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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2020 Bosnia and Herzegovina Junior BMO TST
2020 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
2
1
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JBMO TST Bosnia and Herzegovina P2
A board
n
×
n
n \times n
n
×
n
is divided into
n
2
n^2
n
2
unit squares and a number is written in each unit square. Such a board is called interesting if the following conditions hold:
∘
\circ
∘
In all unit squares below the main diagonal, the number
0
0
0
is written;
∘
\circ
∘
Positive integers are written in all other unit squares.
∘
\circ
∘
When we look at the sums in all
n
n
n
rows, and the sums in all
n
n
n
columns, those
2
n
2n
2
n
numbers are actually the numbers
1
,
2
,
.
.
.
,
2
n
1,2,...,2n
1
,
2
,
...
,
2
n
(not necessarily in that order).
a
)
a)
a
)
Determine the largest number that can appear in a
6
×
6
6 \times 6
6
×
6
interesting board.
b
)
b)
b
)
Prove that there is no interesting board of dimensions
7
×
7
7\times 7
7
×
7
.
4
1
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JBMO TST Bosnia and Herzegovina P4
Determine the largest positive integer
n
n
n
such that the following statement holds: If
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
a_1,a_2,a_3,a_4,a_5,a_6
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
are six distinct positive integers less than or equal to
n
n
n
, then there exist
3
3
3
distinct positive integers ,from these six, say
a
,
b
,
c
a,b,c
a
,
b
,
c
s.t.
a
b
>
c
,
b
c
>
a
,
c
a
>
b
ab>c,bc>a,ca>b
ab
>
c
,
b
c
>
a
,
c
a
>
b
.
3
1
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JBMO TST Bosnia and Herzegovina 2020 P3
The angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
of triangle
A
B
C
ABC
A
BC
(
A
B
>
B
C
AB>BC
A
B
>
BC
) cuts the circumcircle of that triangle in
K
K
K
. The foot of the perpendicular from
K
K
K
to
A
B
AB
A
B
is
N
N
N
, and
P
P
P
is the midpoint of
B
N
BN
BN
. The line through
P
P
P
parallel to
B
C
BC
BC
cuts line
B
K
BK
B
K
in
T
T
T
. Prove that the line
N
T
NT
NT
passes through the midpoint of
A
C
AC
A
C
.
1
1
Hide problems
JBMO TST Bosnia and Herzegovina 2020 P1
Determine all four-digit numbers
a
b
c
d
‾
\overline{abcd}
ab
c
d
which are perfect squares and for which the equality holds:
a
b
‾
=
3
⋅
c
d
‾
+
1
\overline{ab}=3 \cdot \overline{cd} + 1
ab
=
3
⋅
c
d
+
1
.