MathDB
Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2018 Bosnia and Herzegovina Team Selection Test
2018 Bosnia and Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(3)
4
1
Hide problems
Bosnia and Herzegovina 2018 TST Day 2 Problem 1
Every square of
1000
×
1000
1000 \times 1000
1000
×
1000
board is colored black or white. It is known that exists one square
10
×
10
10 \times 10
10
×
10
such that all squares inside it are black and one square
10
×
10
10 \times 10
10
×
10
such that all squares inside are white. For every square
K
K
K
10
×
10
10 \times 10
10
×
10
we define its power
m
(
K
)
m(K)
m
(
K
)
as an absolute value of difference between number of white and black squares
1
×
1
1 \times 1
1
×
1
in square
K
K
K
. Let
T
T
T
be a square
10
×
10
10 \times 10
10
×
10
which has minimum power among all squares
10
×
10
10 \times 10
10
×
10
in this board. Determine maximal possible value of
m
(
T
)
m(T)
m
(
T
)
3
1
Hide problems
Bosnia and Herzegovina 2018 TST Day 1 Problem 3
Find all values of positive integers
a
a
a
and
b
b
b
such that it is possible to put
a
a
a
ones and
b
b
b
zeros in every of vertices in polygon with
a
+
b
a+b
a
+
b
sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring
0
0
0
and
1
1
1
switch places and in every other vertices other than those two numbers remain the same.
1
1
Hide problems
Bosnia and Herzegovina 2018 TST Day 1 Problem 1
In acute triangle
A
B
C
ABC
A
BC
(
A
B
<
A
C
)
(AB < AC)
(
A
B
<
A
C
)
let
D
D
D
,
E
E
E
and
F
F
F
be foots of perpedicular from
A
A
A
,
B
B
B
and
C
C
C
to
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively. Let
P
P
P
and
Q
Q
Q
be points on line
E
F
EF
EF
such that
D
P
⊥
E
F
DP \perp EF
D
P
⊥
EF
and
B
Q
=
C
Q
BQ=CQ
BQ
=
CQ
. Prove that
∠
A
D
P
=
∠
P
B
Q
\angle ADP = \angle PBQ
∠
A
D
P
=
∠
PBQ