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Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1999 Greece National Olympiad
1999 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
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Dividing the circle with segments connecting n points
On a circle are given
n
≥
3
n\ge 3
n
≥
3
points. At most, how many parts can the segments with the endpoints at these
n
n
n
points divide the interior of the circle into?
3
1
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Circumcircle of NKL passes through midpoint of BC
In an acute-angled triangle
A
B
C
ABC
A
BC
,
A
D
,
B
E
AD,BE
A
D
,
BE
and
C
F
CF
CF
are the altitudes and
H
H
H
the orthocentre. Lines
E
F
EF
EF
and
B
C
BC
BC
meet at
N
N
N
. The line passing through
D
D
D
and parallel to
F
E
FE
FE
meets lines
A
B
AB
A
B
and
A
C
AC
A
C
at
K
K
K
and
L
L
L
, respectively. Prove that the circumcircle of the triangle
N
K
L
NKL
N
K
L
bisects the side
B
C
BC
BC
.
2
1
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Right triangle with integer lengths
A right triangle has integer side lengths, and the sum of its area and the length of one of its legs equals
75
75
75
. Find the side lengths of the triangle.
1
1
Hide problems
f(x^2)f(y^2) is at least f(xy)^2 for quadratic f
Let
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x)=ax^2+bx+c
f
(
x
)
=
a
x
2
+
b
x
+
c
, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are nonnegative real numbers, not all equal to zero. Prove that
f
(
x
y
)
2
≤
f
(
x
2
)
f
(
y
2
)
f(xy)^2\le f(x^2)f(y^2)
f
(
x
y
)
2
≤
f
(
x
2
)
f
(
y
2
)
for all real numbers
x
,
y
x,y
x
,
y
.