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T5
BMT 2021 General Tiebreaker p5
BMT 2021 General Tiebreaker p5
Source:
September 27, 2023
algebra
Problem Statement
Let
r
,
s
,
t
,
u
r, s, t, u
r
,
s
,
t
,
u
be the distinct roots of the polynomial
x
4
+
2
x
3
+
3
x
2
+
3
x
+
5
x^4 + 2x^3 + 3x^2 + 3x + 5
x
4
+
2
x
3
+
3
x
2
+
3
x
+
5
. For
n
≥
1
n \ge 1
n
≥
1
, define
s
n
=
r
n
+
s
n
+
t
n
+
u
n
s_n = r^n + s^n + t^n + u^n
s
n
=
r
n
+
s
n
+
t
n
+
u
n
and
t
n
=
s
1
+
s
2
+
.
.
.
+
s
n
t_n = s_1 + s_2 + ...+ s_n
t
n
=
s
1
+
s
2
+
...
+
s
n
. Compute
t
4
+
2
t
3
+
3
t
2
+
3
t
1
+
5
t_4 + 2t_3 + 3t_2 + 3t_1 + 5
t
4
+
2
t
3
+
3
t
2
+
3
t
1
+
5
.
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