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2018 Stanford Mathematics Tournament
2018 Stanford Mathematics Tournament
Part of
Stanford Mathematics Tournament
Subcontests
(11)
10
1
Hide problems
SMT 2018 Geometry #10 2 incircles
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
13
AB = 13
A
B
=
13
,
A
C
=
14
AC = 14
A
C
=
14
, and
B
C
=
15
BC = 15
BC
=
15
, and let
Γ
\Gamma
Γ
be its incircle with incenter
I
I
I
. Let
D
D
D
and
E
E
E
be the points of tangency between
Γ
\Gamma
Γ
and
B
C
BC
BC
and
A
C
AC
A
C
respectively, and let
ω
\omega
ω
be the circle inscribed in
C
D
I
E
CDIE
C
D
I
E
. If
Q
Q
Q
is the intersection point between
Γ
\Gamma
Γ
and
ω
\omega
ω
and
P
P
P
is the intersection point between
C
Q
CQ
CQ
and
ω
\omega
ω
, compute the length of
P
Q
P Q
PQ
.
9
1
Hide problems
SMT 2018 Geometry #9 cyclic quad
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
3
A
B
=
2
A
D
3AB = 2AD
3
A
B
=
2
A
D
and
B
C
=
C
D
BC = CD
BC
=
C
D
. The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
X
X
X
. Let
E
E
E
be a point on
A
D
AD
A
D
such that
D
E
=
A
B
DE = AB
D
E
=
A
B
and
Y
Y
Y
be the point of intersection of lines
A
C
AC
A
C
and
B
E
BE
BE
. If the area of triangle
A
B
Y
ABY
A
B
Y
is
5
5
5
, then what is the area of quadrilateral
D
E
Y
X
DEY X
D
E
Y
X
?
8
1
Hide problems
SMT 2018 Geometry #8 right triangle
Let
A
B
C
ABC
A
BC
be a right triangle with
∠
A
C
B
=
9
0
o
\angle ACB = 90^o
∠
A
CB
=
9
0
o
,
B
C
=
16
BC = 16
BC
=
16
, and
A
C
=
12
AC = 12
A
C
=
12
. Let the angle bisectors of
∠
B
A
C
\angle BAC
∠
B
A
C
and
∠
A
B
C
\angle ABC
∠
A
BC
intersect
B
C
BC
BC
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
respectively. Let
A
D
AD
A
D
and
B
E
BE
BE
intersect at
I
I
I
, and let the circle centered at
I
I
I
passing through
C
C
C
intersect
A
B
AB
A
B
at
P
P
P
and
Q
Q
Q
such that
A
Q
<
A
P
AQ < AP
A
Q
<
A
P
. Compute the area of quadrilateral
D
P
Q
E
DP QE
D
PQE
.
7
1
Hide problems
SMT 2018 Geometry #7 3d
Two equilateral triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
, each with side length
1
1
1
, are drawn in
2
2
2
parallel planes such that when one plane is projected onto the other, the vertices of the triangles form a regular hexagon
A
F
B
D
C
E
AF BDCE
A
FB
D
CE
. Line segments
A
E
AE
A
E
,
A
F
AF
A
F
,
B
F
BF
BF
,
B
D
BD
B
D
,
C
D
,
CD,
C
D
,
and
C
E
CE
CE
are drawn, and suppose that each of these segments also has length
1
1
1
. Compute the volume of the resulting solid that is formed.
6
1
Hide problems
SMT 2018 Geometry #6
In
△
A
B
\vartriangle AB
△
A
B
C,
A
B
=
3
AB = 3
A
B
=
3
,
A
C
=
6
,
AC = 6,
A
C
=
6
,
and
D
D
D
is drawn on
B
C
BC
BC
such that
A
D
AD
A
D
is the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
.
D
D
D
is reflected across
A
B
AB
A
B
to a point
E
E
E
, and suppose that
A
C
AC
A
C
and
B
E
BE
BE
are parallel. Compute
C
E
CE
CE
.
1
Hide problems
2018 SMT Team Round - Short Answers - Stanford Math Tournament
p1. Suppose
A
A
A
and
B
B
B
are points in the plane lying on the parabola
y
=
x
2
y = x^2
y
=
x
2
, and the
x
x
x
-coordinates of
A
A
A
and
B
B
B
are
−
29
-29
−
29
and
51
51
51
, respectively. Let
C
C
C
be the point where line
A
B
AB
A
B
intersects the y-axis. What is the
y
y
y
-coordinate of
C
C
C
? p2. Cindy has a collection of identical rectangular prisms. She stacks them, end to end, to form
1
1
1
longer rectangular prism. Suppose that joining
11
11
11
of them will form a rectangular prism with
3
3
3
times the surface area of an individual rectangular prism. How many will she need to join end to end to form a rectangular prism with
9
9
9
times the surface area? p3. A lattice point is a point
(
a
,
b
)
(a, b)
(
a
,
b
)
on the Cartesian plane where a and b are integers. Compute the number of lattice points in the interior and on the boundary of the triangle with vertices at
(
0
,
0
)
(0, 0)
(
0
,
0
)
,
(
0
,
20
)
(0, 20)
(
0
,
20
)
, and
(
18
,
0
)
(18, 0)
(
18
,
0
)
. p4. Let
1
=
a
1
<
a
2
<
a
3
<
.
.
.
<
a
k
=
n
1 = a_1 < a_2 < a_3 < ... < a_k = n
1
=
a
1
<
a
2
<
a
3
<
...
<
a
k
=
n
be the positive divisors of n in increasing order. If
n
=
a
3
3
−
a
2
3
n = a_3^3 - a^3_2
n
=
a
3
3
−
a
2
3
, what is n? p5. A point
(
x
0
,
y
0
)
(x_0, y_0)
(
x
0
,
y
0
)
with integer coordinates is a primitive point of a circle if for some pair of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
, the line
a
x
+
b
y
=
1
ax + by = 1
a
x
+
b
y
=
1
intersects the circle at
(
x
0
,
y
0
)
(x_0, y_0)
(
x
0
,
y
0
)
. How many primitive points are there of the circle centered at
(
2
,
−
3
)
(2, -3)
(
2
,
−
3
)
with radius
5
5
5
? p6. Three distinct points are chosen uniformly at random from the vertices of a regular
2018
2018
2018
-gon. What is the probability that the triangle formed by these points is a right triangle? p7. Consider any
5
5
5
points placed on the surface of a cube of side length
2
2
2
centered at the origin. Let
m
x
m_x
m
x
be the minimum distance between the
x
x
x
coordinates of any of the
5
5
5
points,
m
y
m_y
m
y
be the minimum distance between y coordinates, and
m
z
m_z
m
z
be the minimum distance between
z
z
z
coordinates. What is the maximum value of
m
x
+
m
y
+
m
z
m_x + m_y + m_z
m
x
+
m
y
+
m
z
? p8. Eddy has two blank cubes
A
A
A
and
B
B
B
and a marker. Eddy is allowed to draw a total of
36
36
36
dots on cubes
A
A
A
and
B
B
B
to turn them into dice, where each side has an equal probability of appearing, and each side has at least one dot on it. Eddy then rolls dice
A
A
A
twice and dice
B
B
B
once and computes the product of the three numbers. Given that Eddy draws dots on the two dice to maximize his expected product, what is his expected product? p9. Let
A
B
C
D
ABCD
A
BC
D
be a square. Point
E
E
E
is chosen inside the square such that
A
E
=
6
AE = 6
A
E
=
6
. Point
F
F
F
is chosen outside the square such that
B
E
=
B
F
=
2
5
BE = BF = 2\sqrt5
BE
=
BF
=
2
5
,
∠
A
B
F
=
∠
C
B
E
\angle ABF = \angle CBE
∠
A
BF
=
∠
CBE
, and
A
E
B
F
AEBF
A
EBF
is cyclic. Compute the area of
A
B
C
D
ABCD
A
BC
D
. p10. Find the total number of sets of nonnegative integers
(
w
,
x
,
y
,
z
)
(w, x, y, z)
(
w
,
x
,
y
,
z
)
where
w
≤
x
≤
y
≤
z
w \le x \le y \le z
w
≤
x
≤
y
≤
z
such that
5
w
+
3
x
+
y
+
z
=
100
5w + 3x + y + z = 100
5
w
+
3
x
+
y
+
z
=
100
. p11. Let
f
(
k
)
f(k)
f
(
k
)
be a function defined by the following rules: (a)
f
(
k
)
f(k)
f
(
k
)
is multiplicative. In other words, if
g
c
d
(
a
,
b
)
=
1
gcd(a, b) = 1
g
c
d
(
a
,
b
)
=
1
, then
f
(
a
b
)
=
f
(
a
)
⋅
f
(
b
)
f(ab) = f(a) \cdot f(b)
f
(
ab
)
=
f
(
a
)
⋅
f
(
b
)
, (b)
f
(
p
k
)
=
k
f(p^k) = k
f
(
p
k
)
=
k
for primes
p
=
2
,
3
p = 2, 3
p
=
2
,
3
and all
k
>
0
k > 0
k
>
0
, (c)
f
(
p
k
)
=
0
f(p^k) = 0
f
(
p
k
)
=
0
for primes
p
>
3
p > 3
p
>
3
and all
k
>
0
k > 0
k
>
0
, and (d)
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
. For example,
f
(
12
)
=
2
f(12) = 2
f
(
12
)
=
2
and
f
(
160
)
=
0
f(160) = 0
f
(
160
)
=
0
. Evaluate
∑
k
=
1
∞
f
(
k
)
k
.
\sum_{k=1}^{\infty}\frac{f(k)}{k}.
∑
k
=
1
∞
k
f
(
k
)
.
p12. Consider all increasing arithmetic progressions of the form
1
a
\frac{1}{a}
a
1
,
1
b
\frac{1}{b}
b
1
,
1
c
\frac{1}{c}
c
1
such that
a
,
b
,
c
∈
N
a, b, c \in N
a
,
b
,
c
∈
N
and
g
c
d
(
a
,
b
,
c
)
=
1
gcd(a, b, c) = 1
g
c
d
(
a
,
b
,
c
)
=
1
. Find the sum of all possible values of
1
a
\frac{1}{a}
a
1
. p13. In
△
A
B
C
\vartriangle ABC
△
A
BC
, let
D
,
E
D, E
D
,
E
, and
F
F
F
be the feet of the altitudes drawn from
A
,
B
A, B
A
,
B
, and
C
C
C
respectively. Let
P
P
P
and
Q
Q
Q
be points on line
E
F
EF
EF
such that
B
P
BP
BP
is perpendicular to
E
F
EF
EF
and
C
Q
CQ
CQ
is perpendicular to
E
F
EF
EF
. If
P
Q
=
2018
P Q = 2018
PQ
=
2018
and
D
E
=
D
F
+
4
DE = DF + 4
D
E
=
D
F
+
4
, find
D
E
DE
D
E
. p14. Let
A
A
A
and
B
B
B
be two points chosen independently and uniformly at random inside the unit circle centered at
O
O
O
. Compute the expected area of
△
A
B
O
\vartriangle ABO
△
A
BO
. p15. Suppose that
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are positive integers satisfying
25
a
b
+
25
a
c
+
b
2
=
14
b
c
25ab + 25ac + b^2 = 14bc
25
ab
+
25
a
c
+
b
2
=
14
b
c
4
b
c
+
4
b
d
+
9
c
2
=
31
c
d
4bc + 4bd + 9c^2 = 31cd
4
b
c
+
4
b
d
+
9
c
2
=
31
c
d
9
c
d
+
9
c
a
+
25
d
2
=
95
d
a
9cd + 9ca + 25d^2 = 95da
9
c
d
+
9
c
a
+
25
d
2
=
95
d
a
5
d
a
+
5
d
b
+
20
a
2
=
16
a
b
5da + 5db + 20a^2 = 16ab
5
d
a
+
5
d
b
+
20
a
2
=
16
ab
Compute
a
b
+
b
c
+
c
d
+
d
a
.
\frac{a}{b} +\frac{b}{c} +\frac{c}{d} +\frac{d}{a}.
b
a
+
c
b
+
d
c
+
a
d
.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
4
2
Hide problems
SMT 2018 Geometry #4 regular dodecagon
Let
a
1
,
a
2
,
.
.
.
,
a
12
a_1, a_2, ..., a_{12}
a
1
,
a
2
,
...
,
a
12
be the vertices of a regular dodecagon
D
1
D_1
D
1
(
12
12
12
-gon). The four vertices
a
1
a_1
a
1
,
a
4
a_4
a
4
,
a
7
a_7
a
7
,
a
10
a_{10}
a
10
form a square, as do the four vertices
a
2
a_2
a
2
,
a
5
a_5
a
5
,
a
8
a_8
a
8
,
a
11
a_{11}
a
11
and
a
3
a_3
a
3
,
a
6
a_6
a
6
,
a
9
a_9
a
9
,
a
12
a_{12}
a
12
. Let
D
2
D_2
D
2
be the polygon formed by the intersection of these three squares. If we let
[
A
]
[A]
[
A
]
denotes the area of polygon
A
A
A
, compute
[
D
2
]
[
D
1
]
\frac{[D_2]}{[D_1]}
[
D
1
]
[
D
2
]
.
F_{n (k+1)} = b_n F_{n k} + c_n F_{n (k-1)} , Fibonacci numbers
Let
F
k
F_k
F
k
denote the series of Fibonacci numbers shifted back by one index, so that
F
0
=
1
F_0 = 1
F
0
=
1
,
F
1
=
1
,
F_1 = 1,
F
1
=
1
,
and
F
k
+
1
=
F
k
+
F
k
−
1
F_{k+1} = F_k +F_{k-1}
F
k
+
1
=
F
k
+
F
k
−
1
. It is known that for any fixed
n
≥
1
n \ge 1
n
≥
1
there exist real constants
b
n
b_n
b
n
,
c
n
c_n
c
n
such that the following recurrence holds for all
k
≥
1
k \ge 1
k
≥
1
:
F
n
⋅
(
k
+
1
)
=
b
n
⋅
F
n
⋅
k
+
c
n
⋅
F
n
⋅
(
k
−
1
)
.
F_{n\cdot (k+1)} = b_n \cdot F_{n \cdot k} + c_n \cdot F_{n\cdot (k-1)}.
F
n
⋅
(
k
+
1
)
=
b
n
⋅
F
n
⋅
k
+
c
n
⋅
F
n
⋅
(
k
−
1
)
.
Prove that
∣
c
n
∣
=
1
|c_n| = 1
∣
c
n
∣
=
1
for all
n
≥
1
n \ge 1
n
≥
1
.
5
2
Hide problems
AB + BC + CD + DA + AC + BD)^2 > 2|AC^3 - BC^3| + 2|BD^3 - AD^3| - (AB + CD)^3
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
and diagonals
A
C
AC
A
C
,
B
D
BD
B
D
. Suppose that all sides of the quadrilateral have length greater than
1
1
1
, and that the difference between any side and diagonal is less than 1. Prove that the following inequality holds
(
A
B
+
B
C
+
C
D
+
D
A
+
A
C
+
B
D
)
2
>
2
∣
A
C
3
−
B
C
3
∣
+
2
∣
B
D
3
−
A
D
3
∣
−
(
A
B
+
C
D
)
3
(AB + BC + CD + DA + AC + BD)^2 > 2|AC^3 - BC^3| + 2|BD^3 - AD^3| - (AB + CD)^3
(
A
B
+
BC
+
C
D
+
D
A
+
A
C
+
B
D
)
2
>
2∣
A
C
3
−
B
C
3
∣
+
2∣
B
D
3
−
A
D
3
∣
−
(
A
B
+
C
D
)
3
SMT 2018 Geometry #5
In
△
A
B
C
\vartriangle ABC
△
A
BC
,
∠
A
B
C
=
7
5
o
\angle ABC = 75^o
∠
A
BC
=
7
5
o
and
∠
B
A
C
\angle BAC
∠
B
A
C
is obtuse. Points
D
D
D
and
E
E
E
are on
A
C
AC
A
C
and
B
C
BC
BC
, respectively, such that
A
B
B
C
=
D
E
E
C
\frac{AB}{BC} = \frac{DE}{EC}
BC
A
B
=
EC
D
E
and
∠
D
E
C
=
∠
E
D
C
\angle DEC = \angle EDC
∠
D
EC
=
∠
E
D
C
. Compute
∠
D
E
C
\angle DEC
∠
D
EC
in degrees.
3
3
Hide problems
a shape in the Cartesian coordinate plane with area greater than 1
Show that if
A
A
A
is a shape in the Cartesian coordinate plane with area greater than
1
1
1
, then there are distinct points
(
a
,
b
)
(a, b)
(
a
,
b
)
,
(
c
,
d
)
(c, d)
(
c
,
d
)
in
A
A
A
where
a
−
c
=
2
x
+
5
y
a - c = 2x + 5y
a
−
c
=
2
x
+
5
y
and
b
−
d
=
x
+
3
y
b - d = x + 3y
b
−
d
=
x
+
3
y
where
x
,
y
x, y
x
,
y
are integers.
SMT 2018 Geometry #3 area
Let
A
B
C
ABC
A
BC
be a triangle and
D
D
D
be a point such that
A
A
A
and
D
D
D
are on opposite sides of
B
C
BC
BC
. Give that
∠
A
C
D
=
7
5
o
\angle ACD = 75^o
∠
A
C
D
=
7
5
o
,
A
C
=
2
AC = 2
A
C
=
2
,
B
D
=
6
BD =\sqrt6
B
D
=
6
, and
A
D
AD
A
D
is an angle bisector of both
△
A
B
C
\vartriangle ABC
△
A
BC
and
△
B
C
D
\vartriangle BCD
△
BC
D
, find the area of quadrilateral
A
B
D
C
ABDC
A
B
D
C
.
SMT 2018 Geometry Tiebreaker #3
A triangle has side lengths of
7
7
7
,
8
8
8
, and
9
9
9
. Find the radius of the largest possible semicircle inscribed in the triangle.
2
3
Hide problems
a game played on the integers in the closed interval [1, n]
Consider a game played on the integers in the closed interval
[
1
,
n
]
[1, n]
[
1
,
n
]
. The game begins with some tokens placed in
[
1
,
n
]
[1, n]
[
1
,
n
]
. At each turn, tokens are added or removed from
[
1
,
n
]
[1, n]
[
1
,
n
]
using the following rule: For each integer
k
∈
[
1
,
n
]
k \in [1, n]
k
∈
[
1
,
n
]
, if exactly one of
k
−
1
k - 1
k
−
1
and
k
+
1
k + 1
k
+
1
has a token, place a token at
k
k
k
for the next turn, otherwise leave k blank for the next turn. We call a position static if no changes to the interval occur after one turn. For instance, the trivial position with no tokens is static because no tokens are added or removed after a turn (because there are no tokens). Find all non-trivial static positions.
SMT 2018 Geometry #12 trapezoid
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with
A
B
AB
A
B
parallel to
C
D
CD
C
D
and perpendicular to
B
C
BC
BC
. Let
M
M
M
be a point on
B
C
BC
BC
such that
∠
A
M
B
=
∠
D
M
C
\angle AMB = \angle DMC
∠
A
MB
=
∠
D
MC
. If
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
24
BC = 24
BC
=
24
, and
C
D
=
4
CD = 4
C
D
=
4
, what is the value of
A
M
+
M
D
AM + MD
A
M
+
M
D
?
SMT 2018 Geometry Tiebreaker #2 3D
What is the largest possible height of a right cylinder with radius
3
3
3
that can fit in a cube with side length
12
12
12
?
1
3
Hide problems
7 does not divide a + b if 7 divides a^2 + b^2 + 1
Prove that if
7
7
7
divides
a
2
+
b
2
+
1
a^2 + b^2 + 1
a
2
+
b
2
+
1
, then
7
7
7
does not divide
a
+
b
a + b
a
+
b
.
SMT 2018 Geometry #1 semicircle
Consider a semi-circle with diameter
A
B
AB
A
B
. Let points
C
C
C
and
D
D
D
be on diameter
A
B
AB
A
B
such that
C
D
CD
C
D
forms the base of a square inscribed in the semicircle. Given that
C
D
=
2
CD = 2
C
D
=
2
, compute the length of
A
B
AB
A
B
.
SMT 2018 Geometry Tiebreaker #1
Point
E
E
E
is on side
C
D
CD
C
D
of rectangle
A
B
C
D
ABCD
A
BC
D
such that
C
E
D
E
=
2
5
.
\frac{CE}{DE} =\frac{2}{5}.
D
E
CE
=
5
2
.
If the area of triangle
B
C
E
BCE
BCE
is
30
30
30
, what is the area of rectangle
A
B
C
D
ABCD
A
BC
D
?