MathDB
Problems
Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia And Herzegovina - Regional Olympiad
2010 Bosnia And Herzegovina - Regional Olympiad
2010 Bosnia And Herzegovina - Regional Olympiad
Part of
Bosnia And Herzegovina - Regional Olympiad
Subcontests
(4)
4
4
Show problems
3
2
Hide problems
Regional Olympiad - FBH 2010 Grade 9 Problem 3
If
a
a
a
and
b
b
b
are positive integers such that
a
b
∣
a
2
+
b
2
ab \mid a^2+b^2
ab
∣
a
2
+
b
2
prove that
a
=
b
a=b
a
=
b
Regional Olympiad - FBH 2010 Grade 11 Problem 3
Let
n
n
n
be an odd positive integer bigger than
1
1
1
. Prove that
3
n
+
1
3^n+1
3
n
+
1
is not divisible with
n
n
n
2
3
Hide problems
Regional Olympiad - FBH 2010 Grade 9 Problem 2
In convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
O
O
O
at angle
9
0
∘
90^{\circ}
9
0
∘
. Let
K
K
K
,
L
L
L
,
M
M
M
and
N
N
N
be orthogonal projections of point
O
O
O
to sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
and
D
A
DA
D
A
of quadrilateral
A
B
C
D
ABCD
A
BC
D
. Prove that
K
L
M
N
KLMN
K
L
MN
is cyclic
Regional Olympiad - FBH 2010 Grade 10 Problem 2
It is given acute triangle
A
B
C
ABC
A
BC
with orthocenter at point
H
H
H
. Prove that
A
H
⋅
h
a
+
B
H
⋅
h
b
+
C
H
⋅
h
c
=
a
2
+
b
2
+
c
2
2
AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}
A
H
⋅
h
a
+
B
H
⋅
h
b
+
C
H
⋅
h
c
=
2
a
2
+
b
2
+
c
2
where
a
a
a
,
b
b
b
and
c
c
c
are sides of a triangle, and
h
a
h_a
h
a
,
h
b
h_b
h
b
and
h
c
h_c
h
c
altitudes of
A
B
C
ABC
A
BC
Regional Olympiad - FBH 2010 Grade 11 Problem 2
Angle bisector from vertex
A
A
A
of acute triangle
A
B
C
ABC
A
BC
intersects side
B
C
BC
BC
in point
D
D
D
, and circumcircle of
A
B
C
ABC
A
BC
in point
E
E
E
(different from
A
A
A
). Let
F
F
F
and
G
G
G
be foots of perpendiculars from point
D
D
D
to sides
A
B
AB
A
B
and
A
C
AC
A
C
. Prove that area of quadrilateral
A
E
F
G
AEFG
A
EFG
is equal to the area of triangle
A
B
C
ABC
A
BC
1
4
Show problems