MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece National Olympiad
2022 Greece National Olympiad
2022 Greece National Olympiad
Part of
Greece National Olympiad
Subcontests
(4)
4
1
Hide problems
How many elements can a good set contain?
Let
Q
n
Q_n
Q
n
be the set of all
n
n
n
-tuples
x
=
(
x
1
,
…
,
x
n
)
x=(x_1,\ldots,x_n)
x
=
(
x
1
,
…
,
x
n
)
with
x
i
∈
{
0
,
1
,
2
}
x_i \in \{0,1,2 \}
x
i
∈
{
0
,
1
,
2
}
,
i
=
1
,
2
,
…
,
n
i=1,2,\ldots,n
i
=
1
,
2
,
…
,
n
. A triple
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
(where
x
=
(
x
1
,
x
2
,
…
,
x
n
)
x=(x_1,x_2,\ldots,x_n)
x
=
(
x
1
,
x
2
,
…
,
x
n
)
,
y
=
(
y
1
,
y
2
,
…
,
y
n
)
y=(y_1,y_2,\ldots,y_n)
y
=
(
y
1
,
y
2
,
…
,
y
n
)
,
z
=
(
z
1
,
z
2
,
…
,
z
n
)
z=(z_1,z_2,\ldots,z_n)
z
=
(
z
1
,
z
2
,
…
,
z
n
)
) of distinct elements of
Q
n
Q_n
Q
n
is called a good triple, if there exists at least one
i
∈
{
1
,
2
,
…
,
n
}
i \in \{1,2, \ldots, n \}
i
∈
{
1
,
2
,
…
,
n
}
, for which
{
x
i
,
y
i
,
z
i
}
=
{
0
,
1
,
2
}
\{x_i,y_i,z_i \}=\{0,1,2 \}
{
x
i
,
y
i
,
z
i
}
=
{
0
,
1
,
2
}
. A subset
A
A
A
of
Q
n
Q_n
Q
n
will be called a good subset, if any three elements of
A
A
A
form a good triple. Prove that every good subset of
Q
n
Q_n
Q
n
contains at most
2
⋅
(
3
2
)
n
2 \cdot \left(\frac{3}{2}\right)^n
2
⋅
(
2
3
)
n
elements.
3
1
Hide problems
Classical-looking inequality
The positive real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfy the equality
a
+
b
c
+
c
d
+
d
b
+
1
a
b
2
c
2
d
2
=
18.
a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.
a
+
b
c
+
c
d
+
d
b
+
a
b
2
c
2
d
2
1
=
18.
Find the maximum possible value of
a
a
a
.
2
1
Hide problems
Number Theory involving divisors
Let
n
>
4
n>4
n
>
4
be a positive integer, which is divisible by
4
4
4
. We denote by
A
n
A_n
A
n
the sum of the odd positive divisors of
n
n
n
. We also denote
B
n
B_n
B
n
the sum of the even positive divisors of
n
n
n
, excluding the number
n
n
n
itself. Find the least possible value of the expression
f
(
n
)
=
B
n
−
2
A
n
,
f(n)=B_n-2A_n,
f
(
n
)
=
B
n
−
2
A
n
,
for all possible values of
n
n
n
, as well as for which positive integers
n
n
n
this minimum value is attained.
1
1
Hide problems
Three circles concur
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
. Let
D
,
E
D,E
D
,
E
be points on the segment
B
C
BC
BC
such that
B
D
=
B
A
BD=BA
B
D
=
B
A
and
C
E
=
C
A
CE=CA
CE
=
C
A
. If
K
K
K
is the circumcenter of triangle
A
D
E
ADE
A
D
E
,
F
F
F
is the intersection of lines
A
D
,
K
C
AD,KC
A
D
,
K
C
and
G
G
G
is the intersection of lines
A
E
,
K
B
AE,KB
A
E
,
K
B
, then prove that the circumcircle of triangle
K
D
E
KDE
KD
E
(let it be
c
1
c_1
c
1
), the circle with center the point
F
F
F
and radius
F
E
FE
FE
(let it be
c
2
c_2
c
2
) and the circle with center
G
G
G
and radius
G
D
GD
G
D
(let it be
c
3
c_3
c
3
) concur on a point which lies on the line
A
K
AK
A
K
.