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International Contests
Junior Balkan MO
1998 Junior Balkan MO
1998 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
3
1
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find all positive integers such that x^y = y^{x-y}
Find all pairs of positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that x^y \equal{} y^{x \minus{} y}. Albania
1
1
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Prove that 111...11222...225 is a perfect square
Prove that the number
111
…
11
⏟
1997
22
…
22
⏟
1998
5
\underbrace{111\ldots 11}_{1997}\underbrace{22\ldots 22}_{1998}5
1997
111
…
11
1998
22
…
22
5
(which has 1997 of 1-s and 1998 of 2-s) is a perfect square.
2
1
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JBMO geometry problem
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon such that
A
B
=
A
E
=
C
D
=
1
AB=AE=CD=1
A
B
=
A
E
=
C
D
=
1
,
∠
A
B
C
=
∠
D
E
A
=
9
0
∘
\angle ABC=\angle DEA=90^\circ
∠
A
BC
=
∠
D
E
A
=
9
0
∘
and
B
C
+
D
E
=
1
BC+DE=1
BC
+
D
E
=
1
. Compute the area of the pentagon. Greece
4
1
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mod 16
Do there exist 16 three digit numbers, using only three different digits in all, so that the all numbers give different residues when divided by 16? Bulgaria