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International Contests
Junior Balkan MO
2014 Junior Balkan MO
2014 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
2
1
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AREA in terms of S.
Consider an acute triangle
A
B
C
ABC
A
BC
of area
S
S
S
. Let
C
D
⊥
A
B
CD \perp AB
C
D
⊥
A
B
(
D
∈
A
B
D \in AB
D
∈
A
B
),
D
M
⊥
A
C
DM \perp AC
D
M
⊥
A
C
(
M
∈
A
C
M \in AC
M
∈
A
C
) and
D
N
⊥
B
C
DN \perp BC
D
N
⊥
BC
(
N
∈
B
C
N \in BC
N
∈
BC
). Denote by
H
1
H_1
H
1
and
H
2
H_2
H
2
the orthocentres of the triangles
M
N
C
MNC
MNC
, respectively
M
N
D
MND
MN
D
. Find the area of the quadrilateral
A
H
1
B
H
2
AH_1BH_2
A
H
1
B
H
2
in terms of
S
S
S
.
4
1
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GAME
For a positive integer
n
n
n
, two payers
A
A
A
and
B
B
B
play the following game: Given a pile of
s
s
s
stones, the players take turn alternatively with
A
A
A
going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of
n
n
n
stones. The winner is the one who takes the last stone. Assuming both
A
A
A
and
B
B
B
play perfectly, for how many values of
s
s
s
the player
A
A
A
cannot win?
3
1
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JBMO 2014 #3 -- Inequality
For positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
with
a
b
c
=
1
abc=1
ab
c
=
1
prove that
(
a
+
1
b
)
2
+
(
b
+
1
c
)
2
+
(
c
+
1
a
)
2
≥
3
(
a
+
b
+
c
+
1
)
\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)
(
a
+
b
1
)
2
+
(
b
+
c
1
)
2
+
(
c
+
a
1
)
2
≥
3
(
a
+
b
+
c
+
1
)
1
1
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JBMO 2014 #1 -- Equation with Prime Powers.
Find all triples of primes
(
p
,
q
,
r
)
(p,q,r)
(
p
,
q
,
r
)
satisfying
3
p
4
−
5
q
4
−
4
r
2
=
26
3p^{4}-5q^{4}-4r^{2}=26
3
p
4
−
5
q
4
−
4
r
2
=
26
.