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Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2005 Bosnia and Herzegovina Team Selection Test
2005 Bosnia and Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
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perfect cube of an integer
Let
a
a
a
,
b
b
b
and
c
c
c
are integers such that
a
b
+
b
c
+
c
a
=
3
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3
b
a
+
c
b
+
a
c
=
3
. Prove that
a
b
c
abc
ab
c
is a perfect cube of an integer.
5
1
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Prove that permutation is identical
If for an arbitrary permutation
(
a
1
,
a
2
,
.
.
.
,
a
n
)
(a_1,a_2,...,a_n)
(
a
1
,
a
2
,
...
,
a
n
)
of set
1
,
2
,
.
.
.
,
n
{1,2,...,n}
1
,
2
,
...
,
n
holds
a
k
2
a
k
+
1
≤
k
+
2
\frac{{a_k}^2}{a_{k+1}}\leq k+2
a
k
+
1
a
k
2
≤
k
+
2
,
k
=
1
,
2
,
.
.
.
,
n
−
1
k=1,2,...,n-1
k
=
1
,
2
,
...
,
n
−
1
, prove that
a
k
=
k
a_k=k
a
k
=
k
for
k
=
1
,
2
,
.
.
.
,
n
k=1,2,...,n
k
=
1
,
2
,
...
,
n
4
1
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Three points collinear
On the line which contains diameter
P
Q
PQ
PQ
of circle
k
(
S
,
r
)
k(S,r)
k
(
S
,
r
)
, point
A
A
A
is chosen outside the circle such that tangent
t
t
t
from point
A
A
A
touches the circle in point
T
T
T
. Tangents on circle
k
k
k
in points
P
P
P
and
Q
Q
Q
are
p
p
p
and
q
q
q
, respectively. If
P
T
∩
q
=
N
PT \cap q={N}
PT
∩
q
=
N
and
Q
T
∩
p
=
M
QT \cap p={M}
QT
∩
p
=
M
, prove that points
A
A
A
,
M
M
M
and
N
N
N
are collinear.
3
1
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Maximal value of least common multiples
Let
n
n
n
be a positive integer such that
n
≥
2
n \geq 2
n
≥
2
. Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
be
n
n
n
distinct positive integers and
S
i
S_i
S
i
sum of all numbers between them except
x
i
x_i
x
i
for
i
=
1
,
2
,
.
.
.
,
n
i=1,2,...,n
i
=
1
,
2
,
...
,
n
. Let
f
(
x
1
,
x
2
,
.
.
.
,
x
n
)
=
G
C
D
(
x
1
,
S
1
)
+
G
C
D
(
x
2
,
S
2
)
+
.
.
.
+
G
C
D
(
x
n
,
S
n
)
x
1
+
x
2
+
.
.
.
+
x
n
.
f(x_1,x_2,...,x_n)=\frac{GCD(x_1,S_1)+GCD(x_2,S_2)+...+GCD(x_n,S_n)}{x_1+x_2+...+x_n}.
f
(
x
1
,
x
2
,
...
,
x
n
)
=
x
1
+
x
2
+
...
+
x
n
GC
D
(
x
1
,
S
1
)
+
GC
D
(
x
2
,
S
2
)
+
...
+
GC
D
(
x
n
,
S
n
)
.
Determine maximal value of
f
(
x
1
,
x
2
,
.
.
.
,
x
n
)
f(x_1,x_2,...,x_n)
f
(
x
1
,
x
2
,
...
,
x
n
)
, while
(
x
1
,
x
2
,
.
.
.
,
x
n
)
(x_1,x_2,...,x_n)
(
x
1
,
x
2
,
...
,
x
n
)
is an element of set which consists from all
n
n
n
-tuples of distinct positive integers.
2
1
Hide problems
Another very common inequality
If
a
1
a_1
a
1
,
a
2
a_2
a
2
and
a
3
a_3
a
3
are nonnegative real numbers for which
a
1
+
a
2
+
a
3
=
1
a_1+a_2+a_3=1
a
1
+
a
2
+
a
3
=
1
, then prove the inequality
a
1
a
2
+
a
2
a
3
+
a
3
a
1
≤
1
3
a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}
a
1
a
2
+
a
2
a
3
+
a
3
a
1
≤
3
1
1
1
Hide problems
Orthocenter, midpoints and angle bisectors
Let
H
H
H
be an orthocenter of an acute triangle
A
B
C
ABC
A
BC
. Prove that midpoints of
A
B
AB
A
B
and
C
H
CH
C
H
and intersection point of angle bisectors of
∠
C
A
H
\angle CAH
∠
C
A
H
and
∠
C
B
H
\angle CBH
∠
CB
H
lie on the same line.