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Sweden Contests
Swedish Mathematical Competition
2023 Swedish Mathematical Competition
4
4
Part of
2023 Swedish Mathematical Competition
Problems
(1)
f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}) , f(x, y) <= xy
Source: 2023 Swedish Mathematical Competition p4
3/24/2024
Let
f
f
f
be a function that associates a positive integer
(
x
,
y
)
(x, y)
(
x
,
y
)
with each pair of positive integers
f
(
x
,
y
)
f(x, y)
f
(
x
,
y
)
. Suppose that
f
(
x
,
y
)
≤
x
y
f(x, y) \le xy
f
(
x
,
y
)
≤
x
y
for all positive integers
x
x
x
,
y
y
y
. Show that there are
2023
2023
2023
different pairs
(
x
1
,
y
1
)
(x_1, y_1)
(
x
1
,
y
1
)
,
.
.
.
...
...
,
(
x
2023
,
y
2023
(x_{2023}, y_{2023}
(
x
2023
,
y
2023
) such that
f
(
x
1
,
y
1
)
=
f
(
x
2
,
y
2
)
=
.
.
.
.
=
f
(
x
2023
,
y
2023
)
.
f(x_1, y_1) = f(x_2, y_2) = ....= f(x_{2023}, y_{2023}).
f
(
x
1
,
y
1
)
=
f
(
x
2
,
y
2
)
=
....
=
f
(
x
2023
,
y
2023
)
.
algebra
combinatorics