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Romanian Masters of Mathematics Collection
2021 Romanian Master of Mathematics
1
1
Part of
2021 Romanian Master of Mathematics
Problems
(1)
How many points is too many
Source: RMM 2021/1
10/13/2021
Let
T
1
,
T
2
,
T
3
,
T
4
T_1, T_2, T_3, T_4
T
1
,
T
2
,
T
3
,
T
4
be pairwise distinct collinear points such that
T
2
T_2
T
2
lies between
T
1
T_1
T
1
and
T
3
T_3
T
3
, and
T
3
T_3
T
3
lies between
T
2
T_2
T
2
and
T
4
T_4
T
4
. Let
ω
1
\omega_1
ω
1
be a circle through
T
1
T_1
T
1
and
T
4
T_4
T
4
; let
ω
2
\omega_2
ω
2
be the circle through
T
2
T_2
T
2
and internally tangent to
ω
1
\omega_1
ω
1
at
T
1
T_1
T
1
; let
ω
3
\omega_3
ω
3
be the circle through
T
3
T_3
T
3
and externally tangent to
ω
2
\omega_2
ω
2
at
T
2
T_2
T
2
; and let
ω
4
\omega_4
ω
4
be the circle through
T
4
T_4
T
4
and externally tangent to
ω
3
\omega_3
ω
3
at
T
3
T_3
T
3
. A line crosses
ω
1
\omega_1
ω
1
at
P
P
P
and
W
W
W
,
ω
2
\omega_2
ω
2
at
Q
Q
Q
and
R
R
R
,
ω
3
\omega_3
ω
3
at
S
S
S
and
T
T
T
, and
ω
4
\omega_4
ω
4
at
U
U
U
and
V
V
V
, the order of these points along the line being
P
,
Q
,
R
,
S
,
T
,
U
,
V
,
W
P,Q,R,S,T,U,V,W
P
,
Q
,
R
,
S
,
T
,
U
,
V
,
W
. Prove that
P
Q
+
T
U
=
R
S
+
V
W
PQ + TU = RS + VW
PQ
+
T
U
=
RS
+
VW
Geza Kos, Hungary
geometry
RMM