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India STEMS
2024 India STEMS
STEMS 2024 Math Cat A
P2
P2
Part of
STEMS 2024 Math Cat A
Problems
(1)
find all (p,q) that forms Z^2
Source: STEMS 2024, CAT B - P1, CAT A - P2
12/17/2023
Let
S
=
Z
×
Z
S = \mathbb Z \times \mathbb Z
S
=
Z
×
Z
. A subset
P
P
P
of
S
S
S
is called nice if[*]
(
a
,
b
)
∈
P
⟹
(
b
,
a
)
∈
P
(a, b) \in P \implies (b, a) \in P
(
a
,
b
)
∈
P
⟹
(
b
,
a
)
∈
P
[*]
(
a
,
b
)
(a, b)
(
a
,
b
)
,
(
c
,
d
)
∈
P
⟹
(
a
+
c
,
b
−
d
)
∈
P
(c, d)\in P \implies (a + c, b - d) \in P
(
c
,
d
)
∈
P
⟹
(
a
+
c
,
b
−
d
)
∈
P
Find all
(
p
,
q
)
∈
S
(p, q) \in S
(
p
,
q
)
∈
S
so that if
(
p
,
q
)
∈
P
(p, q) \in P
(
p
,
q
)
∈
P
for some nice set
P
P
P
then
P
=
S
P = S
P
=
S
.
combinatorics