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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia and Herzegovina EGMO Team Selection Test
2017 Bosnia and Herzegovina EGMO TST
2017 Bosnia and Herzegovina EGMO TST
Part of
Bosnia and Herzegovina EGMO Team Selection Test
Subcontests
(4)
4
1
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Bosnia and Herzegovina EGMO TST 2017 Problem 4
Let
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
and
e
e
e
be distinct positive real numbers such that
a
2
+
b
2
+
c
2
+
d
2
+
e
2
=
a
b
+
a
c
+
a
d
+
a
e
+
b
c
+
b
d
+
b
e
+
c
d
+
c
e
+
d
e
a^2+b^2+c^2+d^2+e^2=ab+ac+ad+ae+bc+bd+be+cd+ce+de
a
2
+
b
2
+
c
2
+
d
2
+
e
2
=
ab
+
a
c
+
a
d
+
a
e
+
b
c
+
b
d
+
b
e
+
c
d
+
ce
+
d
e
a
)
a)
a
)
Prove that among these
5
5
5
numbers there exists triplet such that they cannot be sides of a triangle
b
)
b)
b
)
Prove that, for
a
)
a)
a
)
, there exists at least
6
6
6
different triplets
3
1
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Bosnia and Herzegovina EGMO TST 2017 Problem 3
For positive integer
n
n
n
we define
f
(
n
)
f(n)
f
(
n
)
as sum of all of its positive integer divisors (including
1
1
1
and
n
n
n
). Find all positive integers
c
c
c
such that there exists strictly increasing infinite sequence of positive integers
n
1
,
n
2
,
n
3
,
.
.
.
n_1, n_2,n_3,...
n
1
,
n
2
,
n
3
,
...
such that for all
i
∈
N
i \in \mathbb{N}
i
∈
N
holds
f
(
n
i
)
−
n
i
=
c
f(n_i)-n_i=c
f
(
n
i
)
−
n
i
=
c
2
1
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Bosnia and Herzegovina EGMO TST 2017 Problem 2
It is given triangle
A
B
C
ABC
A
BC
and points
P
P
P
and
Q
Q
Q
on sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
P
Q
∣
∣
B
C
PQ\mid\mid BC
PQ
∣∣
BC
. Let
X
X
X
and
Y
Y
Y
be intersection points of lines
B
Q
BQ
BQ
and
C
P
CP
CP
with circumcircle
k
k
k
of triangle
A
P
Q
APQ
A
PQ
, and
D
D
D
and
E
E
E
intersection points of lines
A
X
AX
A
X
and
A
Y
AY
A
Y
with side
B
C
BC
BC
. If
2
⋅
D
E
=
B
C
2\cdot DE=BC
2
⋅
D
E
=
BC
, prove that circle
k
k
k
contains intersection point of angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
with
B
C
BC
BC
1
1
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Bosnia and Herzegovina EGMO TST 2017 Problem 1
It is given sequence wih length of
2017
2017
2017
which consists of first
2017
2017
2017
positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be
k
k
k
. From given sequence we form a new sequence of length 2017, such that first
k
k
k
elements of new sequence are same as first
k
k
k
elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element
1
1
1
.