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LMT
2023 LMT Fall
14
14
Part of
2023 LMT Fall
Problems
(2)
2023 Fall LMT Team #14 sum i gcd(i ,100).
Source:
2/4/2024
Find
∑
i
=
1
100
i
gcd
(
i
,
100
)
.
\sum^{100}_{i=1}i \gcd(i ,100).
i
=
1
∑
100
i
g
cd
(
i
,
100
)
.
number theory
2023 Fall LMT Speed p14 , [ACD]+[BCD]
Source:
3/2/2024
In obtuse triangle
A
B
C
ABC
A
BC
with
A
B
=
7
AB = 7
A
B
=
7
,
B
C
=
20
BC = 20
BC
=
20
, and
C
A
=
15
C A = 15
C
A
=
15
, let point
D
D
D
be the foot of the altitude from
C
C
C
to line
A
B
AB
A
B
. Evaluate
[
A
C
D
]
+
[
B
C
D
]
[ACD]+[BCD]
[
A
C
D
]
+
[
BC
D
]
. (Note that
[
X
Y
Z
]
[XY Z]
[
X
Y
Z
]
means the area of triangle
X
Y
Z
XY Z
X
Y
Z
.)Proposed by Jonathan Liu
geometry