
1
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Convex quadrilateral divided
Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.
3
1
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Function on integers
Find all functions
f
:
Z
↦
Z
f :\mathbb{Z}\mapsto\mathbb{Z}
f
:
Z
↦
Z
such that following conditions holds:
a
)
a)
a
)
f
(
n
)
⋅
f
(
−
n
)
=
f
(
n
2
)
f(n) \cdot f(-n)=f(n^2)
f
(
n
)
⋅
f
(
−
n
)
=
f
(
n
2
)
for all
n
∈
Z
n\in\mathbb{Z}
n
∈
Z
b
)
b)
b
)
f
(
m
+
n
)
=
f
(
m
)
+
f
(
n
)
+
2
m
n
f(m+n)=f(m)+f(n)+2mn
f
(
m
+
n
)
=
f
(
m
)
+
f
(
n
)
+
2
mn
for all
m
,
n
∈
Z
m,n\in\mathbb{Z}
m
,
n
∈
Z
2
1
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Prove that it is constant
Let
A
B
AB
A
B
and
F
D
FD
F
D
be chords in circle, which does not intersect and
P
P
P
point on arc
A
B
AB
A
B
which does not contain chord
F
D
FD
F
D
. Lines
P
F
PF
PF
and
P
D
PD
P
D
intersect chord
A
B
AB
A
B
in
Q
Q
Q
and
R
R
R
. Prove that
A
Q
∗
R
B
Q
R
\frac{AQ* RB}{QR}
QR
A
Q
∗
RB
is constant, while point
P
P
P
moves along the ray
A
B
AB
A
B
.
1
1
Hide problems
Distinct prime numbers
a
)
a)
a
)
Let
p
p
p
and
q
q
q
be distinct prime numbers such that
p
+
q
2
p+q^2
p
+
q
2
divides
p
2
+
q
p^2+q
p
2
+
q
. Prove that
p
+
q
2
p+q^2
p
+
q
2
divides
p
q
−
1
pq-1
pq
−
1
.
b
)
b)
b
)
Find all prime numbers
p
p
p
such that
p
+
121
p+121
p
+
121
divides
p
2
+
11
p^2+11
p
2
+
11
.