MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1995 Greece National Olympiad
1995 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
Hide problems
Labelling the intersection points of lines
Given are the lines
l
1
,
l
2
,
…
,
l
k
l_1,l_2,\ldots ,l_k
l
1
,
l
2
,
…
,
l
k
in the plane, no two of which are parallel and no three of which are concurrent. For which
k
k
k
can one label the intersection points of these lines by
1
,
2
,
…
,
k
−
1
1, 2,\ldots , k-1
1
,
2
,
…
,
k
−
1
so that in each of the given lines all the labels appear exactly once?
1
1
Hide problems
-5^4 + 5^5 + 5^n and 2^4 + 2^7 + 2^n perfect squares
Find all positive integers
n
n
n
such that
−
5
4
+
5
5
+
5
n
-5^4 + 5^5 + 5^n
−
5
4
+
5
5
+
5
n
is a perfect square. Do the same for
2
4
+
2
7
+
2
n
.
2^4 + 2^7 + 2^n.
2
4
+
2
7
+
2
n
.
3
1
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ax^4+bx^3+cx^2+dx+e=0 has at least one real root [Greece 1995]
If the equation
a
x
2
+
(
c
−
b
)
x
+
(
e
−
d
)
=
0
ax^2+(c-b)x+(e-d)=0
a
x
2
+
(
c
−
b
)
x
+
(
e
−
d
)
=
0
has real roots greater than
1
1
1
, prove that the equation
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
=
0
ax^4+bx^3+cx^2+dx+e=0
a
x
4
+
b
x
3
+
c
x
2
+
d
x
+
e
=
0
has at least one real root.
2
1
Hide problems
Altitude
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
and let
D
D
D
be a point on
B
C
BC
BC
such that the incircle of
A
B
D
ABD
A
B
D
and the excircle of
A
D
C
ADC
A
D
C
corresponding to
A
A
A
have the same radius. Prove that this radius is equal to one quarter of the altitude from
B
B
B
of triangle
A
B
C
ABC
A
BC
.