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Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1996 Greece National Olympiad
1996 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
1
1
Hide problems
a_{n+2}/ a_n = 1/ 4 , a_{k+1}/a_k+ a_{n+1}/a_n}=1 , geometric progression
Let
a
n
a_n
a
n
be a sequence of positive numbers such that: i)
a
n
+
2
a
n
=
1
4
\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}
a
n
a
n
+
2
=
4
1
, for every
n
∈
N
⋆
n\in\mathbb{N}^{\star}
n
∈
N
⋆
ii)
a
k
+
1
a
k
+
a
n
+
1
a
n
=
1
\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1
a
k
a
k
+
1
+
a
n
a
n
+
1
=
1
, for every
k
,
n
∈
N
⋆
k,n\in\mathbb{N}^{\star}
k
,
n
∈
N
⋆
with
∣
k
−
n
∣
≠
1
|k-n|\neq 1
∣
k
−
n
∣
=
1
. (a) Prove that
(
a
n
)
(a_n)
(
a
n
)
is a geometric progression. (n) Prove that exists
t
>
0
t>0
t
>
0
, such that
a
n
+
1
≤
1
2
a
n
+
t
\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t
a
n
+
1
≤
2
1
a
n
+
t
4
1
Hide problems
Number of functions
Find the number of functions
f
:
{
1
,
2
,
.
.
.
,
n
}
→
{
1995
,
1996
}
f : \{1, 2, . . . , n\} \to \{1995, 1996\}
f
:
{
1
,
2
,
...
,
n
}
→
{
1995
,
1996
}
such that
f
(
1
)
+
f
(
2
)
+
.
.
.
+
f
(
1996
)
f(1) + f(2) + ... + f(1996)
f
(
1
)
+
f
(
2
)
+
...
+
f
(
1996
)
is odd.
2
1
Hide problems
easy geometry
Let
A
B
C
ABC
A
BC
be an acute triangle,
A
D
,
B
E
,
C
Z
AD,BE,CZ
A
D
,
BE
,
CZ
its altitudes and
H
H
H
its orthocenter. Let
A
I
,
A
Θ
AI,A \Theta
A
I
,
A
Θ
be the internal and external bisectors of angle
A
A
A
. Let
M
,
N
M,N
M
,
N
be the midpoints of
B
C
,
A
H
BC,AH
BC
,
A
H
, respectively. Prove that: (a)
M
N
MN
MN
is perpendicular to
E
Z
EZ
EZ
(b) if
M
N
MN
MN
cuts the segments
A
I
,
A
Θ
AI,A \Theta
A
I
,
A
Θ
at the points
K
,
L
K,L
K
,
L
, then KL\equal{}AH
3
1
Hide problems
fourth power of an integer
Prove that among
81
81
81
natural numbers whose prime divisors are in the set
{
2
,
3
,
5
}
\{2, 3, 5\}
{
2
,
3
,
5
}
there exist four numbers whose product is the fourth power of an integer.