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Taiwan APMO Prelininary
2019 Taiwan APMO Preliminary Test
P4
2019 Taiwan APMO Preliminary P4
2019 Taiwan APMO Preliminary P4
Source: 2019 Taiwan APMO Preliminary
August 23, 2020
algebra
floor function
Sequence
recursive
Problem Statement
We define a sequence
a
n
{a_n}
a
n
:
a
1
=
1
,
a
n
+
1
=
a
n
+
n
2
,
n
=
1
,
2
,
.
.
.
a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...
a
1
=
1
,
a
n
+
1
=
a
n
+
n
2
,
n
=
1
,
2
,
...
(1)Find
⌊
a
2019
⌋
\lfloor a_{2019}\rfloor
⌊
a
2019
⌋
(2)Find
⌊
a
1
2
⌋
+
⌊
a
2
2
⌋
+
.
.
.
+
⌊
a
20
2
⌋
\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor
⌊
a
1
2
⌋
+
⌊
a
2
2
⌋
+
...
+
⌊
a
20
2
⌋
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