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Stanford Mathematics Tournament
2019 Stanford Mathematics Tournament
2019 Stanford Mathematics Tournament
Part of
Stanford Mathematics Tournament
Subcontests
(11)
10
1
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SMT 2019 Geometry #10
Let
A
B
C
ABC
A
BC
be an acute triangle with
B
C
=
48
BC = 48
BC
=
48
. Let
M
M
M
be the midpoint of
B
C
BC
BC
, and let
D
D
D
and
E
E
E
be the feet of the altitudes drawn from
B
B
B
and
C
C
C
to
A
C
AC
A
C
and
A
B
AB
A
B
respectively. Let
P
P
P
be the intersection between the line through
A
A
A
parallel to
B
C
BC
BC
and line
D
E
DE
D
E
. If
A
P
=
10
AP = 10
A
P
=
10
, compute the length of
P
M
PM
PM
,
9
1
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SMT 2019 Geometry #9
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
∠
A
B
C
=
∠
C
D
A
=
4
5
o
\angle ABC = \angle CDA = 45^o
∠
A
BC
=
∠
C
D
A
=
4
5
o
,
A
B
=
7
AB = 7
A
B
=
7
, and
B
D
=
25
BD = 25
B
D
=
25
. If
A
C
AC
A
C
is perpendicular to
C
D
CD
C
D
, compute the length of
B
C
BC
BC
.
8
1
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SMT 2019 Geometry #8
In the coordinate plane, a point
A
A
A
is chosen on the line
y
=
3
2
x
y =\frac32 x
y
=
2
3
x
in the first quadrant. Two perpendicular lines
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
intersect at A where
ℓ
1
\ell_1
ℓ
1
has slope
m
>
1
m > 1
m
>
1
. Let
ℓ
1
\ell_1
ℓ
1
intersect the
x
x
x
-axis at
B
B
B
, and
ℓ
2
\ell_2
ℓ
2
intersects the
x
x
x
and
y
y
y
axes at
C
C
C
and
D
D
D
, respectively. Suppose that line
B
D
BD
B
D
has slope
−
m
-m
−
m
and
B
D
=
2
BD = 2
B
D
=
2
. Compute the length of
C
D
CD
C
D
.
7
1
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SMT 2019 Geometry #7
Let
G
G
G
be the centroid of triangle
A
B
C
ABC
A
BC
with
A
B
=
9
AB = 9
A
B
=
9
,
B
C
=
10
,
BC = 10,
BC
=
10
,
and
A
C
=
17
AC = 17
A
C
=
17
. Denote
D
D
D
as the midpoint of
B
C
BC
BC
. A line through
G
G
G
parallel to
B
C
BC
BC
intersects
A
B
AB
A
B
at
M
M
M
and
A
C
AC
A
C
at
N
N
N
. If
B
G
BG
BG
intersects
C
M
CM
CM
at
E
E
E
and
C
G
CG
CG
intersects
B
N
BN
BN
at
F
F
F
, compute the area of triangle
D
E
F
DEF
D
EF
.
6
1
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SMT 2019 Geometry #6
Let the altitude of
△
A
B
C
\vartriangle ABC
△
A
BC
from
A
A
A
intersect the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
at
D
D
D
. Let
E
E
E
be a point on line
A
D
AD
A
D
such that
E
≠
A
E \ne A
E
=
A
and
A
D
=
D
E
AD = DE
A
D
=
D
E
. If
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
14
BC = 14
BC
=
14
, and
A
C
=
15
AC = 15
A
C
=
15
, what is the area of quadrilateral
B
D
C
E
BDCE
B
D
CE
?
5
1
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SMT 2019 Geometry #5 prism
The bases of a right hexagonal prism are regular hexagons of side length
s
>
0
s > 0
s
>
0
, and the prism has height
h
h
h
. The prism contains some water, and when it is placed on a flat surface with a hexagonal face on the bottom, the water has depth
s
3
4
\frac{s\sqrt3}{4}
4
s
3
. The water depth doesn’t change when the prism is turned so that a rectangular face is on the bottom. Compute
h
s
\frac{h}{s}
s
h
.
4
1
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SMT 2019 Geometry #4 intersecting circles
Let
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
be two circles intersecting at points
P
P
P
and
Q
Q
Q
. The tangent line closer to
Q
Q
Q
touches
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
at
M
M
M
and
N
N
N
respectively. If
P
Q
=
3
P Q = 3
PQ
=
3
,
Q
N
=
2
QN = 2
QN
=
2
, and
M
N
=
P
N
MN = P N
MN
=
PN
, what is
Q
M
2
QM^2
Q
M
2
?
3
2
Hide problems
SMT 2019 Geometry #3
In triangle
A
B
C
ABC
A
BC
with
A
B
=
10
AB = 10
A
B
=
10
, let
D
D
D
be a point on side BC such that
A
D
AD
A
D
bisects
∠
B
A
C
\angle BAC
∠
B
A
C
. If
C
D
B
D
=
2
\frac{CD}{BD} = 2
B
D
C
D
=
2
and the area of
A
B
C
ABC
A
BC
is
50
50
50
, compute the value of
∠
B
A
D
\angle BAD
∠
B
A
D
in degrees.
SMT 2019 Geometry Tiebreaker #3 tangent circles
Right triangle
A
B
C
ABC
A
BC
with
∠
A
B
C
=
9
0
o
\angle ABC = 90^o
∠
A
BC
=
9
0
o
is inscribed in a circle
ω
1
\omega_1
ω
1
with radius
3
3
3
. A circle
ω
2
\omega_2
ω
2
tangent to
A
B
AB
A
B
,
B
C
BC
BC
, and
ω
1
\omega_1
ω
1
has radius
2
2
2
. Compute the area of
△
A
B
C
\vartriangle ABC
△
A
BC
.
2
2
Hide problems
SMT 2019 Geometry #2 cat and mouse
A cat and mouse live on a house mapped out by the points
(
−
1
,
0
)
(-1, 0)
(
−
1
,
0
)
,
(
−
1
,
2
)
(-1, 2)
(
−
1
,
2
)
,
(
0
,
3
)
(0, 3)
(
0
,
3
)
,
(
1
,
2
)
(1, 2)
(
1
,
2
)
,
(
1
,
0
)
(1, 0)
(
1
,
0
)
. The cat starts at the top of the house (point
(
0
,
3
)
(0, 3)
(
0
,
3
)
) and the mouse starts at the origin (0, 0). Both start running clockwise around the house at the same time. If the cat runs at
12
12
12
units a minute and the mouse at 9 units a minute, how many laps around the house will the cat run before it catches the mouse?
SMT 2019 Geometry Tiebreaker #2
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with
A
B
=
8
AB = 8
A
B
=
8
and
B
C
=
6
BC = 6
BC
=
6
. Point
E
E
E
is outside of the rectangle such that
C
E
=
D
E
CE = DE
CE
=
D
E
. Point
D
D
D
is reflected over line
A
E
AE
A
E
so that its image,
D
′
D'
D
′
, lies on the interior of the rectangle. Point
D
′
D'
D
′
is then reflected over diagonal
A
C
AC
A
C
, and its image lies on side
A
B
AB
A
B
. What is the length of
D
E
DE
D
E
?
1
2
Hide problems
SMT 2019 Geometry #1 semicircle
Let
A
B
C
D
ABCD
A
BC
D
be a unit square. A semicircle with diameter
A
B
AB
A
B
is drawn so that it lies outside of the square. If
E
E
E
is the midpoint of arc
A
B
AB
A
B
of the semicircle, what is the area of triangle
C
D
E
CDE
C
D
E
SMT 2019 Geometry Tiebreaker #1
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
∠
D
A
B
=
∠
A
B
C
=
12
0
o
\angle DAB = \angle ABC = 120^o
∠
D
A
B
=
∠
A
BC
=
12
0
o
. If
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
2
BC = 2
BC
=
2
, and
A
D
=
4
AD = 4
A
D
=
4
, what is the length of
C
D
CD
C
D
?
1
Hide problems
2019 SMT Team Round - Stanford Math Tournament
p1. Given
x
+
y
=
7
x + y = 7
x
+
y
=
7
, find the value of x that minimizes
4
x
2
+
12
x
y
+
9
y
2
4x^2 + 12xy + 9y^2
4
x
2
+
12
x
y
+
9
y
2
. p2. There are real numbers
b
b
b
and
c
c
c
such that the only
x
x
x
-intercept of
8
y
=
x
2
+
b
x
+
c
8y = x^2 + bx + c
8
y
=
x
2
+
b
x
+
c
equals its
y
y
y
-intercept. Compute
b
+
c
b + c
b
+
c
.p3. Consider the set of
5
5
5
digit numbers
A
B
C
D
E
ABCDE
A
BC
D
E
(with
A
≠
0
A \ne 0
A
=
0
) such that
A
+
B
=
C
A+B = C
A
+
B
=
C
,
B
+
C
=
D
B+C = D
B
+
C
=
D
, and
C
+
D
=
E
C + D = E
C
+
D
=
E
. What’s the size of this set? p4. Let
D
D
D
be the midpoint of
B
C
BC
BC
in
△
A
B
C
\vartriangle ABC
△
A
BC
. A line perpendicular to D intersects
A
B
AB
A
B
at
E
E
E
. If the area of
△
A
B
C
\vartriangle ABC
△
A
BC
is four times that of the area of
△
B
D
E
\vartriangle BDE
△
B
D
E
, what is
∠
A
C
B
\angle ACB
∠
A
CB
in degrees? p5. Define the sequence
c
0
,
c
1
,
.
.
.
c_0, c_1, ...
c
0
,
c
1
,
...
with
c
0
=
2
c_0 = 2
c
0
=
2
and
c
k
=
8
c
k
−
1
+
5
c_k = 8c_{k-1} + 5
c
k
=
8
c
k
−
1
+
5
for
k
>
0
k > 0
k
>
0
. Find
lim
k
→
∞
c
k
8
k
\lim_{k \to \infty} \frac{c_k}{8^k}
lim
k
→
∞
8
k
c
k
. p6. Find the maximum possible value of
∣
n
2
+
4
n
+
5
−
n
2
+
2
n
+
5
∣
|\sqrt{n^2 + 4n + 5} - \sqrt{n^2 + 2n + 5}|
∣
n
2
+
4
n
+
5
−
n
2
+
2
n
+
5
∣
. p7. Let
f
(
x
)
=
sin
8
(
x
)
+
cos
8
(
x
)
+
3
8
sin
4
(
2
x
)
f(x) = \sin^8 (x) + \cos^8(x) + \frac38 \sin^4 (2x)
f
(
x
)
=
sin
8
(
x
)
+
cos
8
(
x
)
+
8
3
sin
4
(
2
x
)
. Let
f
(
n
)
f^{(n)}
f
(
n
)
(x) be the
n
n
n
th derivative of
f
f
f
. What is the largest integer
a
a
a
such that
2
a
2^a
2
a
divides
f
(
2020
)
(
1
5
o
)
f^{(2020)}(15^o)
f
(
2020
)
(
1
5
o
)
? p8. Let
R
n
R^n
R
n
be the set of vectors
(
x
1
,
x
2
,
.
.
.
,
x
n
)
(x_1, x_2, ..., x_n)
(
x
1
,
x
2
,
...
,
x
n
)
where
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
are all real numbers. Let
∣
∣
(
x
1
,
.
.
.
,
x
n
)
∣
∣
||(x_1, . . . , x_n)||
∣∣
(
x
1
,
...
,
x
n
)
∣∣
denote
x
1
2
+
.
.
.
+
x
n
2
\sqrt{x^2_1 +... + x^2_n}
x
1
2
+
...
+
x
n
2
. Let
S
S
S
be the set in
R
9
R^9
R
9
given by
S
=
{
(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
3
,
1
=
∣
∣
x
∣
∣
=
∣
∣
y
−
x
∣
∣
=
∣
∣
z
−
y
∣
∣
}
.
S = \{(x, y, z) : x, y, z \in R^3 , 1 = ||x|| = ||y - x|| = ||z -y||\}.
S
=
{(
x
,
y
,
z
)
:
x
,
y
,
z
∈
R
3
,
1
=
∣∣
x
∣∣
=
∣∣
y
−
x
∣∣
=
∣∣
z
−
y
∣∣
}
.
If a point
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
is uniformly at random from
S
S
S
, what is
E
[
∣
∣
z
∣
∣
2
]
E[||z||^2]
E
[
∣∣
z
∣
∣
2
]
? p9. Let
f
(
x
)
f(x)
f
(
x
)
be the unique integer between
0
0
0
and
x
−
1
x - 1
x
−
1
, inclusive, that is equivalent modulo
x
x
x
to
(
∑
i
=
0
2
(
x
−
1
i
)
(
(
x
−
1
−
i
)
!
+
i
!
)
)
\left( \sum^2_{i=0} {{x-1} \choose i} ((x - 1 - i)! + i!) \right)
(
∑
i
=
0
2
(
i
x
−
1
)
((
x
−
1
−
i
)!
+
i
!)
)
. Let
S
S
S
be the set of primes between
3
3
3
and
30
30
30
, inclusive. Find
∑
x
∈
S
f
(
x
)
\sum_{x\in S}^{f(x)}
∑
x
∈
S
f
(
x
)
. p10. In the Cartesian plane, consider a box with vertices
(
0
,
0
)
(0, 0)
(
0
,
0
)
,
(
22
7
,
0
)
\left( \frac{22}{7}, 0\right)
(
7
22
,
0
)
,
(
0
,
24
)
(0, 24)
(
0
,
24
)
,
(
22
7
,
4
)
\left( \frac{22}{7}, 4\right)
(
7
22
,
4
)
. We pick an integer
a
a
a
between
1
1
1
and
24
24
24
, inclusive, uniformly at random. We shoot a puck from
(
0
,
0
)
(0, 0)
(
0
,
0
)
in the direction of
(
22
7
,
a
)
\left( \frac{22}{7}, a\right)
(
7
22
,
a
)
and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at
(
0
,
0
)
(0, 0)
(
0
,
0
)
and when it ends at some vertex of the box? p11. Sarah is buying school supplies and she has
$
2019
\$2019
$2019
. She can only buy full packs of each of the following items. A pack of pens is
$
4
\$4
$4
, a pack of pencils is
$
3
\$3
$3
, and any type of notebook or stapler is
$
1
\$1
$1
. Sarah buys at least
1
1
1
pack of pencils. She will either buy
1
1
1
stapler or no stapler. She will buy at most
3
3
3
college-ruled notebooks and at most
2
2
2
graph paper notebooks. How many ways can she buy school supplies? p12. Let
O
O
O
be the center of the circumcircle of right triangle
A
B
C
ABC
A
BC
with
∠
A
C
B
=
9
0
o
\angle ACB = 90^o
∠
A
CB
=
9
0
o
. Let
M
M
M
be the midpoint of minor arc
A
C
AC
A
C
and let
N
N
N
be a point on line
B
C
BC
BC
such that
M
N
⊥
B
C
MN \perp BC
MN
⊥
BC
. Let
P
P
P
be the intersection of line
A
N
AN
A
N
and the Circle
O
O
O
and let
Q
Q
Q
be the intersection of line
B
P
BP
BP
and
M
N
MN
MN
. If
Q
N
=
2
QN = 2
QN
=
2
and
B
N
=
8
BN = 8
BN
=
8
, compute the radius of the Circle
O
O
O
. p13. Reduce the following expression to a simplified rational
1
1
−
cos
π
9
+
1
1
−
cos
5
π
9
+
1
1
−
cos
7
π
9
\frac{1}{1 - \cos \frac{\pi}{9}}+\frac{1}{1 - \cos \frac{5 \pi}{9}}+\frac{1}{1 - \cos \frac{7 \pi}{9}}
1
−
cos
9
π
1
+
1
−
cos
9
5
π
1
+
1
−
cos
9
7
π
1
p14. Compute the following integral
∫
0
∞
log
(
1
+
e
−
t
)
d
t
\int_0^{\infty} \log (1 + e^{-t})dt
∫
0
∞
lo
g
(
1
+
e
−
t
)
d
t
. p15. Define
f
(
n
)
f(n)
f
(
n
)
to be the maximum possible least-common-multiple of any sequence of positive integers which sum to
n
n
n
. Find the sum of all possible odd
f
(
n
)
f(n)
f
(
n
)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.