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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2024 Greece Junior Math Olympiad
2024 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
2
1
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junior perpenicularity, 2 circles related
Consider an acute triangle
A
B
C
ABC
A
BC
and it's circumcircle
ω
\omega
ω
. With center
A
A
A
, we construct a circle
γ
\gamma
γ
that intersects arc
A
B
AB
A
B
of circle
ω
\omega
ω
, that doesn't contain
C
C
C
, at point
D
D
D
and arc
A
C
AC
A
C
, that doesn't contain
B
B
B
, at point
E
E
E
. Suppose that the intersection point
K
K
K
of lines
B
E
BE
BE
and
C
D
CD
C
D
lies on circle
γ
\gamma
γ
. Prove that line
A
K
AK
A
K
is perpendicular on line
B
C
BC
BC
.
3
1
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16 pos.divisors of 2024 on 4x4 table such that each line / row multiple of 3
Examine if we can put the sixteen positive divisors of
2024
2024
2024
on the cells of the table shown such that the sum of the four numbers of any line or row to be a multiple of
3
3
3
. \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular}
4
1
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x^2+y^2+z^2+xy+yz+zx=6xyz diophantine
Prove that there are infinite triples of positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
x
2
+
y
2
+
z
2
+
x
y
+
y
z
+
z
x
=
6
x
y
z
.
x^2+y^2+z^2+xy+yz+zx=6xyz.
x
2
+
y
2
+
z
2
+
x
y
+
yz
+
z
x
=
6
x
yz
.
1
1
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(k+l+m)^2>= 3 (kl+lm+mk), a >= 3b^2 if a(x+y+z)=b(xy+yz+zx)=xyz
a) Prove that for all real numbers
k
,
l
,
m
k,l,m
k
,
l
,
m
holds :
(
k
+
l
+
m
)
2
≥
3
(
k
l
+
l
m
+
m
k
)
(k+l+m)^2 \ge 3 (kl+lm+mk)
(
k
+
l
+
m
)
2
≥
3
(
k
l
+
l
m
+
mk
)
When does equality holds?b) If
x
,
y
,
z
x,y,z
x
,
y
,
z
are positive real numbers and
a
,
b
a,b
a
,
b
real numbers such that
a
(
x
+
y
+
z
)
=
b
(
x
y
+
y
z
+
z
x
)
=
x
y
z
,
a(x+y+z)=b(xy+yz+zx)=xyz,
a
(
x
+
y
+
z
)
=
b
(
x
y
+
yz
+
z
x
)
=
x
yz
,
prove that
a
≥
3
b
2
a \ge 3b^2
a
≥
3
b
2
. When does equality holds?