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National and Regional Contests
China Contests
ASDAN Math Tournament
2019 ASDAN Math Tournament
2019 ASDAN Math Tournament
Part of
ASDAN Math Tournament
Subcontests
(10)
10
1
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2019 Geometry # 10
Regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
has side length
1
1
1
. Given that
P
P
P
is a point inside
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
, compute the minimum of
A
P
3
+
C
P
+
D
P
+
E
P
3
AP \sqrt3 + CP + DP + EP\sqrt3
A
P
3
+
CP
+
D
P
+
EP
3
.
9
1
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2019 Geometry # 9
Consider triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with circumradius
R
=
10
R = 10
R
=
10
, inradius
r
=
2
r = 2
r
=
2
and semi-perimeter
S
=
18
S = 18
S
=
18
. Let
I
I
I
be the incenter, and we extend
A
I
AI
A
I
,
B
I
BI
B
I
and
C
I
CI
C
I
to intersect the circumcircle at
D
,
E
D, E
D
,
E
and
F
F
F
respectively. Compute the area of
△
D
E
F
\vartriangle DEF
△
D
EF
.
8
1
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2019 Geometry # 8
Let triangle
△
A
E
F
\vartriangle AEF
△
A
EF
be inscribed in a square
A
B
C
D
ABCD
A
BC
D
such that
E
E
E
lies on
B
C
BC
BC
and
F
F
F
lies on
C
D
CD
C
D
. If
∠
E
A
F
=
4
5
o
\angle EAF = 45^o
∠
E
A
F
=
4
5
o
and
∠
B
E
A
=
7
0
o
\angle BEA = 70^o
∠
BE
A
=
7
0
o
, compute
∠
C
F
E
\angle CF E
∠
CFE
.
7
1
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2019 Geometry # 7
Consider a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with
A
B
=
7
AB = 7
A
B
=
7
,
B
C
=
8
BC = 8
BC
=
8
,
C
A
=
9
CA = 9
C
A
=
9
, and area
12
5
12\sqrt5
12
5
. We draw squares on each sides, namely
B
C
D
2
D
1
BCD_2D_1
BC
D
2
D
1
,
C
A
E
2
E
1
CAE_2E_1
C
A
E
2
E
1
and
A
B
F
2
F
1
ABF_2F_1
A
B
F
2
F
1
, so that the interiors of the squares do not intersect the interior of the triangle. What is the area of
△
D
2
E
2
F
2
\vartriangle D_2E_2F_2
△
D
2
E
2
F
2
?
6
1
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2019 Geometry # 6
Consider a triangle
△
A
C
E
\vartriangle ACE
△
A
CE
with
∠
A
C
E
=
4
5
o
\angle ACE = 45^o
∠
A
CE
=
4
5
o
and
∠
C
E
A
=
7
5
o
\angle CEA = 75^o
∠
CE
A
=
7
5
o
. Define points
Q
,
R
Q, R
Q
,
R
, and
P
P
P
such that
A
Q
AQ
A
Q
,
C
R
CR
CR
, and
E
P
EP
EP
are the altitudes of
△
A
C
E
\vartriangle ACE
△
A
CE
. Let
H
H
H
be the intersection of
A
Q
AQ
A
Q
,
C
R
CR
CR
, and
E
P
EP
EP
. Next define points
B
,
D
B, D
B
,
D
, and
F
F
F
as follows. Extend
E
P
EP
EP
to point
B
B
B
such that
B
P
=
H
P
BP = HP
BP
=
H
P
, extend
A
Q
AQ
A
Q
to point
D
D
D
such that
D
Q
=
H
Q
DQ = HQ
D
Q
=
H
Q
, and extend
C
R
CR
CR
to point
F
F
F
such that
F
R
=
H
R
F R = HR
FR
=
H
R
. Finally, lengths
C
H
=
2
CH = 2
C
H
=
2
,
A
H
=
2
AH =\sqrt2
A
H
=
2
, and
E
H
=
3
−
1
EH =\sqrt3 - 1
E
H
=
3
−
1
. Compute the area of hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
.
5
1
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2019 Geometry # 5
Trapezoid
A
B
C
D
ABCD
A
BC
D
has properties
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
,
A
B
=
15
AB = 15
A
B
=
15
,
C
D
=
27
CD = 27
C
D
=
27
, and
B
C
=
A
D
=
10
BC = AD = 10
BC
=
A
D
=
10
. A smaller trapezoid
E
F
G
H
EF GH
EFG
H
is drawn within
A
B
C
D
ABCD
A
BC
D
with
A
B
∥
E
F
AB\parallel EF
A
B
∥
EF
,
B
C
∥
F
G
BC\parallel F G
BC
∥
FG
,
C
D
∥
G
H
CD\parallel GH
C
D
∥
G
H
, and
D
A
∥
H
E
DA\parallel HE
D
A
∥
H
E
such that each edge in
A
B
C
D
ABCD
A
BC
D
is a distance
2
2
2
away from the corresponding edge in
E
F
G
H
EF GH
EFG
H
. Compute the area of
E
F
G
H
EF GH
EFG
H
.
4
1
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2019 Geometry # 4
Suppose
Z
,
Y
Z, Y
Z
,
Y
, and
W
W
W
are points on a circle such that lengths
Z
Y
=
Y
W
ZY = Y W
Z
Y
=
YW
. Extend
Z
Y
ZY
Z
Y
and let
X
X
X
be a point on
Z
Y
ZY
Z
Y
where
Z
Y
=
Y
X
ZY = Y X
Z
Y
=
Y
X
. If
X
W
XW
X
W
is a tangent of the circle, what is
∠
W
X
Y
\angle W XY
∠
W
X
Y
?
3
3
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2019 Geometry # 3
Consider an equilateral triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with side length
1
1
1
. Let
D
D
D
and
E
E
E
lie on segments
A
B
AB
A
B
and
A
C
AC
A
C
respectively such that
∠
A
D
E
=
3
0
o
\angle ADE = 30^o
∠
A
D
E
=
3
0
o
and
D
E
DE
D
E
is tangent to the incircle of
△
A
B
C
\vartriangle ABC
△
A
BC
. Compute the perimeter of
△
A
D
E
\vartriangle ADE
△
A
D
E
.
2019 Discrete Tiebreaker #3
5
5
5
monkeys,
5
5
5
snakes, and
5
5
5
tigers are standing in line at the local grocery store, with animals of the same species being indistinguishable. A monkey stands at the front of the line and a tiger stands at the end of the line. Unfortunately, monkeys and tigers are sworn enemies, so monkeys and tigers cannot stand in adjacent places in line. Compute the number of possible arrangements of the line.
2019 Geometry Tiebreaker #3
Consider a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with
B
C
=
10
BC = 10
BC
=
10
. An excircle is a circle that is tangent to one side of the triangle as well as the extensions of the other two sides; suppose that the excircle opposite vertex
B
B
B
has center
I
2
I_2
I
2
and exradius
r
2
=
11
r_2 = 11
r
2
=
11
, and suppose that the excircle opposite vertex
C
C
C
has center
I
3
I_3
I
3
and exradius
r
3
=
13
r_3 = 13
r
3
=
13
. Compute
I
2
I
3
I_2I_3
I
2
I
3
.
2
3
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2019 Discrete Tiebreaker #2
Let
P
1
,
P
2
,
…
,
P
720
P_1,P_2,\dots,P_{720}
P
1
,
P
2
,
…
,
P
720
denote the integers whose digits are a permutation of
123456
123456
123456
, arranged in ascending order (so
P
1
=
123456
P_1=123456
P
1
=
123456
,
P
2
=
123465
P_2=123465
P
2
=
123465
, and
P
720
=
654321
P_{720}=654321
P
720
=
654321
). What is
P
144
P_{144}
P
144
?
2019 Geometry #2
A square and a line intersect at a
4
5
o
45^o
4
5
o
angle. The line bisects the square into two unequal pieces such that the area of one piece is twice that of the other. If the square has side length
6
6
6
, compute the length of the cut due to the line. https://cdn.artofproblemsolving.com/attachments/6/4/2eb33fb9766497d25d342001cdbae9a7ffd4b4.png
2019 Geometry Tiebreaker # 2
Consider a triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with
A
B
=
5
AB = 5
A
B
=
5
and
B
C
=
4
BC = 4
BC
=
4
. Let
G
G
G
be the centroid of the triangle, and let
P
P
P
lie on line
A
G
AG
A
G
such that
A
G
=
G
P
AG = GP
A
G
=
GP
and
P
≠
A
P\ne A
P
=
A
. Suppose that
P
P
P
lies on the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
. Compute
C
A
CA
C
A
.
1
3
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2019 Discrete Tiebreaker #1
What is the greatest positive integer
x
x
x
for which
2
2
x
+
1
+
2
2^{2^x+1}+2
2
2
x
+
1
+
2
is divisible by
17
17
17
?
2019 Geometry # 1
A square
A
B
C
D
ABCD
A
BC
D
and point
E
E
E
are drawn in a plane such that lengths
D
E
<
B
E
DE < BE
D
E
<
BE
and
△
A
C
E
\vartriangle ACE
△
A
CE
is equilateral. Compute
∠
B
A
E
\angle BAE
∠
B
A
E
.
2019 Geometry Tiebreaker # 1
A kite is a quadrilateral with
2
2
2
pairs of equal adjacent sides. Given a cyclic kite with side lengths
3
3
3
and
4
4
4
, compute the distance between the intersection of its diagonals and the center of the circle circumscribing it.