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Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
1997 Bosnia and Herzegovina Team Selection Test
1997 Bosnia and Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
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Bosnia and Herzegovina TST 1997 Day 2 Problem 3
Let
k
k
k
,
m
m
m
and
n
n
n
be integers such that
1
<
n
≤
m
−
1
≤
k
1<n \leq m-1 \leq k
1
<
n
≤
m
−
1
≤
k
. Find maximum size of subset
S
S
S
of set
{
1
,
2
,
.
.
.
,
k
}
\{1,2,...,k\}
{
1
,
2
,
...
,
k
}
such that sum of any
n
n
n
different elements from
S
S
S
is not:
a
)
a)
a
)
equal to
m
m
m
,
b
)
b)
b
)
exceeding
m
m
m
5
1
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Bosnia and Herzegovina TST 1997 Day 2 Problem 2
a
)
a)
a
)
Prove that for all positive integers
n
n
n
exists a set
M
n
M_n
M
n
of positive integers with exactly
n
n
n
elements and:
i
)
i)
i
)
Arithmetic mean of arbitrary non-empty subset of
M
n
M_n
M
n
is integer
i
i
)
ii)
ii
)
Geometric mean of arbitrary non-empty subset of
M
n
M_n
M
n
is integer
i
i
i
)
iii)
iii
)
Both arithmetic mean and geometry mean of arbitrary non-empty subset of
M
n
M_n
M
n
is integer
b
)
b)
b
)
Does there exist infinite set
M
M
M
of positive integers such that arithmetic mean of arbitrary non-empty subset of
M
M
M
is integer
4
1
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Bosnia and Herzegovina TST 1997 Day 2 Problem 1
a
)
a)
a
)
In triangle
A
B
C
ABC
A
BC
let
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
be touching points of incircle
A
B
C
ABC
A
BC
with
B
A
BA
B
A
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively. Let
l
1
l_1
l
1
,
l
2
l_2
l
2
and
l
3
l_3
l
3
be lenghts of arcs
B
1
C
1
B_1C_1
B
1
C
1
,
A
1
C
1
A_1C_1
A
1
C
1
,
B
1
A
1
B_1A_1
B
1
A
1
of incircle
A
B
C
ABC
A
BC
, respectively, which does not contain points
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
, respectively. Does the following inequality hold:
a
l
1
+
b
l
2
+
c
l
3
≥
9
3
π
\frac{a}{l_1}+\frac{b}{l_2}+\frac{c}{l_3} \geq \frac{9\sqrt{3}}{\pi}
l
1
a
+
l
2
b
+
l
3
c
≥
π
9
3
b
)
b)
b
)
Tetrahedron
A
B
C
D
ABCD
A
BC
D
has three pairs of equal opposing sides. Find length of height of tetrahedron in function od lengths of sides
3
1
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Bosnia and Herzegovina TST 1997 Day 1 Problem 3
It is given function
f
:
A
→
R
f : A \rightarrow \mathbb{R}
f
:
A
→
R
,
(
A
⊆
R
)
(A\subseteq \mathbb{R})
(
A
⊆
R
)
such that
f
(
x
+
y
)
=
f
(
x
)
⋅
f
(
y
)
−
f
(
x
y
)
+
1
;
(
∀
x
,
y
∈
A
)
f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)
f
(
x
+
y
)
=
f
(
x
)
⋅
f
(
y
)
−
f
(
x
y
)
+
1
;
(
∀
x
,
y
∈
A
)
If
f
:
A
→
R
f : A \rightarrow \mathbb{R}
f
:
A
→
R
,
(
N
⊆
A
⊆
R
)
(\mathbb{N} \subseteq A\subseteq \mathbb{R})
(
N
⊆
A
⊆
R
)
is solution of given functional equation, prove that:
f
(
n
)
=
{
c
n
+
1
−
1
c
−
1
,
∀
n
∈
N
,
c
≠
1
n
+
1
,
∀
n
∈
N
,
c
=
1
f(n)=\begin{cases} \frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\ n+1 \text{, } \forall n \in \mathbb{N}, c = 1 \end{cases}
f
(
n
)
=
{
c
−
1
c
n
+
1
−
1
,
∀
n
∈
N
,
c
=
1
n
+
1
,
∀
n
∈
N
,
c
=
1
where
c
=
f
(
1
)
−
1
c=f(1)-1
c
=
f
(
1
)
−
1
a
)
a)
a
)
Solve given functional equation for
A
=
N
A=\mathbb{N}
A
=
N
b
)
b)
b
)
With
A
=
Q
A=\mathbb{Q}
A
=
Q
, find all functions
f
f
f
which are solutions of the given functional equation and also
f
(
1997
)
≠
f
(
1998
)
f(1997) \neq f(1998)
f
(
1997
)
=
f
(
1998
)
2
1
Hide problems
Bosnia and Herzegovina TST 1997 Day 1 Problem 2
In isosceles triangle
A
B
C
ABC
A
BC
with base side
A
B
AB
A
B
, on side
B
C
BC
BC
it is given point
M
M
M
. Let
O
O
O
be a circumcenter and
S
S
S
incenter of triangle
A
B
C
ABC
A
BC
. Prove that
S
M
∣
∣
A
C
⇔
O
M
⊥
B
S
SM \mid \mid AC \Leftrightarrow OM \perp BS
SM
∣∣
A
C
⇔
OM
⊥
BS
1
1
Hide problems
Bosnia and Herzegovina TST 1997 Day 1 Problem 1
Solve system of equation
8
(
x
3
+
y
3
+
z
3
)
=
73
8(x^3+y^3+z^3)=73
8
(
x
3
+
y
3
+
z
3
)
=
73
2
(
x
2
+
y
2
+
z
2
)
=
3
(
x
y
+
y
z
+
z
x
)
2(x^2+y^2+z^2)=3(xy+yz+zx)
2
(
x
2
+
y
2
+
z
2
)
=
3
(
x
y
+
yz
+
z
x
)
x
y
z
=
1
xyz=1
x
yz
=
1
in set
R
3
\mathbb{R}^3
R
3