MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece Team Selection Test
2018 Greece Team Selection Test
2018 Greece Team Selection Test
Part of
Greece Team Selection Test
Subcontests
(3)
3
1
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The identity gives perfect square
Find all functions
f
:
Z
>
0
↦
Z
>
0
f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}
f
:
Z
>
0
↦
Z
>
0
such that
x
f
(
x
)
+
(
f
(
y
)
)
2
+
2
x
f
(
y
)
xf(x)+(f(y))^2+2xf(y)
x
f
(
x
)
+
(
f
(
y
)
)
2
+
2
x
f
(
y
)
is perfect square for all positive integers
x
,
y
x,y
x
,
y
. **This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.
2
1
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Concyclic points!
A triangle
A
B
C
ABC
A
BC
is inscribed in a circle
(
C
)
(C)
(
C
)
.Let
G
G
G
the centroid of
△
A
B
C
\triangle ABC
△
A
BC
. We draw the altitudes
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
of the given triangle .Rays
A
G
AG
A
G
and
G
D
GD
G
D
meet (C) at
M
M
M
and
N
N
N
.Prove that points
F
,
E
,
M
,
N
F,E,M,N
F
,
E
,
M
,
N
are concyclic.
1
1
Hide problems
Simple inequality
If
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive real numbers such that
x
+
y
+
z
=
9
x
y
z
.
x + y + z = 9xyz.
x
+
y
+
z
=
9
x
yz
.
Prove that:
x
x
2
+
2
y
z
+
2
+
y
y
2
+
2
z
x
+
2
+
z
z
2
+
2
x
y
+
2
≥
1.
\frac {x} {\sqrt {x^2+2yz+2}}+\frac {y} {\sqrt {y^2+2zx+2}}+\frac {z} {\sqrt {z^2+2xy+2}}\ge 1.
x
2
+
2
yz
+
2
x
+
y
2
+
2
z
x
+
2
y
+
z
2
+
2
x
y
+
2
z
≥
1.
Consider when equality applies.