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Korea Contests
Korea Junior Mathematics Olympiad
2016 Korea Junior Math Olympiad
1
prove integer
prove integer
Source: 2016 KJMO #1
November 13, 2016
number theory
algebra
Problem Statement
positive reals
a
1
,
a
2
,
.
.
.
a_1, a_2, . . .
a
1
,
a
2
,
...
satisfying (i)
a
n
+
1
=
a
1
2
⋅
a
2
2
⋅
.
.
.
⋅
a
n
2
−
3
a_{n+1}=a_1^2\cdot a_2^2 \cdot . . . \cdot a_n^2-3
a
n
+
1
=
a
1
2
⋅
a
2
2
⋅
...
⋅
a
n
2
−
3
(all positive integers
n
n
n
) (ii)
1
2
(
a
1
+
a
2
−
1
)
\frac{1}{2}(a_1+\sqrt{a_2-1})
2
1
(
a
1
+
a
2
−
1
)
is positive integer. prove that
1
2
(
a
1
⋅
a
2
⋅
.
.
.
⋅
a
n
+
a
n
+
1
−
1
)
\frac{1}{2}(a_1 \cdot a_2 \cdot . . . \cdot a_n + \sqrt{a_{n+1}-1})
2
1
(
a
1
⋅
a
2
⋅
...
⋅
a
n
+
a
n
+
1
−
1
)
is positive integer
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