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Simon Marais Mathematical Competition
2024 Simon Marais Mathematical Competition
B3
B3
Part of
2024 Simon Marais Mathematical Competition
Problems
(1)
g: lines -> points, L_3 connects g(L_1), g(L_2) => g(L_3) is their intersection
Source: SMMC 2024 B3
10/12/2024
Let
L
\mathcal{L}
L
be the set of all lines in the plane and let
P
\mathcal{P}
P
be the set of all points in the plane. Determine whether there exists a function
g
:
L
→
P
g : \mathcal{L} \to \mathcal{P}
g
:
L
→
P
such that for any two distinct non-parallel lines
ℓ
1
,
ℓ
2
∈
L
\ell_1, \ell_2 \in \mathcal{L}
ℓ
1
,
ℓ
2
∈
L
, we have
(
a
)
(a)
(
a
)
g
(
ℓ
1
)
≠
g
(
ℓ
2
)
g(\ell_1) \neq g(\ell_2)
g
(
ℓ
1
)
=
g
(
ℓ
2
)
, and
(
b
)
(b)
(
b
)
if
ℓ
3
\ell_3
ℓ
3
is the line passing through
g
(
ℓ
1
)
g(\ell_1)
g
(
ℓ
1
)
and
g
(
ℓ
2
)
g(\ell_2)
g
(
ℓ
2
)
, then
g
(
ℓ
3
)
g(\ell_3)
g
(
ℓ
3
)
is the intersection of
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
.
geometry