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2020 Princeton University Math Competition
13
13
Part of
2020 Princeton University Math Competition
Problems
(1)
2020 PUMaC Team 13
Source:
1/1/2022
Will and Lucas are playing a game. Will claims that he has a polynomial
f
f
f
with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute
i
i
i
for the value of
f
(
i
)
f(i)
f
(
i
)
, starting from minute
1
1
1
. If Will is telling the truth, he will report
f
(
i
)
f(i)
f
(
i
)
. Otherwise, he will randomly and uniformly pick a positive integer from the range
[
1
,
(
i
+
1
)
!
]
[1,(i+1)!]
[
1
,
(
i
+
1
)!]
. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round
20
20
20
is
a
/
b
a/b
a
/
b
. If the prime factorization of
b
b
b
is
p
1
e
1
.
.
.
p
k
e
k
p_1^{e_1}... p_k^{e_k}
p
1
e
1
...
p
k
e
k
, determine the sum
∑
i
=
1
k
e
i
\sum_{i=1}^{k} e_i
∑
i
=
1
k
e
i
.
algebra
combinatorics