MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2008 Greece National Olympiad
2008 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
Hide problems
Inequality for n integers
If
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots , a_n
a
1
,
a
2
,
…
,
a
n
are positive integers and
k
=
max
{
a
1
,
…
,
a
n
}
k = \max\{a_1, \ldots, a_n\}
k
=
max
{
a
1
,
…
,
a
n
}
,
t
=
min
{
a
1
,
…
,
a
n
}
t = \min\{a_1,\ldots, a_n\}
t
=
min
{
a
1
,
…
,
a
n
}
, prove the inequality
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
a
1
+
a
2
+
⋯
+
a
n
)
k
n
t
≥
a
1
a
2
⋯
a
n
.
\left(\frac{a_1^2+a_2^2+\cdots+a_n^2}{a_1+a_2+\cdots+a_n}\right)^{\frac{kn}{t}} \geq a_1a_2\cdots a_n.
(
a
1
+
a
2
+
⋯
+
a
n
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
t
kn
≥
a
1
a
2
⋯
a
n
.
When does equality hold?
3
1
Hide problems
Compute the area of the concave quadrilateral ABHC
A triangle
A
B
C
ABC
A
BC
with orthocenter
H
H
H
is inscribed in a circle with center
K
K
K
and radius
1
1
1
, where the angles at
B
B
B
and
C
C
C
are non-obtuse. If the lines
H
K
HK
HK
and
B
C
BC
BC
meet at point
S
S
S
such that
S
K
(
S
K
−
S
H
)
=
1
SK(SK -SH) = 1
S
K
(
S
K
−
S
H
)
=
1
, compute the area of the concave quadrilateral
A
B
H
C
ABHC
A
B
H
C
.
2
1
Hide problems
On the equation x^8 + 2^{2^x+2} = p (x ∈ Z, p ∈ P)
Find all integers
x
x
x
and prime numbers
p
p
p
satisfying
x
8
+
2
2
x
+
2
=
p
x^8 + 2^{2^x+2} = p
x
8
+
2
2
x
+
2
=
p
.
1
1
Hide problems
A computer which generates integers from reals
A computer generates all pairs of real numbers
x
,
y
∈
(
0
,
1
)
x, y \in (0, 1)
x
,
y
∈
(
0
,
1
)
for which the numbers
a
=
x
+
m
y
a = x+my
a
=
x
+
m
y
and
b
=
y
+
m
x
b = y+mx
b
=
y
+
m
x
are both integers, where
m
m
m
is a given positive integer. Finding one such pair
(
x
,
y
)
(x, y)
(
x
,
y
)
takes
5
5
5
seconds. Find
m
m
m
if the computer needs
595
595
595
seconds to find all possible ordered pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
.