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15
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2020 PUMaC Team 15
Source:
January 1, 2022
floor function
algebra
Problem Statement
Suppose that f is a function
f
:
R
≥
0
→
R
f : R_{\ge 0} \to R
f
:
R
≥
0
→
R
so that for all
x
,
y
∈
R
≥
0
x, y \in R_{\ge 0}
x
,
y
∈
R
≥
0
(nonnegative reals) we have that
f
(
x
)
+
f
(
y
)
=
f
(
x
+
y
+
x
y
)
+
f
(
x
)
f
(
y
)
.
f(x)+f(y) = f(x+y+xy)+f(x)f(y).
f
(
x
)
+
f
(
y
)
=
f
(
x
+
y
+
x
y
)
+
f
(
x
)
f
(
y
)
.
Given that
f
(
3
5
)
=
1
2
f\left(\frac{3}{5} \right) = \frac12
f
(
5
3
)
=
2
1
and
f
(
1
)
=
3
f(1) = 3
f
(
1
)
=
3
, determine
⌊
log
2
(
−
f
(
1
0
2021
−
1
)
)
⌋
.
\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.
⌊
lo
g
2
(
−
f
(
1
0
2021
−
1
))⌋
.
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