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Stanford Mathematics Tournament
2021 Stanford Mathematics Tournament
2021 Stanford Mathematics Tournament
Part of
Stanford Mathematics Tournament
Subcontests
(20)
10
1
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SMT 2021 Geometry #10
In acute
△
A
B
C
\vartriangle ABC
△
A
BC
, let points
D
D
D
,
E
,
E,
E
,
and
F
F
F
be the feet of the altitudes of the triangle from
A
A
A
,
B
B
B
,and
C
C
C
, respectively. The area of
△
A
E
F
\vartriangle AEF
△
A
EF
is
1
1
1
, the area of
△
C
D
E
\vartriangle CDE
△
C
D
E
is
2
2
2
, and the area of
△
B
F
D
\vartriangle BF D
△
BF
D
is
2
−
3
2 -\sqrt3
2
−
3
. What is the area of
△
D
E
F
\vartriangle DEF
△
D
EF
?
9
1
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SMT 2021 Geometry #9
Rectangle
A
B
C
D
ABCD
A
BC
D
has an area of 30. Four circles of radius
r
1
=
2
r_1 = 2
r
1
=
2
,
r
2
=
3
r_2 = 3
r
2
=
3
,
r
3
=
5
r_3 = 5
r
3
=
5
, and
r
4
=
4
r_4 = 4
r
4
=
4
are centered on the four vertices
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
respectively. Two pairs of external tangents are drawn for the circles at A and
C
C
C
and for the circles at
B
B
B
and
D
D
D
. These four tangents intersect to form a quadrilateral
W
X
Y
Z
W XY Z
W
X
Y
Z
where
W
X
‾
\overline{W X}
W
X
and
Y
Z
‾
\overline{Y Z}
Y
Z
lie on the tangents through the circles on
A
A
A
and
C
C
C
. If
W
X
‾
+
Y
Z
‾
=
20
\overline{W X} + \overline{Y Z} = 20
W
X
+
Y
Z
=
20
, find the area of quadrilateral
W
X
Y
Z
W XY Z
W
X
Y
Z
. https://cdn.artofproblemsolving.com/attachments/5/a/cb3b3457f588a15ffb4c875b1646ef2aec8d11.png
8
1
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SMT 2021 Geometry #8
In triangle
△
A
B
C
\vartriangle ABC
△
A
BC
,
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
7
BC = 7
BC
=
7
, and
C
A
=
8
CA = 8
C
A
=
8
. Let
E
E
E
and
F
F
F
be the feet of the altitudes from
B
B
B
and
C
C
C
, respectively, and let
M
M
M
be the midpoint of
B
C
BC
BC
. The area of triangle
M
E
F
MEF
MEF
can be expressed as
a
b
c
\frac{a \sqrt{b}}{c}
c
a
b
for positive integers
a
a
a
,
b
b
b
, and
c
c
c
such that the greatest common divisor of
a
a
a
and
c
c
c
is
1
1
1
and
b
b
b
is not divisible by the square of any prime. Compute
a
+
b
+
c
a + b + c
a
+
b
+
c
.
7
1
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SMT 2021 Geometry #7
An
n
n
n
-sided regular polygon with side length
1
1
1
is rotated by
18
0
o
n
\frac{180^o}{n}
n
18
0
o
about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a
2
n
2n
2
n
-sided regular polygon with side length
1
−
t
a
n
2
1
0
o
2
\frac{1-tan^2 10^o}{2}
2
1
−
t
a
n
2
1
0
o
. What is the value of
n
n
n
?
6
1
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SMT 2021 Geometry #6
⊙
A
\odot A
⊙
A
, centered at point
A
A
A
, has radius
14
14
14
and
⊙
B
\odot B
⊙
B
, centered at point
B
B
B
, has radius
15
15
15
.
A
B
=
13
AB = 13
A
B
=
13
. The circles intersect at points
C
C
C
and
D
D
D
. Let
E
E
E
be a point on
⊙
A
\odot A
⊙
A
, and
F
F
F
be the point where line
E
C
EC
EC
intersects
⊙
B
\odot B
⊙
B
, again. Let the midpoints of
D
E
DE
D
E
and
D
F
DF
D
F
be
M
M
M
and
N
N
N
, respectively. Lines
A
M
AM
A
M
and
B
N
BN
BN
intersect at point
G
G
G
. If point
E
E
E
is allowed to move freely on
⊙
A
\odot A
⊙
A
, what is the radius of the locus of
G
G
G
?
5
1
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SMT 2021 Geometry #5
Let
A
B
C
D
ABCD
A
BC
D
be a square of side length
1
1
1
, and let
E
E
E
and
F
F
F
be on the lines
A
B
AB
A
B
and
A
D
AD
A
D
, respectively, so that
B
B
B
lies between
A
A
A
and
E
E
E
, and
D
D
D
lies between
A
A
A
and
F
F
F
. Suppose that
∠
B
C
E
=
2
0
o
\angle BCE = 20^o
∠
BCE
=
2
0
o
and
∠
D
C
F
=
2
5
o
\angle DCF = 25^o
∠
D
CF
=
2
5
o
. Find the area of triangle
△
E
A
F
\vartriangle EAF
△
E
A
F
.
4
1
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SMT 2021 Geometry #4
△
A
0
B
0
C
0
\vartriangle A_0B_0C_0
△
A
0
B
0
C
0
has side lengths
A
0
B
0
=
13
A_0B_0 = 13
A
0
B
0
=
13
,
B
0
C
0
=
14
B_0C_0 = 14
B
0
C
0
=
14
, and
C
0
A
0
=
15
C_0A_0 = 15
C
0
A
0
=
15
.
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
is inscribed in the incircle of
△
A
0
B
0
C
0
\vartriangle A_0B_0C_0
△
A
0
B
0
C
0
such that it is similar to the first triangle. Beginning with
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
, the same steps are repeated to construct
△
A
2
B
2
C
2
\vartriangle A_2B_2C_2
△
A
2
B
2
C
2
, and so on infinitely many times. What is the value of
∑
i
=
0
∞
A
i
B
i
\sum_{i=0}^{\infty} A_iB_i
∑
i
=
0
∞
A
i
B
i
?
3
2
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SMT 2021 Geometry #3
If
r
r
r
is a rational number, let
f
(
r
)
=
(
1
−
r
2
1
+
r
2
,
2
r
1
+
r
2
)
f(r) = \left( \frac{1-r^2}{1+r^2} , \frac{2r}{1+r^2} \right)
f
(
r
)
=
(
1
+
r
2
1
−
r
2
,
1
+
r
2
2
r
)
. Then the images of
f
f
f
forms a curve in the
x
y
xy
x
y
plane. If
f
(
1
/
3
)
=
p
1
f(1/3) = p_1
f
(
1/3
)
=
p
1
and
f
(
2
)
=
p
2
f(2) = p_2
f
(
2
)
=
p
2
, what is the distance along the curve between
p
1
p_1
p
1
and
p
2
p_2
p
2
?
SMT 2021 Geometry Tiebreaker #3
In quadrilateral
A
B
C
D
ABCD
A
BC
D
,
C
D
=
14
CD = 14
C
D
=
14
,
∠
B
A
D
=
10
5
o
\angle BAD = 105^o
∠
B
A
D
=
10
5
o
,
∠
A
C
D
=
3
5
o
\angle ACD = 35^o
∠
A
C
D
=
3
5
o
, and
∠
A
C
B
=
4
0
o
\angle ACB = 40^o
∠
A
CB
=
4
0
o
. Let the midpoint of
C
D
CD
C
D
be
M
M
M
. Points
P
P
P
and
Q
Q
Q
lie on
A
M
→
\overrightarrow{AM}
A
M
and
B
M
→
\overrightarrow{BM}
BM
, respectively, such that
∠
A
P
B
=
4
0
o
\angle AP B = 40^o
∠
A
PB
=
4
0
o
and
∠
A
Q
B
=
4
0
o
\angle AQB = 40^o
∠
A
QB
=
4
0
o
.
P
B
P B
PB
intersects
C
D
CD
C
D
at point
R
R
R
and
Q
A
QA
Q
A
intersects
C
D
CD
C
D
at point
S
S
S
. If
C
R
=
2
CR = 2
CR
=
2
, what is the length of
S
M
SM
SM
?
2
2
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SMT 2021 Geometry #2
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with bases
A
B
=
50
AB = 50
A
B
=
50
and
C
D
=
125
CD = 125
C
D
=
125
, and legs
A
D
=
45
AD = 45
A
D
=
45
and
B
C
=
60
BC = 60
BC
=
60
. Find the area of the intersection between the circle centered at
B
B
B
with radius
B
D
BD
B
D
and the circle centered at
D
D
D
with radius
B
D
BD
B
D
. Express your answer as a common fraction in simplest radical form and in terms of
π
\pi
π
.
SMT 2021 Geometry Tiebreaker #2
If two points are picked randomly on the perimeter of a square, what is the probability that the distance between those points is less than the side length of the square?
1
2
Hide problems
SMT 2021 Geometry #1
A paper rectangle
A
B
C
D
ABCD
A
BC
D
has
A
B
=
8
AB = 8
A
B
=
8
and
B
C
=
6
BC = 6
BC
=
6
. After corner
B
B
B
is folded over diagonal
A
C
AC
A
C
, what is
B
D
BD
B
D
?
SMT 2021 Geometry Tiebreaker #1
What is the radius of the largest circle centered at
(
2
,
2
)
(2, 2)
(
2
,
2
)
that is completely bounded within the parabola
y
=
x
2
−
4
x
+
5
y = x^2 - 4x + 5
y
=
x
2
−
4
x
+
5
?
R9
1
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2021 SMT Guts Round 9 - final- p33-36 - Stanford Math Tournament
p33. Lines
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
have slopes
m
1
m_1
m
1
and
m
2
m_2
m
2
such that
0
<
m
2
<
m
1
0 < m_2 < m_1
0
<
m
2
<
m
1
.
ℓ
1
′
\ell'_1
ℓ
1
′
and
ℓ
2
′
\ell'_2
ℓ
2
′
are the reflections of
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
about the line
ℓ
3
\ell_3
ℓ
3
defined by
y
=
x
y = x
y
=
x
. Let
A
=
ℓ
1
∩
ℓ
2
=
(
5
,
4
)
A = \ell_1 \cap \ell_2 = (5, 4)
A
=
ℓ
1
∩
ℓ
2
=
(
5
,
4
)
,
B
=
ℓ
1
∩
ℓ
3
B = \ell_1 \cap \ell_3
B
=
ℓ
1
∩
ℓ
3
,
C
=
ℓ
1
′
∩
ℓ
2
′
C = \ell'_1 \cap \ell'_2
C
=
ℓ
1
′
∩
ℓ
2
′
and
D
=
ℓ
2
∩
ℓ
3
D = \ell_2 \cap \ell_3
D
=
ℓ
2
∩
ℓ
3
. If
4
−
5
m
1
−
5
−
4
m
1
=
m
2
\frac{4-5m_1}{-5-4m_1} = m_2
−
5
−
4
m
1
4
−
5
m
1
=
m
2
and
(
1
+
m
1
2
)
(
1
+
m
2
2
)
(
1
−
m
1
)
2
(
1
−
m
2
)
2
=
41
\frac{(1+m^2_1)(1+m^2_2)}{(1-m_1)^2(1-m_2)^2} = 41
(
1
−
m
1
)
2
(
1
−
m
2
)
2
(
1
+
m
1
2
)
(
1
+
m
2
2
)
=
41
, compute the area of quadrilateral
A
B
C
D
ABCD
A
BC
D
. p34. Suppose
S
(
m
,
n
)
=
∑
i
=
1
m
(
−
1
)
i
i
n
S(m, n) = \sum^m_{i=1}(-1)^ii^n
S
(
m
,
n
)
=
∑
i
=
1
m
(
−
1
)
i
i
n
. Compute the remainder when
S
(
2020
,
4
)
S(2020, 4)
S
(
2020
,
4
)
is divided by
S
(
1010
,
2
)
S(1010, 2)
S
(
1010
,
2
)
. p35. Let
N
N
N
be the number of ways to place the numbers
1
,
2
,
.
.
.
,
12
1, 2, ..., 12
1
,
2
,
...
,
12
on a circle such that every pair of adjacent numbers has greatest common divisor
1
1
1
. What is
N
/
144
N/144
N
/144
? (Arrangements that can be rotated to yield each other are the same). p36. Compute the series
∑
n
=
1
∞
(
−
1
)
n
−
1
(
2
n
2
)
=
1
(
2
2
)
−
1
(
4
2
)
+
1
(
6
2
)
−
1
(
8
2
)
−
1
(
10
2
)
+
1
(
12
2
)
+
.
.
.
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{{2n \choose 2}} =\frac{1}{{2 \choose 2}} - \frac{1}{{4 \choose 2}} +\frac{1}{{6 \choose 2}} -\frac{1}{{8 \choose 2}} -\frac{1}{{10 \choose 2}}+\frac{1}{{12 \choose 2}} +...
∑
n
=
1
∞
(
2
2
n
)
(
−
1
)
n
−
1
=
(
2
2
)
1
−
(
2
4
)
1
+
(
2
6
)
1
−
(
2
8
)
1
−
(
2
10
)
1
+
(
2
12
)
1
+
...
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R8
1
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2021 SMT Guts Round 8 p29-32 - Stanford Math Tournament
p29. Consider pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
. How many paths are there from vertex
A
A
A
to vertex
E
E
E
where no edge is repeated and does not go through
E
E
E
. p30. Let
a
1
,
a
2
,
.
.
.
a_1, a_2, ...
a
1
,
a
2
,
...
be a sequence of positive real numbers such that
∑
n
=
1
∞
a
n
=
4
\sum^{\infty}_{n=1} a_n = 4
∑
n
=
1
∞
a
n
=
4
. Compute the maximum possible value of
∑
n
=
1
∞
a
n
2
n
\sum^{\infty}_{n=1}\frac{\sqrt{a_n}}{2^n}
∑
n
=
1
∞
2
n
a
n
(assume this always converges). p31. Define function
f
(
x
)
=
x
4
+
4
f(x) = x^4 + 4
f
(
x
)
=
x
4
+
4
. Let
P
=
∏
k
=
1
2021
f
(
4
k
−
1
)
f
(
4
k
−
3
)
.
P =\prod^{2021}_{k=1} \frac{f(4k - 1)}{f(4k - 3)}.
P
=
k
=
1
∏
2021
f
(
4
k
−
3
)
f
(
4
k
−
1
)
.
Find the remainder when
P
P
P
is divided by
1000
1000
1000
. p32. Reduce the following expression to a simplified rational:
cos
7
π
9
+
cos
7
5
π
9
+
cos
7
7
π
9
\cos^7 \frac{\pi}{9} + \cos^7 \frac{5\pi}{9}+ \cos^7 \frac{7\pi}{9}
cos
7
9
π
+
cos
7
9
5
π
+
cos
7
9
7
π
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R7
1
Hide problems
2021 SMT Guts Round 7 p25-28 - Stanford Math Tournament
p25. Compute:
∑
i
=
0
∞
(
2
π
)
4
i
+
1
(
4
i
+
1
)
!
∑
i
=
0
∞
(
2
π
)
4
i
+
1
(
4
i
+
3
)
!
\frac{ \sum^{\infty}_{i=0}\frac{(2\pi)^{4i+1}}{(4i+1)!}}{\sum^{\infty}_{i=0}\frac{(2\pi)^{4i+1}}{(4i+3)!}}
∑
i
=
0
∞
(
4
i
+
3
)!
(
2
π
)
4
i
+
1
∑
i
=
0
∞
(
4
i
+
1
)!
(
2
π
)
4
i
+
1
p26. Suppose points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
lie on a circle
ω
\omega
ω
with radius
4
4
4
such that
A
B
C
D
ABCD
A
BC
D
is a quadrilateral with
A
B
=
6
AB = 6
A
B
=
6
,
A
C
=
8
AC = 8
A
C
=
8
,
A
D
=
7
AD = 7
A
D
=
7
. Let
E
E
E
and
F
F
F
be points on
ω
\omega
ω
such that
A
E
AE
A
E
and
A
F
AF
A
F
are respectively the angle bisectors of
∠
B
A
C
\angle BAC
∠
B
A
C
and
∠
D
A
C
\angle DAC
∠
D
A
C
. Compute the area of quadrilateral
A
E
C
F
AECF
A
ECF
. p27. Let
P
(
x
)
=
x
2
−
a
x
+
8
P(x) = x^2 - ax + 8
P
(
x
)
=
x
2
−
a
x
+
8
with a a positive integer, and suppose that
P
P
P
has two distinct real roots
r
r
r
and
s
s
s
. Points
(
r
,
0
)
(r, 0)
(
r
,
0
)
,
(
0
,
s
)
(0, s)
(
0
,
s
)
, and
(
t
,
t
)
(t, t)
(
t
,
t
)
for some positive integer t are selected on the coordinate plane to form a triangle with an area of
2021
2021
2021
. Determine the minimum possible value of
a
+
t
a + t
a
+
t
. p28. A quartic
p
(
x
)
p(x)
p
(
x
)
has a double root at
x
=
−
21
4
x = -\frac{21}{4}
x
=
−
4
21
, and
p
(
x
)
−
1344
x
p(x) - 1344x
p
(
x
)
−
1344
x
has two double roots each
1
4
\frac14
4
1
less than an integer. What are these two double roots? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R6
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Hide problems
2021 SMT Guts Round 6 p21-24 - Stanford Math Tournament
p21. If
f
=
cos
(
sin
(
x
)
)
f = \cos(\sin (x))
f
=
cos
(
sin
(
x
))
. Calculate the sum
∑
n
=
0
2021
f
′
′
(
n
π
)
\sum^{2021}_{n=0} f'' (n \pi)
∑
n
=
0
2021
f
′′
(
nπ
)
. p22. Find all real values of
A
A
A
that minimize the difference between the local maximum and local minimum of
f
(
x
)
=
(
3
x
2
−
4
)
(
x
−
A
+
1
A
)
f(x) = \left(3x^2 - 4\right)\left(x - A + \frac{1}{A}\right)
f
(
x
)
=
(
3
x
2
−
4
)
(
x
−
A
+
A
1
)
. p23. Bessie is playing a game. She labels a square with vertices labeled
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
in clockwise order. There are
7
7
7
possible moves: she can rotate her square
90
90
90
degrees about the center,
180
180
180
degrees about the center,
270
270
270
degrees about the center; or she can flip across diagonal
A
C
AC
A
C
, flip across diagonal
B
D
BD
B
D
, flip the square horizontally (flip the square so that vertices A and B are switched and vertices
C
C
C
and
D
D
D
are switched), or flip the square vertically (vertices
B
B
B
and
C
C
C
are switched, vertices
A
A
A
and
D
D
D
are switched). In how many ways can Bessie arrive back at the original square for the first time in
3
3
3
moves? p24. A positive integer is called happy if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of
5
5
5
-digit happy integers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R5
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Hide problems
2021 SMT Guts Round 5 p17-20 - Stanford Math Tournament
p17. Let the roots of the polynomial
f
(
x
)
=
3
x
3
+
2
x
2
+
x
+
8
=
0
f(x) = 3x^3 + 2x^2 + x + 8 = 0
f
(
x
)
=
3
x
3
+
2
x
2
+
x
+
8
=
0
be
p
,
q
p, q
p
,
q
, and
r
r
r
. What is the sum
1
p
+
1
q
+
1
r
\frac{1}{p} +\frac{1}{q} +\frac{1}{r}
p
1
+
q
1
+
r
1
? p18. Two students are playing a game. They take a deck of five cards numbered
1
1
1
through
5
5
5
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? p19. Compute the sum of all primes
p
p
p
such that
2
p
+
p
2
2^p + p^2
2
p
+
p
2
is also prime. p20. In how many ways can one color the
8
8
8
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R4
1
Hide problems
2021 SMT Guts Round 4 p13-16 - Stanford Math Tournament
p13. Emma has the five letters:
A
,
B
,
C
,
D
,
E
A, B, C, D, E
A
,
B
,
C
,
D
,
E
. How many ways can she rearrange the letters into words? Note that the order of words matter, ie
A
B
C
D
E
ABC DE
A
BC
D
E
and
D
E
A
B
C
DE ABC
D
E
A
BC
are different. p14. Seven students are doing a holiday gift exchange. Each student writes their name on a slip of paper and places it into a hat. Then, each student draws a name from the hat to determine who they will buy a gift for. What is the probability that no student draws himself/herself? p15. We model a fidget spinner as shown below (include diagram) with a series of arcs on circles of radii
1
1
1
. What is the area swept out by the fidget spinner as it’s turned
6
0
o
60^o
6
0
o
? https://cdn.artofproblemsolving.com/attachments/9/8/db27ffce2af68d27eee5903c9f09a36c2a6edf.png p16. Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the sides of a triangle such that
g
c
d
(
a
,
b
)
=
3528
gcd(a, b) = 3528
g
c
d
(
a
,
b
)
=
3528
,
g
c
d
(
b
,
c
)
=
1008
gcd(b, c) = 1008
g
c
d
(
b
,
c
)
=
1008
,
g
c
d
(
a
,
c
)
=
504
gcd(a, c) = 504
g
c
d
(
a
,
c
)
=
504
. Find the value of
a
∗
b
∗
c
a * b * c
a
∗
b
∗
c
. Write your answer as a prime factorization. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R3
1
Hide problems
2021 SMT Guts Round 3 p9-12 - Stanford Math Tournament
p9. The frozen yogurt machine outputs yogurt at a rate of
5
5
5
froyo
3
^3
3
/second. If the bowl is described by
z
=
x
2
+
y
2
z = x^2+y^2
z
=
x
2
+
y
2
and has height
5
5
5
froyos, how long does it take to fill the bowl with frozen yogurt? p10. Prankster Pete and Good Neighbor George visit a street of
2021
2021
2021
houses (each with individual mailboxes) on alternate nights, such that Prankster Pete visits on night
1
1
1
and Good Neighbor George visits on night
2
2
2
, and so on. On each night
n
n
n
that Prankster Pete visits, he drops a packet of glitter in the mailbox of every
n
t
h
n^{th}
n
t
h
house. On each night
m
m
m
that Good Neighbor George visits, he checks the mailbox of every
m
t
h
m^{th}
m
t
h
house, and if there is a packet of glitter there, he takes it home and uses it to complete his art project. After the
202
1
t
h
2021^{th}
202
1
t
h
night, Prankster Pete becomes enraged that none of the houses have yet checked their mail. He then picks three mailboxes at random and takes out a single packet of glitter to dump on George’s head, but notices that all of the mailboxes he visited had an odd number of glitter packets before he took one. In how many ways could he have picked these three glitter packets? Assume that each of these three was from a different house, and that he can only visit houses in increasing numerical order. p11. The taxi-cab length of a line segment with endpoints
(
x
1
,
y
1
)
(x_1, y_1)
(
x
1
,
y
1
)
and
(
x
2
,
y
2
)
(x_2, y_2)
(
x
2
,
y
2
)
is
∣
x
1
−
x
2
∣
+
∣
y
1
−
y
2
∣
|x_1 - x_2| + |y_1- y_2|
∣
x
1
−
x
2
∣
+
∣
y
1
−
y
2
∣
. Given a series of straight line segments connected head-to-tail, the taxi-cab length of this path is the sum of the taxi-cab lengths of its line segments. A goat is on a rope of taxi-cab length
7
2
\frac72
2
7
tied to the origin, and it can’t enter the house, which is the three unit squares enclosed by
(
−
2
,
0
)
(-2, 0)
(
−
2
,
0
)
,
(
0
,
0
)
(0, 0)
(
0
,
0
)
,
(
0
,
−
2
)
(0, -2)
(
0
,
−
2
)
,
(
−
1
,
−
2
)
(-1, -2)
(
−
1
,
−
2
)
,
(
−
1
,
−
1
)
(-1, -1)
(
−
1
,
−
1
)
,
(
−
2
,
−
1
)
(-2, -1)
(
−
2
,
−
1
)
. What is the area of the region the goat can reach? (Note: the rope can’t ”curve smoothly”-it must bend into several straight line segments.) p12. Parabola
P
P
P
,
y
=
a
x
2
+
c
y = ax^2 + c
y
=
a
x
2
+
c
has
a
>
0
a > 0
a
>
0
and
c
<
0
c < 0
c
<
0
. Circle
C
C
C
, which is centered at the origin and lies tangent to
P
P
P
at
P
P
P
’s vertex, intersects
P
P
P
at only the vertex. What is the maximum value of a, possibly in terms of
c
c
c
? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R2
1
Hide problems
2021 SMT Guts Round 2 p5-8 - Stanford Math Tournament
p5. Find the number of three-digit integers that contain at least one
0
0
0
or
5
5
5
. The leading digit of the three-digit integer cannot be zero. p6. What is the sum of the solutions to
x
+
8
5
x
+
7
=
x
+
8
7
x
+
5
\frac{x+8}{5x+7} =\frac{x+8}{7x+5}
5
x
+
7
x
+
8
=
7
x
+
5
x
+
8
p7. Let
B
C
BC
BC
be a diameter of a circle with center
O
O
O
and radius
4
4
4
. Point
A
A
A
is on the circle such that
∠
A
O
B
=
4
5
o
\angle AOB = 45^o
∠
A
OB
=
4
5
o
. Point
D
D
D
is on the circle such that line segment
O
D
OD
O
D
intersects line segment
A
C
AC
A
C
at
E
E
E
and
O
D
OD
O
D
bisects
∠
A
O
C
\angle AOC
∠
A
OC
. Compute the area of
A
D
E
ADE
A
D
E
, which is enclosed by line segments
A
E
AE
A
E
and
E
D
ED
E
D
and minor arc
A
D
AD
A
D
. p8. William is a bacteria farmer. He would like to give his fiance
2021
2021
2021
bacteria as a wedding gift. Since he is an intelligent and frugal bacteria farmer, he would like to add the least amount of bacteria on his favorite infinite plane petri dish to produce those
2021
2021
2021
bacteria. The infinite plane petri dish starts off empty and William can add as many bacteria as he wants each day. Each night, all the bacteria reproduce through binary fission, splitting into two. If he has infinite amount of time before his wedding day, how many bacteria should he add to the dish in total to use the least number of bacteria to accomplish his nuptial goals? PS. You should use hide for answers Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R1
1
Hide problems
2021 SMT Guts Round 1 p1-4 - Stanford Math Tournament
p1. A rectangular pool has diagonal
17
17
17
units and area
120
120
120
units
2
^2
2
. Joey and Rachel start on opposite sides of the pool when Rachel starts chasing Joey. If Rachel runs
5
5
5
units/sec faster than Joey, how long does it take for her to catch him? p2. Alice plays a game with her standard deck of
52
52
52
cards. She gives all of the cards number values where Aces are
1
1
1
’s, royal cards are
10
10
10
’s and all other cards are assigned their face value. Every turn she flips over the top card from her deck and creates a new pile. If the flipped card has value
v
v
v
, she places
12
−
v
12 - v
12
−
v
cards on top of the flipped card. For example: if she flips the
3
3
3
of diamonds then she places
9
9
9
cards on top. Alice continues creating piles until she can no longer create a new pile. If the number of leftover cards is
4
4
4
and there are
5
5
5
piles, what is the sum of the flipped over cards? p3. There are
5
5
5
people standing at
(
0
,
0
)
(0, 0)
(
0
,
0
)
,
(
3
,
0
)
(3, 0)
(
3
,
0
)
,
(
0
,
3
)
(0, 3)
(
0
,
3
)
,
(
−
3
,
0
)
(-3, 0)
(
−
3
,
0
)
, and
(
−
3
,
0
)
(-3, 0)
(
−
3
,
0
)
on a coordinate grid at a time
t
=
0
t = 0
t
=
0
seconds. Each second, every person on the grid moves exactly
1
1
1
unit up, down, left, or right. The person at the origin is infected with covid-
19
19
19
, and if someone who is not infected is at the same lattice point as a person who is infected, at any point in time, they will be infected from that point in time onwards. (Note that this means that if two people run into each other at a non-lattice point, such as
(
0
,
1.5
)
(0, 1.5)
(
0
,
1.5
)
, they will not infect each other.) What is the maximum possible number of infected people after
t
=
7
t = 7
t
=
7
seconds? p4. Kara gives Kaylie a ring with a circular diamond inscribed in a gold hexagon. The diameter of the diamond is
2
2
2
mm. If diamonds cost
$
100
/
m
m
2
\$100/ mm ^2
$100/
m
m
2
and gold costs
$
50
/
m
m
2
\$50 /mm ^2
$50/
m
m
2
, what is the cost of the ring? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
1
Hide problems
2021 SMT Team Round - Stanford Math Tournament
p1. What is the length of the longest string of consecutive prime numbers that divide
224444220
224444220
224444220
? p2. Two circles of radius
r
r
r
are spaced so their centers are
2
r
2r
2
r
apart. If
A
(
r
)
A(r)
A
(
r
)
is the area of the smallest square containing both circles, what is
A
(
r
)
r
2
\frac{A(r)}{r^2}
r
2
A
(
r
)
? p3. Define
e
0
=
1
e_0 = 1
e
0
=
1
and
e
k
=
e
e
k
−
1
e_k = e^{e_{k-1}}
e
k
=
e
e
k
−
1
for
k
≥
1
k \ge 1
k
≥
1
. Compute
∫
e
4
e
6
1
x
(
ln
x
)
(
ln
ln
x
)
(
ln
ln
ln
x
)
d
x
\int^{e_6}_{e_4} \frac{1}{x(\ln x)(\ln \ln x)(\ln \ln \ln x)}dx
∫
e
4
e
6
x
(
l
n
x
)
(
l
n
l
n
x
)
(
l
n
l
n
l
n
x
)
1
d
x
. p4. Compute
∏
n
=
0
2020
∑
k
=
0
2020
(
1
2020
)
k
(
2021
)
n
\prod_{n=0}^{2020}\sum_{k=0}^{2020}\left( \frac{1}{2020}\right)^{k(2021)^n}
∏
n
=
0
2020
∑
k
=
0
2020
(
2020
1
)
k
(
2021
)
n
p5. A potter makes a bowl, beginning with a sphere of clay and then cutting away the top and bottom of the sphere with two parallel cuts that are equidistant from the center. Finally, he hollows out the remaining shape, leaving a base at one end. Assume that the thickness of the bowl is negligible. If the potter wants the bowl to hold a volume that is
13
27
\frac{13}{27}
27
13
of the volume of the sphere he started with, the distance from the center at which he makes his cuts should be what fraction of the radius? p6. Let
A
B
AB
A
B
be a line segment with length
2
+
2
2 + \sqrt2
2
+
2
. A circle
ω
\omega
ω
with radius
1
1
1
is drawn such that it passes through the end point
B
B
B
of the line segment and its center
O
O
O
lies on the line segment
A
B
AB
A
B
. Let
C
C
C
be a point on circle
ω
\omega
ω
such that
A
C
=
B
C
AC = BC
A
C
=
BC
. What is the size of angle
C
A
B
CAB
C
A
B
in degrees? p7. Find all possible values of
sin
x
\sin x
sin
x
such that
4
sin
(
6
x
)
=
5
sin
(
2
x
)
4 \sin(6x) = 5 \sin(2x)
4
sin
(
6
x
)
=
5
sin
(
2
x
)
. p8. Frank the frog sits on the first lily pad in an infinite line of lily pads. Each lily pad besides the one first one is randomly assigned a real number from
0
0
0
to
1
1
1
. Franks starting lily pad is assigned
0
0
0
. Frank will jump forward to the next lily pad as long as the next pad’s number is greater than his current pad’s number. For example, if the first few lily pads including Frank’s are numbered
0
0
0
,
.
4
.4
.4
,
.
72
.72
.72
,
.
314
.314
.314
, Frank will jump forward twice, visiting a total of
3
3
3
lily pads. What is the expected number of lily pads that Frank will visit? p9. For positive integers
n
n
n
and
k
k
k
with
k
≤
n
k \le n
k
≤
n
, let
f
(
n
,
k
)
=
∑
j
=
0
k
−
1
j
(
k
−
1
j
)
(
n
−
k
+
1
k
−
j
)
.
f(n, k) = \sum^{k-1}_{j=0} j {{k -1} \choose j}{{n - k + 1} \choose {k - j}}.
f
(
n
,
k
)
=
j
=
0
∑
k
−
1
j
(
j
k
−
1
)
(
k
−
j
n
−
k
+
1
)
.
Compute the sum of the prime factors of
f
(
4
,
4
)
+
f
(
5
,
4
)
+
f
(
6
,
4
)
+
.
.
.
+
f
(
2021
,
4
)
f(4, 4) + f(5, 4) + f(6, 4) + ... + f(2021, 4)
f
(
4
,
4
)
+
f
(
5
,
4
)
+
f
(
6
,
4
)
+
...
+
f
(
2021
,
4
)
. p10.
△
A
B
C
\vartriangle ABC
△
A
BC
has side lengths
A
B
=
5
AB = 5
A
B
=
5
,
A
C
=
10
AC = 10
A
C
=
10
, and
B
C
=
9
BC = 9
BC
=
9
. The median of
△
A
B
C
\vartriangle ABC
△
A
BC
from
A
A
A
intersects the circumcircle of the triangle again at point
D
D
D
. What is
B
D
+
C
D
BD + CD
B
D
+
C
D
? p11. A subset of five distinct positive integers is chosen uniformly at random from the set
{
1
,
2
,
.
.
.
,
11
}
\{1, 2, ... , 11\}
{
1
,
2
,
...
,
11
}
. The probability that the subset does not contain three consecutive integers can be written as
m
/
n
m/n
m
/
n
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m + n
m
+
n
. p12. Compute
∫
−
1
1
(
x
2
+
x
+
1
−
x
2
)
2
d
x
\int_{-1}^{1} (x^2 + x +\sqrt{1 - x^2})^2dx
∫
−
1
1
(
x
2
+
x
+
1
−
x
2
)
2
d
x
. p13. Three friends, Xander, Yulia, and Zoe, have each planned to visit the same cafe one day. If each person arrives at the cafe at a random time between
2
2
2
PM and
3
3
3
PM and stays for
15
15
15
minutes, what is the probability that all three friends will be there at the same time at some point? p14. Jim the Carpenter starts with a wooden rod of length
1
1
1
unit. Jim will cut the middle
1
3
\frac13
3
1
of the rod and remove it, creating
2
2
2
smaller rods of length
1
3
\frac13
3
1
. He repeats this process, randomly choosing a rod to split into
2
2
2
smaller rods. Thus, after two such splits, Jim will have
3
3
3
rods of length
1
3
\frac13
3
1
,
1
9
\frac19
9
1
, and
1
9
\frac19
9
1
. After
3
3
3
splits, Jim will either have
4
4
4
rods of lengths
1
9
\frac19
9
1
,
1
9
\frac19
9
1
,
1
9
\frac19
9
1
,
1
9
\frac19
9
1
or
1
3
\frac13
3
1
,
1
9
\frac19
9
1
,
1
27
\frac{1}{27}
27
1
,
1
27
\frac{1}{27}
27
1
. , What is the expected value of the total length of the rods after
5
5
5
splits? p15. Robin is at an archery range. There are
10
10
10
targets, each of varying difficulty. If Robin spends
t
t
t
seconds concentrating on target
n
n
n
where
1
≤
n
≤
10
1 \le n \le 10
1
≤
n
≤
10
, he has a probability
p
=
1
−
e
−
t
/
n
p = 1 - e^{-t/n}
p
=
1
−
e
−
t
/
n
of hitting the target. Hitting target
n
n
n
gives him
n
n
n
points and he is alloted a total of
60
60
60
seconds. If he has at most one attempt on each target, and he allots time to concentrate on each target optimally to maximize his point total, what is the expected value of the number of points Robin will get? (Assume no time is wasted between shots). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.