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2001 JBMO ShortLists
5
5
Part of
2001 JBMO ShortLists
Problems
(1)
At least one square between T_n-1 and T_n - JBMO Shortlist
Source:
10/30/2010
Let
x
k
=
k
(
k
+
1
)
2
x_k=\frac{k(k+1)}{2}
x
k
=
2
k
(
k
+
1
)
for all integers
k
≥
1
k\ge 1
k
≥
1
. Prove that for any integer
n
≥
10
n \ge 10
n
≥
10
, between the numbers
A
=
x
1
+
x
2
+
…
+
x
n
−
1
A=x_1+x_2 + \ldots + x_{n-1}
A
=
x
1
+
x
2
+
…
+
x
n
−
1
and
B
=
A
+
x
n
B=A+x_n
B
=
A
+
x
n
there is at least one square.
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number theory