MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2018 Greece National Olympiad
2018 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(3)
3
1
Hide problems
Polynomial and absolute value
Let
n
,
m
n,m
n
,
m
be positive integers such that
n
<
m
n<m
n
<
m
and
a
1
,
a
2
,
.
.
.
,
a
m
a_1, a_2, ..., a_m
a
1
,
a
2
,
...
,
a
m
be different real numbers. (a) Find all polynomials
P
P
P
with real coefficients and degree at most
n
n
n
such that:
∣
P
(
a
i
)
−
P
(
a
j
)
∣
=
∣
a
i
−
a
j
∣
|P(a_i)-P(a_j)|=|a_i-a_j|
∣
P
(
a
i
)
−
P
(
a
j
)
∣
=
∣
a
i
−
a
j
∣
for all
i
,
j
=
{
1
,
2
,
.
.
.
,
m
}
i,j=\{1, 2, ..., m\}
i
,
j
=
{
1
,
2
,
...
,
m
}
such that
i
<
j
i<j
i
<
j
. (b) If
n
,
m
≥
2
n,m\ge 2
n
,
m
≥
2
does there exist a polynomial
Q
Q
Q
with real coefficients and degree
n
n
n
such that:
∣
Q
(
a
i
)
−
Q
(
a
j
)
∣
<
∣
a
i
−
a
j
∣
|Q(a_i)-Q(a_j)|<|a_i-a_j|
∣
Q
(
a
i
)
−
Q
(
a
j
)
∣
<
∣
a
i
−
a
j
∣
for all
i
,
j
=
{
1
,
2
,
.
.
.
,
m
}
i,j=\{1, 2, ..., m\}
i
,
j
=
{
1
,
2
,
...
,
m
}
such that
i
<
j
i<j
i
<
j
Edit: See #3
2
1
Hide problems
Collinear if and only if
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
and
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
the circumscribed circle. Let
D
,
E
D, E
D
,
E
be points in the small arcs
A
C
,
A
B
AC, AB
A
C
,
A
B
respectively. Let
K
K
K
be the intersection point of
B
D
,
C
E
BD,CE
B
D
,
CE
and
N
N
N
the second common point of the circumscribed circles of the triangles
B
K
E
BKE
B
K
E
and
C
K
D
CKD
C
KD
. Prove that
A
,
K
,
N
A, K, N
A
,
K
,
N
are collinear if and only if
K
K
K
belongs to the symmedian of
A
B
C
ABC
A
BC
passing from
A
A
A
.
1
1
Hide problems
First term is a square
Let
(
x
n
)
,
n
∈
N
(x_n), n\in\mathbb{N}
(
x
n
)
,
n
∈
N
be a sequence such that
x
n
+
1
=
3
x
n
3
+
x
n
,
∀
n
∈
N
x_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N}
x
n
+
1
=
3
x
n
3
+
x
n
,
∀
n
∈
N
and
x
1
=
a
b
x_1=\frac{a}{b}
x
1
=
b
a
where
a
,
b
a,b
a
,
b
are positive integers such that
3
∤
b
3\not|b
3
∣
b
. If
x
m
x_m
x
m
is a square of a rational number for some positive integer
m
m
m
, prove that
x
1
x_1
x
1
is also a square of a rational number.