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2021 Latvia TST
1.2
1.2
Part of
2021 Latvia TST
Problems
(1)
Cute construction problem
Source: Latvian TST 2021 Day 1 P2
3/12/2021
Prove it is possible to find
2
2021
2^{2021}
2
2021
different pairs of positive integers
(
a
i
,
b
i
)
(a_i,b_i)
(
a
i
,
b
i
)
such that:
1
a
i
b
i
+
1
a
2
b
2
+
…
+
1
a
2
2021
b
2
2021
=
1
\frac{1}{a_ib_i}+\frac{1}{a_2b_2} + \ldots + \frac{1}{a_{2^{2021}}b_{2^{2021}}} = 1
a
i
b
i
1
+
a
2
b
2
1
+
…
+
a
2
2021
b
2
2021
1
=
1
a
1
+
a
2
+
…
a
2
2021
+
b
1
+
b
2
+
…
+
b
2
2021
=
3
2022
a_1+a_2 +\ldots a_{2^{2021}} +b_1+b_2 + \ldots +b_{2^{2021}} = 3^{2022}
a
1
+
a
2
+
…
a
2
2021
+
b
1
+
b
2
+
…
+
b
2
2021
=
3
2022
Note: Pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
and
(
c
,
d
)
(c,d)
(
c
,
d
)
are different if
a
≠
c
a \neq c
a
=
c
or
b
≠
d
b \neq d
b
=
d
combinatorics