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2022 Stanford Mathematics Tournament
2022 Stanford Mathematics Tournament
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Stanford Mathematics Tournament
Subcontests
(11)
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4
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4
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3
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1
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1
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2022 SMT Team Round - Stanford Math Tournament
p1. Square
A
B
C
D
ABCD
A
BC
D
has side length
2
2
2
. Let the midpoint of
B
C
BC
BC
be
E
E
E
. What is the area of the overlapping region between the circle centered at
E
E
E
with radius
1
1
1
and the circle centered at
D
D
D
with radius
2
2
2
? (You may express your answer using inverse trigonometry functions of noncommon values.) p2. Find the number of times
f
(
x
)
=
2
f(x) = 2
f
(
x
)
=
2
occurs when
0
≤
x
≤
2022
π
0 \le x \le 2022 \pi
0
≤
x
≤
2022
π
for the function
f
(
x
)
=
2
x
(
c
o
s
(
x
)
+
1
)
f(x) = 2^x(cos(x) + 1)
f
(
x
)
=
2
x
(
cos
(
x
)
+
1
)
. p3. Stanford is building a new dorm for students, and they are looking to offer
2
2
2
room configurations:
∙
\bullet
∙
Configuration
A
A
A
: a one-room double, which is a square with side length of
x
x
x
,
∙
\bullet
∙
Configuration
B
B
B
: a two-room double, which is two connected rooms, each of them squares with a side length of
y
y
y
. To make things fair for everyone, Stanford wants a one-room double (rooms of configuration
A
A
A
) to be exactly
1
1
1
m
2
^2
2
larger than the total area of a two-room double. Find the number of possible pairs of side lengths
(
x
,
y
)
(x, y)
(
x
,
y
)
, where
x
∈
N
x \in N
x
∈
N
,
y
∈
N
y \in N
y
∈
N
, such that
x
−
y
<
2022
x - y < 2022
x
−
y
<
2022
. p4. The island nation of Ur is comprised of
6
6
6
islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands
A
A
A
and
B
B
B
if each island chooses the other). Then, all islands connected by bridges together form an island-state. What is the expected number of island-states Ur is divided into? p5. Let
a
,
b
,
a, b,
a
,
b
,
and
c
c
c
be the roots of the polynomial
x
3
−
3
x
2
−
4
x
+
5
x^3 - 3x^2 - 4x + 5
x
3
−
3
x
2
−
4
x
+
5
. Compute
a
4
+
b
4
a
+
b
+
b
4
+
c
4
b
+
c
+
c
4
+
a
4
c
+
a
\frac{a^4 + b^4}{a + b}+\frac{b^4 + c^4}{b + c}+\frac{c^4 + a^4}{c + a}
a
+
b
a
4
+
b
4
+
b
+
c
b
4
+
c
4
+
c
+
a
c
4
+
a
4
. p6. Carol writes a program that finds all paths on an 10 by 2 grid from cell (1, 1) to cell (10, 2) subject to the conditions that a path does not visit any cell more than once and at each step the path can go up, down, left, or right from the current cell, excluding moves that would make the path leave the grid. What is the total length of all such paths? (The length of a path is the number of cells it passes through, including the starting and ending cells.) p7. Consider the sequence of integers an defined by
a
1
=
1
a_1 = 1
a
1
=
1
,
a
p
=
p
a_p = p
a
p
=
p
for prime
p
p
p
and
a
m
n
=
m
a
n
+
n
a
m
a_{mn} = ma_n + na_m
a
mn
=
m
a
n
+
n
a
m
for
m
,
n
>
1
m, n > 1
m
,
n
>
1
. Find the smallest
n
n
n
such that
a
n
2
2022
\frac{a_n^2}{2022}
2022
a
n
2
is a perfect power of
3
3
3
. p8. Let
△
A
B
C
\vartriangle ABC
△
A
BC
be a triangle whose
A
A
A
-excircle,
B
B
B
-excircle, and
C
C
C
-excircle have radii
R
A
R_A
R
A
,
R
B
R_B
R
B
, and
R
C
R_C
R
C
, respectively. If
R
A
R
B
R
C
=
384
R_AR_BR_C = 384
R
A
R
B
R
C
=
384
and the perimeter of
△
A
B
C
\vartriangle ABC
△
A
BC
is
32
32
32
, what is the area of
△
A
B
C
\vartriangle ABC
△
A
BC
? p9. Consider the set
S
S
S
of functions
f
:
{
1
,
2
,
.
.
.
,
16
}
→
{
1
,
2
,
.
.
.
,
243
}
f : \{1, 2, . . . , 16\} \to \{1, 2, . . . , 243\}
f
:
{
1
,
2
,
...
,
16
}
→
{
1
,
2
,
...
,
243
}
satisfying: (a)
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
(b)
f
(
n
2
)
=
n
2
f
(
n
)
f(n^2) = n^2f(n)
f
(
n
2
)
=
n
2
f
(
n
)
, (c)
n
∣
f
(
n
)
n |f(n)
n
∣
f
(
n
)
, (d)
f
(
l
c
m
(
m
,
n
)
)
f
(
g
c
d
(
m
,
n
)
)
=
f
(
m
)
f
(
n
)
f(lcm(m, n))f(gcd(m, n)) = f(m)f(n)
f
(
l
c
m
(
m
,
n
))
f
(
g
c
d
(
m
,
n
))
=
f
(
m
)
f
(
n
)
. If
∣
S
∣
|S|
∣
S
∣
can be written as
p
1
ℓ
1
⋅
p
2
ℓ
2
⋅
.
.
.
⋅
p
k
ℓ
k
p^{\ell_1}_1 \cdot p^{\ell_2}_2 \cdot ... \cdot p^{\ell_k}_k
p
1
ℓ
1
⋅
p
2
ℓ
2
⋅
...
⋅
p
k
ℓ
k
where
p
i
p_i
p
i
are distinct primes, compute
p
1
ℓ
1
+
p
2
ℓ
2
+
.
.
.
+
p
k
ℓ
k
p_1\ell_1+p_2\ell_2+. . .+p_k\ell_k
p
1
ℓ
1
+
p
2
ℓ
2
+
...
+
p
k
ℓ
k
. p10. You are given that
log
10
2
≈
0.3010
\log_{10}2 \approx 0.3010
lo
g
10
2
≈
0.3010
and that the first (leftmost) two digits of
2
1000
2^{1000}
2
1000
are 10. Compute the number of integers
n
n
n
with
1000
≤
n
≤
2000
1000 \le n \le 2000
1000
≤
n
≤
2000
such that
2
n
2^n
2
n
starts with either the digit
8
8
8
or
9
9
9
(in base
10
10
10
). p11. Let
O
O
O
be the circumcenter of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
M
M
M
be the midpoint of
B
C
BC
BC
, and let
E
E
E
and
F
F
F
be the feet of the altitudes from
B
B
B
and
C
C
C
, respectively, onto the opposite sides.
E
F
EF
EF
intersects
B
C
BC
BC
at
P
P
P
. The line passing through
O
O
O
and perpendicular to
B
C
BC
BC
intersects the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
at
L
L
L
(on the major arc
B
C
BC
BC
) and
N
N
N
, and intersects
B
C
BC
BC
at
M
M
M
. Point
Q
Q
Q
lies on the line
L
A
LA
L
A
such that
O
Q
OQ
OQ
is perpendicular to
A
P
AP
A
P
. Given that
∠
B
A
C
=
6
0
o
\angle BAC = 60^o
∠
B
A
C
=
6
0
o
and
∠
A
M
C
=
6
0
o
\angle AMC = 60^o
∠
A
MC
=
6
0
o
, compute
O
Q
/
A
P
OQ/AP
OQ
/
A
P
. p12. Let
T
T
T
be the isosceles triangle with side lengths
5
,
5
,
6
5, 5, 6
5
,
5
,
6
. Arpit and Katherine simultaneously choose points
A
A
A
and
K
K
K
within this triangle, and compute
d
(
A
,
K
)
d(A, K)
d
(
A
,
K
)
, the squared distance between the two points. Suppose that Arpit chooses a random point
A
A
A
within
T
T
T
. Katherine plays the (possibly randomized) strategy which given Arpit’s strategy minimizes the expected value of
d
(
A
,
K
)
d(A, K)
d
(
A
,
K
)
. Compute this value. p13. For a regular polygon
S
S
S
with
n
n
n
sides, let
f
(
S
)
f(S)
f
(
S
)
denote the regular polygon with
2
n
2n
2
n
sides such that the vertices of
S
S
S
are the midpoints of every other side of
f
(
S
)
f(S)
f
(
S
)
. Let
f
(
k
)
(
S
)
f^{(k)}(S)
f
(
k
)
(
S
)
denote the polygon that results after applying f a total of k times. The area of
lim
k
→
∞
f
(
k
)
(
P
)
\lim_{k \to \infty} f^{(k)}(P)
lim
k
→
∞
f
(
k
)
(
P
)
where
P
P
P
is a pentagon of side length
1
1
1
, can be expressed as
a
+
b
c
d
π
m
\frac{a+b\sqrt{c}}{d}\pi^m
d
a
+
b
c
π
m
for some positive integers
a
,
b
,
c
,
d
,
m
a, b, c, d, m
a
,
b
,
c
,
d
,
m
where
d
d
d
is not divisible by the square of any prime and
d
d
d
does not share any positive divisors with
a
a
a
and
b
b
b
. Find
a
+
b
+
c
+
d
+
m
a + b + c + d + m
a
+
b
+
c
+
d
+
m
. p14. Consider the function
f
(
m
)
=
∑
n
=
0
∞
(
n
−
m
)
2
(
2
n
)
!
f(m) = \sum_{n=0}^{\infty}\frac{(n - m)^2}{(2n)!}
f
(
m
)
=
∑
n
=
0
∞
(
2
n
)!
(
n
−
m
)
2
. This function can be expressed in the form
f
(
m
)
=
a
m
e
+
b
m
4
e
f(m) = \frac{a_m}{e} +\frac{b_m}{4}e
f
(
m
)
=
e
a
m
+
4
b
m
e
for sequences of integers
{
a
m
}
m
≥
1
\{a_m\}_{m\ge 1}
{
a
m
}
m
≥
1
,
{
b
m
}
m
≥
1
\{b_m\}_{m\ge 1}
{
b
m
}
m
≥
1
. Determine
lim
n
→
∞
2022
b
m
a
m
\lim_{n \to \infty}\frac{2022b_m}{a_m}
lim
n
→
∞
a
m
2022
b
m
. p15. In
△
A
B
C
\vartriangle ABC
△
A
BC
, let
G
G
G
be the centroid and let the circumcenters of
△
B
C
G
\vartriangle BCG
△
BCG
,
△
C
A
G
\vartriangle CAG
△
C
A
G
, and
△
A
B
G
\vartriangle ABG
△
A
BG
be
I
,
J
I, J
I
,
J
, and
K
K
K
, respectively. The line passing through
I
I
I
and the midpoint of
B
C
BC
BC
intersects
K
J
KJ
K
J
at
Y
Y
Y
. If the radius of circle
K
K
K
is
5
5
5
, the radius of circle
J
J
J
is
8
8
8
, and
A
G
=
6
AG = 6
A
G
=
6
, what is the length of
K
Y
KY
K
Y
?PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.