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International Contests
Balkan MO
1993 Balkan MO
1993 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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Equation has non-trivial solution iff $m=p$ (Balkan 1993)
Let
p
p
p
be a prime and
m
≥
2
m \geq 2
m
≥
2
be an integer. Prove that the equation
x
p
+
y
p
2
=
(
x
+
y
2
)
m
\frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m
2
x
p
+
y
p
=
(
2
x
+
y
)
m
has a positive integer solution
(
x
,
y
)
≠
(
1
,
1
)
(x, y) \neq (1, 1)
(
x
,
y
)
=
(
1
,
1
)
if and only if
m
=
p
m = p
m
=
p
.Romania
3
1
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Two circles externally tangent again!
Circles
C
1
\mathcal C_1
C
1
and
C
2
\mathcal C_2
C
2
with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, are externally tangent at point
λ
\lambda
λ
. A circle
C
\mathcal C
C
with center
O
O
O
touches
C
1
\mathcal C_1
C
1
at
A
A
A
and
C
2
\mathcal C_2
C
2
at
B
B
B
so that the centers
O
1
O_1
O
1
,
O
2
O_2
O
2
lie inside
C
C
C
. The common tangent to
C
1
\mathcal C_1
C
1
and
C
2
\mathcal C_2
C
2
at
λ
\lambda
λ
intersects the circle
C
\mathcal C
C
at
K
K
K
and
L
L
L
. If
D
D
D
is the midpoint of the segment
K
L
KL
K
L
, show that
∠
O
1
O
O
2
=
∠
A
D
B
\angle O_1OO_2 = \angle ADB
∠
O
1
O
O
2
=
∠
A
D
B
. Greece
2
1
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A monotone positive integer
A positive integer given in decimal representation
a
n
a
n
−
1
…
a
1
a
0
‾
\overline{ a_na_{n-1} \ldots a_1a_0 }
a
n
a
n
−
1
…
a
1
a
0
is called monotone if
a
n
≤
a
n
−
1
≤
⋯
≤
a
0
a_n\leq a_{n-1} \leq \cdots \leq a_0
a
n
≤
a
n
−
1
≤
⋯
≤
a
0
. Determine the number of monotone positive integers with at most 1993 digits.
1
1
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Six real numbers with sum 10
Let
a
,
b
,
c
,
d
,
e
,
f
a,b,c,d,e,f
a
,
b
,
c
,
d
,
e
,
f
be six real numbers with sum 10, such that
(
a
−
1
)
2
+
(
b
−
1
)
2
+
(
c
−
1
)
2
+
(
d
−
1
)
2
+
(
e
−
1
)
2
+
(
f
−
1
)
2
=
6.
(a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6.
(
a
−
1
)
2
+
(
b
−
1
)
2
+
(
c
−
1
)
2
+
(
d
−
1
)
2
+
(
e
−
1
)
2
+
(
f
−
1
)
2
=
6.
Find the maximum possible value of
f
f
f
. Cyprus