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National and Regional Contests
USA Contests
MAA AMC
USAMO
2013 USAMO
2013 USAMO
Part of
USAMO
Subcontests
(3)
6
1
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Special Points on $BC$
Let
A
B
C
ABC
A
BC
be a triangle. Find all points
P
P
P
on segment
B
C
BC
BC
satisfying the following property: If
X
X
X
and
Y
Y
Y
are the intersections of line
P
A
PA
P
A
with the common external tangent lines of the circumcircles of triangles
P
A
B
PAB
P
A
B
and
P
A
C
PAC
P
A
C
, then
(
P
A
X
Y
)
2
+
P
B
⋅
P
C
A
B
⋅
A
C
=
1.
\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.
(
X
Y
P
A
)
2
+
A
B
⋅
A
C
PB
⋅
PC
=
1.
3
1
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Operating on Triangles
Let
n
n
n
be a positive integer. There are
n
(
n
+
1
)
2
\tfrac{n(n+1)}{2}
2
n
(
n
+
1
)
marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing
n
n
n
marks. Initially, each mark has the black side up. An operation is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called admissible if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration
C
C
C
, let
f
(
C
)
f(C)
f
(
C
)
denote the smallest number of operations required to obtain
C
C
C
from the initial configuration. Find the maximum value of
f
(
C
)
f(C)
f
(
C
)
, where
C
C
C
varies over all admissible configurations.
2
1
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Paths around a circle
For a positive integer
n
≥
3
n\geq 3
n
≥
3
plot
n
n
n
equally spaced points around a circle. Label one of them
A
A
A
, and place a marker at
A
A
A
. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of
2
n
2n
2
n
distinct moves available; two from each point. Let
a
n
a_n
a
n
count the number of ways to advance around the circle exactly twice, beginning and ending at
A
A
A
, without repeating a move. Prove that
a
n
−
1
+
a
n
=
2
n
a_{n-1}+a_n=2^n
a
n
−
1
+
a
n
=
2
n
for all
n
≥
4
n\geq 4
n
≥
4
.