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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2018 Greece Junior Math Olympiad
2018 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Circles are tangent
Let
A
B
C
ABC
A
BC
with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
be an acute angled triangle and
c
c
c
its circumcircle. Let
D
D
D
be the point diametrically opposite to
A
A
A
. Point
K
K
K
is on
B
D
BD
B
D
such that
K
B
=
K
C
KB=KC
K
B
=
K
C
. The circle
(
K
,
K
C
)
(K, KC)
(
K
,
K
C
)
intersects
A
C
AC
A
C
at point
E
E
E
. Prove that the circle
(
B
K
E
)
(BKE)
(
B
K
E
)
is tangent to
c
c
c
.
3
1
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Integer fraction implies perfect square
Let
a
a
a
and
b
b
b
be positive integers with
b
b
b
odd, such that the number
(
a
+
b
)
2
+
4
a
a
b
\frac{(a+b)^2+4a}{ab}
ab
(
a
+
b
)
2
+
4
a
is an integer. Prove that
a
a
a
is a perfect square.
2
1
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Rectangle containing no black squares
A
8
×
8
8\times 8
8
×
8
board is given. Seven out of
64
64
64
unit squares are painted black. Suppose that there exists a positive
k
k
k
such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area
k
k
k
containing no black squares. Find the maximum value of
k
k
k
.
1
1
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Both numbers are rational
a) Does there exist a real number
x
x
x
such that
x
+
3
x+\sqrt{3}
x
+
3
and
x
2
+
3
x^2+\sqrt{3}
x
2
+
3
are both rationals? b) Does there exist a real number
y
y
y
such that
y
+
3
y+\sqrt{3}
y
+
3
and
y
3
+
3
y^3+\sqrt{3}
y
3
+
3
are both rationals?