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National and Regional Contests
Greece Contests
Greece Team Selection Test
2019 Greece Team Selection Test
2019 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(3)
1
1
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max and min sum of numbers of overlapping hexagons in a triangular grid
Given an equilateral triangle with sidelength
k
k
k
cm. With lines parallel to it's sides, we split it into
k
2
k^2
k
2
small equilateral triangles with sidelength
1
1
1
cm. This way, a triangular grid is created. In every small triangle of sidelength
1
1
1
cm, we place exactly one integer from
1
1
1
to
k
2
k^2
k
2
(included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths
1
1
1
cm. We shall name as value of the hexagon, the sum of the numbers that lie on the
6
6
6
small equilateral triangles that the hexagon consists of . Find (in terms of the integer
k
>
4
k>4
k
>
4
) the maximum and the minimum value of the sum of the values of all hexagons .
4
1
Hide problems
(y^2+1)f(x)-yf(xy)=yf(x/y) for x,y>0
Find all functions
f
:
(
0
,
∞
)
↦
R
f:(0,\infty)\mapsto\mathbb{R}
f
:
(
0
,
∞
)
↦
R
such that
(
y
2
+
1
)
f
(
x
)
−
y
f
(
x
y
)
=
y
f
(
x
y
)
,
\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}
(
y
2
+
1
)
f
(
x
)
−
y
f
(
x
y
)
=
y
f
(
y
x
)
,
for every
x
,
y
>
0
x,y>0
x
,
y
>
0
.
2
1
Hide problems
Incircle and circumcircle
Let a triangle
A
B
C
ABC
A
BC
inscribed in a circle
Γ
\Gamma
Γ
with center
O
O
O
. Let
I
I
I
the incenter of triangle
A
B
C
ABC
A
BC
and
D
,
E
,
F
D, E, F
D
,
E
,
F
the contact points of the incircle with sides
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
of triangle
A
B
C
ABC
A
BC
respectively . Let also
S
S
S
the foot of the perpendicular line from
D
D
D
to the line
E
F
EF
EF
.Prove that line
S
I
SI
S
I
passes from the antidiametric point
N
N
N
of
A
A
A
in the circle
Γ
\Gamma
Γ
.(
A
N
AN
A
N
is a diametre of the circle
Γ
\Gamma
Γ
).