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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
1996 Bosnia and Herzegovina Team Selection Test
1996 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 1996 Day 2 Problem 3
Let
a
a
a
and
b
b
b
be two integers which are coprime and let
n
n
n
be one variable integer. Determine probability that number of solutions
(
x
,
y
)
(x,y)
(
x
,
y
)
, where
x
x
x
and
y
y
y
are nonnegative integers, of equation
a
x
+
b
y
=
n
ax+by=n
a
x
+
b
y
=
n
is
⌊
n
a
b
⌋
+
1
\left\lfloor \frac{n}{ab} \right\rfloor + 1
⌊
ab
n
⌋
+
1
5
1
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Bosnia and Herzegovina TST 1996 Day 2 Problem 2
Group of
10
10
10
people are buying books. We know the following:
i
)
i)
i
)
Every person bought four different books
i
i
)
ii)
ii
)
Every two persons bought at least one book common for both of them Taking in consideration book which was bought by maximum number of people, determine minimal value of that number
4
1
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Bosnia and Herzegovina TST 1996 Day 2 Problem 1
Solve the functional equation
f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
cos
y
f(x+y)+f(x-y)=2f(x)\cos{y}
f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
cos
y
where
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
and
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
3
1
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Bosnia and Herzegovina TST 1996 Day 1 Problem 3
Let
M
M
M
be a point inside quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
A
B
M
D
ABMD
A
BM
D
is parallelogram. If
∠
C
B
M
=
∠
C
D
M
\angle CBM = \angle CDM
∠
CBM
=
∠
C
D
M
prove that
∠
A
C
D
=
∠
B
C
M
\angle ACD = \angle BCM
∠
A
C
D
=
∠
BCM
2
1
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Bosnia and Herzegovina TST 1996 Day 1 Problem 2
a
)
a)
a
)
Let
m
m
m
and
n
n
n
be positive integers. If
m
>
1
m>1
m
>
1
prove that
n
∣
ϕ
(
m
n
−
1
)
n \mid \phi(m^n-1)
n
∣
ϕ
(
m
n
−
1
)
where
ϕ
\phi
ϕ
is Euler function
b
)
b)
b
)
Prove that number of elements in sequence
1
,
2
,
.
.
.
,
n
1,2,...,n
1
,
2
,
...
,
n
(
n
∈
N
)
(n \in \mathbb{N})
(
n
∈
N
)
, which greatest common divisor with
n
n
n
is
d
d
d
, is
ϕ
(
n
d
)
\phi\left(\frac{n}{d}\right)
ϕ
(
d
n
)
1
1
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Bosnia and Herzegovina TST 1996 Day 1 Problem 1
a
)
a)
a
)
Let
a
a
a
,
b
b
b
and
c
c
c
be positive real numbers. Prove that for all positive integers
m
m
m
holds:
(
a
+
b
)
m
+
(
b
+
c
)
m
+
(
c
+
a
)
m
≤
2
m
(
a
m
+
b
m
+
c
m
)
(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)
(
a
+
b
)
m
+
(
b
+
c
)
m
+
(
c
+
a
)
m
≤
2
m
(
a
m
+
b
m
+
c
m
)
b
)
b)
b
)
Does inequality
a
)
a)
a
)
holds for
1
)
1)
1
)
arbitrary real numbers
a
a
a
,
b
b
b
and
c
c
c
2
)
2)
2
)
any integer
m
m
m