MathDB
Problems
Contests
National and Regional Contests
Romania Contests
IMAR Test
2003 IMAR Test
2003 IMAR Test
Part of
IMAR Test
Subcontests
(4)
4
1
Hide problems
any 2 on an island are friends or enemies, min no of necklaces
On an island live
n
n
n
(
n
≥
2
n \ge 2
n
≥
2
)
x
y
z
xyz
x
yz
s. Any two
x
y
z
xyz
x
yz
s are either friends or enemies. Every
x
y
z
xyz
x
yz
wears a necklace made of colored beads such that any two
x
y
z
xyz
x
yz
s that are befriended have at least one bead of the same color and any two
x
y
z
xyz
x
yz
s that are enemies do not have any common colors in their necklaces. It is also possible for some necklaces not to have any beads. What is the minimum number of colors of beads that is sufficient to manufacture such necklaces regardless on the relationship between the
x
y
z
xyz
x
yz
s?
2
1
Hide problems
s \sqrt3 >= l_a + l_b + l_c, angle bisectors inequality
Prove that in a triangle the following inequality holds:
s
3
≥
ℓ
a
+
ℓ
b
+
ℓ
c
s\sqrt3 \ge \ell_a + \ell_b + \ell_c
s
3
≥
ℓ
a
+
ℓ
b
+
ℓ
c
where
ℓ
a
\ell_a
ℓ
a
is the length of the angle bisector from angle
A
A
A
, and
s
s
s
is the semiperimeter of the triangle
1
1
Hide problems
covering a regular petnagon by open discs with sides as diameters
Prove that the interior of a convex pentagon whose sides are all equal, is not covered by the open disks having the sides of the pentagon as diameter.
3
1
Hide problems
exinscribed circle and its tangency pts
The exinscribed circle of a triangle
A
B
C
ABC
A
BC
corresponding to its vertex
A
A
A
touches the sidelines
A
B
AB
A
B
and
A
C
AC
A
C
in the points
M
M
M
and
P
P
P
, respectively, and touches its side
B
C
BC
BC
in the point
N
N
N
. Show that if the midpoint of the segment
M
P
MP
MP
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
, then the points
O
O
O
,
N
N
N
,
I
I
I
are collinear, where
I
I
I
is the incenter and
O
O
O
is the circumcenter of triangle
A
B
C
ABC
A
BC
.