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International Contests
Junior Balkan MO
2011 Junior Balkan MO
2011 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
4
1
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Jbmo 2011 Problem 4
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and points
E
E
E
and
F
F
F
on sides
A
B
,
C
D
AB,CD
A
B
,
C
D
such that
A
B
A
E
=
C
D
D
F
=
n
\tfrac{AB}{AE}=\tfrac{CD}{DF}=n
A
E
A
B
=
D
F
C
D
=
n
If
S
S
S
is the area of
A
E
F
D
AEFD
A
EF
D
show that
S
≤
A
B
⋅
C
D
+
n
(
n
−
1
)
A
D
2
+
n
2
D
A
⋅
B
C
2
n
2
{S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}
S
≤
2
n
2
A
B
⋅
C
D
+
n
(
n
−
1
)
A
D
2
+
n
2
D
A
⋅
BC
3
1
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Jbmo 2011 Problem 3
Let
n
>
3
n>3
n
>
3
be a positive integer. Equilateral triangle ABC is divided into
n
2
n^2
n
2
smaller congruent equilateral triangles (with sides parallel to its sides). Let
m
m
m
be the number of rhombuses that contain two small equilateral triangles and
d
d
d
the number of rhombuses that contain eight small equilateral triangles. Find the difference
m
−
d
m-d
m
−
d
in terms of
n
n
n
.
2
1
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Jbmo 2011 Problem 2
Find all primes
p
p
p
such that there exist positive integers
x
,
y
x,y
x
,
y
that satisfy
x
(
y
2
−
p
)
+
y
(
x
2
−
p
)
=
5
p
x(y^2-p)+y(x^2-p)=5p
x
(
y
2
−
p
)
+
y
(
x
2
−
p
)
=
5
p
1
1
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Jbmo 2011 Problem 1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
1
abc = 1
ab
c
=
1
. Prove that:
∏
(
a
5
+
a
4
+
a
3
+
a
2
+
a
+
1
)
≥
8
(
a
2
+
a
+
1
)
(
b
2
+
b
+
1
)
(
c
2
+
c
+
1
)
\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)
∏
(
a
5
+
a
4
+
a
3
+
a
2
+
a
+
1
)
≥
8
(
a
2
+
a
+
1
)
(
b
2
+
b
+
1
)
(
c
2
+
c
+
1
)