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Philippine MO
2012 Philippine MO
4
14th Philippine Mathematical Olympiad 2011-National Stage #4
14th Philippine Mathematical Olympiad 2011-National Stage #4
Source:
October 13, 2014
induction
algebra unsolved
algebra
Problem Statement
Let
⋆
\star
⋆
be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers
x
x
x
and
y
y
y
, (i)
(
x
+
1
)
⋆
0
=
(
0
⋆
x
)
+
1
(x + 1)\star 0 = (0\star x) + 1
(
x
+
1
)
⋆
0
=
(
0
⋆
x
)
+
1
(ii)
0
⋆
(
y
+
1
)
=
(
y
⋆
0
)
+
1
0\star (y + 1) = (y\star 0) + 1
0
⋆
(
y
+
1
)
=
(
y
⋆
0
)
+
1
(iii)
(
x
+
1
)
⋆
(
y
+
1
)
=
(
x
⋆
y
)
+
1
(x + 1)\star (y + 1) = (x\star y) + 1
(
x
+
1
)
⋆
(
y
+
1
)
=
(
x
⋆
y
)
+
1
. If
123
⋆
456
=
789
123\star 456 = 789
123
⋆
456
=
789
, find
246
⋆
135
246\star 135
246
⋆
135
.
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