MathDB
Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2018 Bosnia and Herzegovina Junior BMO TST
2018 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
4
1
Hide problems
P4 Bosnia and Herzegovina JBMO TST
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers which satisfy:
a
+
b
+
c
=
2
a+b+c=2
a
+
b
+
c
=
2
a
2
+
b
2
+
c
2
=
2
a^2+b^2+c^2=2
a
2
+
b
2
+
c
2
=
2
Prove that at least one of numbers
∣
a
−
b
∣
,
∣
b
−
c
∣
,
∣
c
−
a
∣
|a-b|, |b-c|, |c-a|
∣
a
−
b
∣
,
∣
b
−
c
∣
,
∣
c
−
a
∣
is greater or equal than
1
1
1
.
3
1
Hide problems
P3 Bosnia and Herzegovina JBMO TST
Let
Γ
\Gamma
Γ
be circumscribed circle of triangle
A
B
C
ABC
A
BC
(
A
B
≠
A
C
)
(AB \neq AC)
(
A
B
=
A
C
)
. Let
O
O
O
be circumcenter of the triangle
A
B
C
ABC
A
BC
. Let
M
M
M
be a point where angle bisector of angle
B
A
C
BAC
B
A
C
intersects
Γ
\Gamma
Γ
. Let
D
D
D
(
D
≠
M
)
(D \neq M)
(
D
=
M
)
be a point where circumscribed circle of the triangle
B
O
M
BOM
BOM
intersects line segment
A
M
AM
A
M
and let
E
E
E
(
E
≠
M
)
(E \neq M)
(
E
=
M
)
be a point where circumscribed circle of triangle
C
O
M
COM
COM
intersects line segment
A
M
AM
A
M
. Prove that
B
D
+
C
E
=
A
M
BD+CE=AM
B
D
+
CE
=
A
M
.
2
1
Hide problems
P2 Bosnia and Herzegovina JBMO TST
Find all integer triples
(
p
,
m
,
n
)
(p,m,n)
(
p
,
m
,
n
)
that satisfy:
p
m
−
n
3
=
27
p^m-n^3=27
p
m
−
n
3
=
27
where
p
p
p
is a prime number.
1
1
Hide problems
P1 Bosnia and Herzegovina JBMO TST
Students are in classroom with
n
n
n
rows. In each row there are
m
m
m
tables. It's given that
m
,
n
≥
3
m,n \geq 3
m
,
n
≥
3
. At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end there were
252
252
252
handshakes. How many students were in the classroom?