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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2011 Greece Junior Math Olympiad
2011 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Inequality
If
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive real numbers with sum
12
12
12
, prove that
x
y
+
y
z
+
z
x
+
3
≥
x
+
y
+
z
\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+ 3 \ge \sqrt{x} +\sqrt{y }+\sqrt{z}
y
x
+
z
y
+
x
z
+
3
≥
x
+
y
+
z
. When equality is valid?
3
1
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Number Theory
If the number
3
n
+
1
3n +1
3
n
+
1
, where n is integer, is multiple of
7
7
7
, find the possible remainders of the following divisions: (a) of
n
n
n
with divisor
7
7
7
, (b) of
n
m
n^{m}
n
m
with divisor
7
7
7
, for all values of the positive integer
m
,
m
>
1
m, m >1
m
,
m
>
1
.
2
1
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Number Theory
We consider the set of four-digit positive integers
x
=
a
b
c
d
‾
x =\overline{abcd}
x
=
ab
c
d
with digits different than zero and pairwise different. We also consider the integers
y
=
d
c
b
a
‾
y = \overline{dcba}
y
=
d
c
ba
and we suppose that
x
>
y
x > y
x
>
y
. Find the greatest and the lowest value of the difference
x
−
y
x-y
x
−
y
, as well as the corresponding four-digit integers
x
,
y
x,y
x
,
y
for which these values are obtained.
1
1
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<A=120^o, median AD perpendicular to side AB (Greece Junior 2011)
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
=
12
0
o
\angle BAC=120^o
∠
B
A
C
=
12
0
o
, which the median
A
D
AD
A
D
is perpendicular to side
A
B
AB
A
B
and intersects the circumscribed circle of triangle
A
B
C
ABC
A
BC
at point
E
E
E
. Lines
B
A
BA
B
A
and
E
C
EC
EC
intersect at
Z
Z
Z
. Prove that a)
Z
D
⊥
B
E
ZD \perp BE
Z
D
⊥
BE
b)
Z
D
=
B
C
ZD=BC
Z
D
=
BC