MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2010 Greece National Olympiad
2010 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
Hide problems
Lines in the plane
On the plane are given k\plus{}n distinct lines , where
k
>
1
k>1
k
>
1
is integer and
n
n
n
is integer as well.Any three of these lines do not pass through the same point . Among these lines exactly
k
k
k
are parallel and all the other
n
n
n
lines intersect each other.All k\plus{}n lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines . If in a such given configuration the minimum number of ''good'' regionrs is
176
176
176
and the maximum number of these regions is
221
221
221
, find
k
k
k
and
n
n
n
. Babis
3
1
Hide problems
Concyclic points
A triangle
A
B
C
ABC
A
BC
is inscribed in a circle
C
(
O
,
R
)
C(O,R)
C
(
O
,
R
)
and has incenter
I
I
I
. Lines
A
I
,
B
I
,
C
I
AI,BI,CI
A
I
,
B
I
,
C
I
meet the circumcircle
(
O
)
(O)
(
O
)
of triangle
A
B
C
ABC
A
BC
at points
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. The circles with diameter
I
D
,
I
E
,
I
F
ID,IE,IF
I
D
,
I
E
,
I
F
meet the sides
B
C
,
C
A
,
A
B
BC,CA, AB
BC
,
C
A
,
A
B
at pairs of points
(
A
1
,
A
2
)
,
(
B
1
,
B
2
)
,
(
C
1
,
C
2
)
(A_1,A_2), (B_1, B_2), (C_1, C_2)
(
A
1
,
A
2
)
,
(
B
1
,
B
2
)
,
(
C
1
,
C
2
)
respectively.Prove that the six points
A
1
,
A
2
,
B
1
,
B
2
,
C
1
,
C
2
A_1,A_2, B_1, B_2, C_1, C_2
A
1
,
A
2
,
B
1
,
B
2
,
C
1
,
C
2
are concyclic. Babis
2
1
Hide problems
Inequality
If
x
,
y
x,y
x
,
y
are positive real numbers with sum
2
a
2a
2
a
, prove that : x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10} When does equality hold ? Babis
1
1
Hide problems
Diophantine Equation
Solve in the integers the diophantine equation
x
4
−
6
x
2
+
1
=
7
⋅
2
y
.
x^4-6x^2+1 = 7 \cdot 2^y.
x
4
−
6
x
2
+
1
=
7
⋅
2
y
.