MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
1998 Greece National Olympiad
1998 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
Hide problems
Prove g(k-1)=g(k)=g(k+1) for non-decreasing g is impossible
Let a function
g
:
N
0
→
N
0
g:\mathbb{N}_0\to\mathbb{N}_0
g
:
N
0
→
N
0
satisfy
g
(
0
)
=
0
g(0)=0
g
(
0
)
=
0
and
g
(
n
)
=
n
−
g
(
g
(
n
−
1
)
)
g(n)=n-g(g(n-1))
g
(
n
)
=
n
−
g
(
g
(
n
−
1
))
for all
n
≥
1
n\ge 1
n
≥
1
. Prove that:a)
g
(
k
)
≥
g
(
k
−
1
)
g(k)\ge g(k-1)
g
(
k
)
≥
g
(
k
−
1
)
for any positive integer
k
k
k
. b) There is no
k
k
k
such that
g
(
k
−
1
)
=
g
(
k
)
=
g
(
k
+
1
)
g(k-1)=g(k)=g(k+1)
g
(
k
−
1
)
=
g
(
k
)
=
g
(
k
+
1
)
.
2
1
Hide problems
Sum of squares of segments is greater than twice the area
For a regular
n
n
n
-gon, let
M
M
M
be the set of the lengths of the segments joining its vertices. Show that the sum of the squares of the elements of
M
M
M
is greater than twice the area of the polygon.
1
1
Hide problems
Infinitely many arithmetic progressions
Prove that for any integer
n
>
3
n>3
n
>
3
there exist infinitely many non-constant arithmetic progressions of length
n
−
1
n-1
n
−
1
whose terms are positive integers whose product is a perfect
n
n
n
-th power.
3
1
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a,b,c non-zero real numbers
Prove that for any non-zero real numbers
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
(
b
+
c
−
a
)
2
(
b
+
c
)
2
+
a
2
+
(
c
+
a
−
b
)
2
(
c
+
a
)
2
+
b
2
+
(
a
+
b
−
c
)
2
(
a
+
b
)
2
+
c
2
≥
3
5
.
\frac{(b+c-a)^2}{(b+c)^2+a^2} + \frac{(c+a-b)^2}{(c+a)^2+b^2} + \frac{(a+b-c)^2}{(a+b)^2+c^2} \geq \frac 35.
(
b
+
c
)
2
+
a
2
(
b
+
c
−
a
)
2
+
(
c
+
a
)
2
+
b
2
(
c
+
a
−
b
)
2
+
(
a
+
b
)
2
+
c
2
(
a
+
b
−
c
)
2
≥
5
3
.