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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia and Herzegovina BMO TST
2022 Bosnia and Herzegovina BMO TST
2022 Bosnia and Herzegovina BMO TST
Part of
Bosnia and Herzegovina BMO TST
Subcontests
(3)
3
1
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Bosnia and Herzegovina 2022 BMO TST P3
Cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in circle
k
k
k
with center
O
O
O
. The angle bisector of
A
B
D
ABD
A
B
D
intersects
A
D
AD
A
D
and
k
k
k
in
K
,
M
K,M
K
,
M
respectively, and the angle bisector of
C
B
D
CBD
CB
D
intersects
C
D
CD
C
D
and
k
k
k
in
L
,
N
L,N
L
,
N
respectively. If
K
L
∥
M
N
KL\parallel MN
K
L
∥
MN
prove that the circumcircle of triangle
M
O
N
MON
MON
bisects segment
B
D
BD
B
D
.
2
1
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Bosnia and Herzegovina 2022 BMO TST P2
Determine all positive integers
A
=
a
n
a
n
−
1
…
a
1
a
0
‾
A= \overline{a_n a_{n-1} \ldots a_1 a_0}
A
=
a
n
a
n
−
1
…
a
1
a
0
such that not all of its digits are equal and no digit is
0
0
0
, and
A
A
A
divides all numbers of the following form:
A
1
=
a
0
a
n
a
n
−
1
…
a
2
a
1
‾
,
A
2
=
a
1
a
0
a
n
…
a
3
a
2
‾
,
…
,
A_1 = \overline{a_0 a_n a_{n-1} \ldots a_2 a_1}, A_2 = \overline{a_1 a_0 a_{n} \ldots a_3 a_2}, \ldots ,
A
1
=
a
0
a
n
a
n
−
1
…
a
2
a
1
,
A
2
=
a
1
a
0
a
n
…
a
3
a
2
,
…
,
A
n
−
1
=
a
n
−
2
a
n
−
3
…
a
0
a
n
a
n
−
1
‾
,
A
n
=
a
n
−
1
a
n
−
2
…
a
1
a
0
a
n
‾
A_{n-1} = \overline{a_{n-2} a_{n-3} \ldots a_0 a_n a_{n-1}}, A_n = \overline{a_{n-1} a_{n-2} \ldots a_1 a_0 a_n}
A
n
−
1
=
a
n
−
2
a
n
−
3
…
a
0
a
n
a
n
−
1
,
A
n
=
a
n
−
1
a
n
−
2
…
a
1
a
0
a
n
.
1
1
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Bosnia and Herzegovina 2022 BMO TST P1
Let
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3, \ldots
a
1
,
a
2
,
a
3
,
…
be an infinite sequence of nonnegative real numbers such that for all positive integers
k
k
k
the following conditions hold:
i
)
i)
i
)
a
k
−
2
a
k
+
1
+
a
k
+
2
≥
0
a_k-2a_{k+1}+a_{k+2} \geq 0
a
k
−
2
a
k
+
1
+
a
k
+
2
≥
0
;
i
i
)
ii)
ii
)
∑
j
=
1
k
a
j
≤
1
\sum_{j=1}^{k} a_j \leq 1
∑
j
=
1
k
a
j
≤
1
. Prove that for all positive integer
k
k
k
holds:
0
≤
a
k
−
a
k
+
1
<
2
k
2
0 \leq a_k - a_{k+1} < \frac{2}{k^2}
0
≤
a
k
−
a
k
+
1
<
k
2
2