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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2015 Bosnia and Herzegovina Junior BMO TST
2015 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
4
1
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Bosnia and Herzegovina JBMO TST 2015 Problem 4
Let
n
n
n
be a positive integer and let
a
1
a_1
a
1
,
a
2
a_2
a
2
,...,
a
n
a_n
a
n
be positive integers from set
{
1
,
2
,
.
.
.
,
n
}
\{1, 2,..., n\}
{
1
,
2
,
...
,
n
}
such that every number from this set occurs exactly once. Is it possible that numbers
a
1
a_1
a
1
,
a
1
+
a
2
,
.
.
.
,
a
1
+
a
2
+
.
.
.
+
a
n
a_1 + a_2 ,..., a_1 + a_2 + ... + a_n
a
1
+
a
2
,
...
,
a
1
+
a
2
+
...
+
a
n
all have different remainders upon division by
n
n
n
, if:
a
)
a)
a
)
n
=
7
n=7
n
=
7
b
)
b)
b
)
n
=
8
n=8
n
=
8
3
1
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Bosnia and Herzegovina JBMO TST 2015 Problem 3
Let
A
D
AD
A
D
be an altitude of triangle
A
B
C
ABC
A
BC
, and let
M
M
M
,
N
N
N
and
P
P
P
be midpoints of
A
B
AB
A
B
,
A
D
AD
A
D
and
B
C
BC
BC
, respectively. Furthermore let
K
K
K
be a foot of perpendicular from point
D
D
D
to line
A
C
AC
A
C
, and let
T
T
T
be point on extension of line
K
D
KD
KD
(over point
D
D
D
) such that
∣
D
T
∣
=
∣
M
N
∣
+
∣
D
K
∣
\mid DT \mid = \mid MN \mid + \mid DK \mid
∣
D
T
∣=∣
MN
∣
+
∣
DK
∣
. If
∣
M
P
∣
=
2
⋅
∣
K
N
∣
\mid MP \mid = 2 \cdot \mid KN \mid
∣
MP
∣=
2
⋅
∣
K
N
∣
, prove that
∣
A
T
∣
=
∣
M
C
∣
\mid AT \mid = \mid MC \mid
∣
A
T
∣=∣
MC
∣
.
2
1
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Bosnia and Herzegovina JBMO TST 2015 Problem 2
Find all triplets of positive integers
a
a
a
,
b
b
b
and
c
c
c
such that
a
≥
b
≥
c
a \geq b \geq c
a
≥
b
≥
c
and
(
1
+
1
a
)
(
1
+
1
b
)
(
1
+
1
c
)
=
2
\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=2
(
1
+
a
1
)
(
1
+
b
1
)
(
1
+
c
1
)
=
2
1
1
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Bosnia and Herzegovina JBMO TST 2015 Problem 1
Solve equation
x
(
x
+
1
)
=
y
(
y
+
4
)
x(x+1) = y(y+4)
x
(
x
+
1
)
=
y
(
y
+
4
)
where
x
x
x
,
y
y
y
are positive integers