MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2020 Greece National Olympiad
2020 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
3
1
Hide problems
no 1 to 2030, replace consecutive (a,b) with (a-b)^{2020}, last remains
On the board there are written in a row, the integers from
1
1
1
until
2030
2030
2030
(included that) in an increasing order. We have the right of ''movement''
K
K
K
: We choose any two numbers
a
,
b
a,b
a
,
b
that are written in consecutive positions and we replace the pair
(
a
,
b
)
(a,b)
(
a
,
b
)
by the number
(
a
−
b
)
2020
(a-b)^{2020}
(
a
−
b
)
2020
. We repeat the movement
K
K
K
, many times until only one number remains written on the board. Determine whether it would be possible, that number to be: (i)
202
0
2020
2020^{2020}
202
0
2020
(ii)
202
1
2020
2021^{2020}
202
1
2020
4
1
Hide problems
\frac{a+b}{a^2+k^2b^2-k^2ab} is a composite number for all a,b \in N+
Find all values of the positive integer
k
k
k
that has the property: There are no positive integers
a
,
b
a,b
a
,
b
such that the expression
A
(
k
,
a
,
b
)
=
a
+
b
a
2
+
k
2
b
2
−
k
2
a
b
A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}
A
(
k
,
a
,
b
)
=
a
2
+
k
2
b
2
−
k
2
ab
a
+
b
is a composite positive number.
2
1
Hide problems
two perpendiculars and a line of a parallelogram are concurrent, equal angles
Given a line segment
A
B
AB
A
B
and a point
C
C
C
lies inside it such that
A
B
=
3
⋅
A
C
AB=3 \cdot AC
A
B
=
3
⋅
A
C
. Construct a parallelogram
A
C
D
E
ACDE
A
C
D
E
such that
A
C
=
D
E
=
C
E
>
A
R
AC=DE=CE>AR
A
C
=
D
E
=
CE
>
A
R
. Let
Z
Z
Z
be a point on
A
C
AC
A
C
such that
∠
A
E
Z
=
∠
A
C
E
=
ω
\angle AEZ=\angle ACE =\omega
∠
A
EZ
=
∠
A
CE
=
ω
. Prove that the line passing through point
B
B
B
and perpendicular on side
E
C
EC
EC
, and the line passing through point
D
D
D
and perpendicular on side
A
B
AB
A
B
, intersect on point , let it be
K
K
K
, lying on line
E
Z
EZ
EZ
.
1
1
Hide problems
P((Q(x))^3)=xP(x)(Q(x))^3 , polynomials
Find all non constant polynomials
P
(
x
)
,
Q
(
x
)
P(x),Q(x)
P
(
x
)
,
Q
(
x
)
with real coefficients such that:
P
(
(
Q
(
x
)
)
3
)
=
x
P
(
x
)
(
Q
(
x
)
)
3
P((Q(x))^3)=xP(x)(Q(x))^3
P
((
Q
(
x
)
)
3
)
=
x
P
(
x
)
(
Q
(
x
)
)
3