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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2001 Greece Junior Math Olympiad
2001 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
3
1
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Combinatorics with weights
We are given
8
8
8
different weights and a balance without a scale. (a) Find the smallest number of weighings necessary to find the heaviest weight. (b) How many weighting is further necessary to find the second heaviest weight?
2
1
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Number Theory
(a) Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of integers satisfying
m
3
−
4
m
n
2
=
8
n
3
−
2
m
2
n
m^3-4mn^2=8n^3-2m^2n
m
3
−
4
m
n
2
=
8
n
3
−
2
m
2
n
(b) Among such pairs find those for which
m
+
n
2
=
3
m+n^2=3
m
+
n
2
=
3
1
1
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Inequality
Let
a
,
b
,
x
,
y
a, b, x, y
a
,
b
,
x
,
y
be positive real numbers such that
a
+
b
=
1
a+b=1
a
+
b
=
1
. Prove that
1
a
x
+
b
y
≤
a
x
+
b
y
\frac{1}{\frac{a}{x}+\frac{b}{y}}\leq ax+by
x
a
+
y
b
1
≤
a
x
+
b
y
and find when equality holds.
4
1
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equal segments in a triangle, angle chasing (Greece Junior 2001)
Let
A
B
C
ABC
A
BC
be a triangle with altitude
A
D
AD
A
D
, angle bisectors
A
E
AE
A
E
and
B
Z
BZ
BZ
that intersecting at point
I
I
I
. From point
I
I
I
let
I
T
IT
I
T
be a perpendicular on
A
C
AC
A
C
. Also let line
(
e
)
(e)
(
e
)
be perpendicular on
A
C
AC
A
C
at point
A
A
A
. Extension of
E
T
ET
ET
intersects line
(
e
)
(e)
(
e
)
at point
K
K
K
. Prove that
A
K
=
A
D
AK=AD
A
K
=
A
D
.