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Girls in Math at Yale
2023 Girls in Math at Yale
2023 Girls in Math at Yale
Part of
Girls in Math at Yale
Subcontests
(3)
2
1
Hide problems
Girls in Math at Yale 2023 Tiebreaker 2 - bee travels in 3D
A bee travels in a series of steps of length
1
1
1
: north, west, north, west, up, south, east, south, east, down. (The bee can move in three dimensions, so north is distinct from up.) There exists a plane
P
P
P
that passes through the midpoints of each step. Suppose we orthogonally project the bee’s path onto the plane
P
P
P
, and let
A
A
A
be the area of the resulting figure. What is
A
2
A^2
A
2
?
1
1
Hide problems
Girls in Math at Yale 2023 Tiebreaker 1: fair coin
Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin?
2
Hide problems
2023 Girls in Math at Yale - Individual Round
p1. Hitori is trying to guess a three-digit integer with three different digits in five guesses to win a new guitar. She guesses
819
819
819
, and is told that exactly one of the digits in her guess is in the answer, but it is in the wrong place. Next, she guesses
217
217
217
, and is told that exactly one of the digits is in the winning number, and it is in the right place. After that, she guesses
362
362
362
, and is told that exactly two of the digits are in the winning number, and exactly one of them is in the right place. Then, she guesses
135
135
135
, and is told that none of the digits are in the winning number. Finally, she guesses correctly. What is the winning number? p2. A three-digit integer
A
2
C
‾
\overline{A2C}
A
2
C
, where
A
A
A
and
C
C
C
are digits, not necessarily distinct, and
A
≠
0
A \ne 0
A
=
0
, is toxic if it is a multiple of
9
9
9
. For example,
126
126
126
is toxic because
126
=
9
⋅
14
126 = 9 \cdot 14
126
=
9
⋅
14
. Find the sum of all toxic integers. p3. Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit multiple of
13
13
13
.” Claire asks, “Is there a two-digit multiple of
13
13
13
greater than or equal to your favorite number such that if you averaged it with your favorite number, and told me the result, I could determine your favorite number?” Cat says, “Yes. However, without knowing that, if I selected a digit of the sum of the squares of the digits of my number at random and told it to you, you would always be unable to determine my favorite number.” Claire says, “Now I know your favorite number!” What is Cat’s favorite number? p4. Define
f
(
x
)
=
log
x
(
2
x
)
f(x) = \log_x(2x)
f
(
x
)
=
lo
g
x
(
2
x
)
for positive
x
x
x
. Compute the product
f
(
4
)
f
(
8
)
f
(
16
)
f
(
32
)
f
(
64
)
f
(
128
)
f
(
256
)
f
(
512
)
f
(
1024
)
f
(
2048
)
.
f(4)f(8)f(16)f(32)f(64)f(128)f(256)f(512)f(1024)f(2048).
f
(
4
)
f
(
8
)
f
(
16
)
f
(
32
)
f
(
64
)
f
(
128
)
f
(
256
)
f
(
512
)
f
(
1024
)
f
(
2048
)
.
p5. Isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
has bases of length
A
B
=
4
AB = 4
A
B
=
4
and
C
D
=
8
CD = 8
C
D
=
8
, and height
6
6
6
. Let
F
F
F
be the midpoint of side
C
D
‾
\overline{CD}
C
D
. Base
A
B
‾
\overline{AB}
A
B
is extended past
B
B
B
to point
E
E
E
so that
B
E
‾
⊥
E
C
‾
\overline{BE} \perp \overline{EC}
BE
⊥
EC
. Also, let
D
E
‾
\overline{DE}
D
E
intersect
A
F
‾
\overline{AF}
A
F
at
G
G
G
,
B
F
‾
\overline{BF}
BF
at
H
H
H
, and
B
C
‾
\overline{BC}
BC
at
I
I
I
. If
[
D
F
G
]
+
[
C
F
H
I
]
−
[
A
G
H
B
]
=
m
n
[DFG] + [CFHI] - [AGHB] = \frac{m}{n}
[
D
FG
]
+
[
CF
H
I
]
−
[
A
G
H
B
]
=
n
m
where
m
m
m
and
n
n
n
are relatively prime positive integers, find
m
+
n
m + n
m
+
n
. p6. Julia writes down all two-digit numbers that can be expressed as the sum of two (not necessarily distinct) positive cubes. For each such number
A
B
‾
\overline{AB}
A
B
, she plots the point
(
A
,
B
)
(A,B)
(
A
,
B
)
on the coordinate plane. She realizes that four of the points form a rhombus. Find the area of this rhombus. p7. When
(
x
3
+
x
+
1
)
24
(x^3 + x + 1)^{24}
(
x
3
+
x
+
1
)
24
is expanded and like terms are combined, what is the sum of the exponents of the terms with an odd coefficient? p8. Given that
3
2023
=
164550...827
3^{2023} = 164550... 827
3
2023
=
164550...827
has
966
966
966
digits, find the number of integers
0
≤
a
≤
2023
0 \le a \le 2023
0
≤
a
≤
2023
such that
3
a
3^a
3
a
has a leftmost digit of
9
9
9
. p9. Triangle
A
B
C
ABC
A
BC
has circumcircle
Ω
\Omega
Ω
. Let the angle bisectors of
∠
C
A
B
\angle CAB
∠
C
A
B
and
∠
C
B
A
\angle CBA
∠
CB
A
meet
Ω
\Omega
Ω
again at
D
D
D
and
E
E
E
. Given that
A
B
=
4
AB = 4
A
B
=
4
,
B
C
=
5
BC = 5
BC
=
5
, and
A
C
=
6
AC = 6
A
C
=
6
, find the product of the side lengths of pentagon
B
A
E
C
D
BAECD
B
A
EC
D
. p10. Let
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
be a sequence of real numbers and
ℓ
\ell
ℓ
be a positive real number such that for all positive integers
k
k
k
,
a
k
=
a
k
+
28
a_k = a_{k+28}
a
k
=
a
k
+
28
and
a
k
+
1
a
k
+
1
=
ℓ
a_k + \frac{1}{a_{k+1}} = \ell
a
k
+
a
k
+
1
1
=
ℓ
. If the sequence
a
n
a_n
a
n
is not constant, find the number of possible values of
ℓ
\ell
ℓ
. p11. The numbers
3
3
3
,
6
6
6
,
9
9
9
, and
12
12
12
(or, in binary,
11
11
11
,
110
110
110
,
1001
1001
1001
,
1100
1100
1100
) form an arithmetic sequence of length four, and each number has exactly two 1s when written in base
2
2
2
. If the longest (nonconstant) arithmetic sequence such that each number in the sequence has exactly ten 1s when written in base
2
2
2
has length
n
n
n
, and the sequence with smallest starting term is
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
n
a_n
a
n
, find
n
+
a
2
n + a_2
n
+
a
2
. p12. Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers greater than 1 that satisfy the inequality
log
x
12
y
8
z
9
(
x
log
x
y
log
y
z
log
z
)
≤
4
,
\log_{\sqrt{x^{12}y^8z^9}}\left( x^{\log x}y^{\log y}z^{\log z} \right) \le 4,
lo
g
x
12
y
8
z
9
(
x
l
o
g
x
y
l
o
g
y
z
l
o
g
z
)
≤
4
,
where
log
(
x
)
\log (x)
lo
g
(
x
)
denotes the base
10
10
10
logarithm. If the maximum value of
log
(
x
5
y
3
z
)
\log \left(\sqrt{x^5y^3z} \right)
lo
g
(
x
5
y
3
z
)
can be expressed as
a
+
b
c
d
\frac{a+b\sqrt{c}}{d}
d
a
+
b
c
, where
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are positive integers such that
c
c
c
is square-free and
g
c
d
(
a
,
b
,
d
)
=
1
gcd(a, b, d) = 1
g
c
d
(
a
,
b
,
d
)
=
1
, find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
2023 Girls in Math at Yale - Team Round
p1. Quarters are worth
25
25
25
cents and weigh
5.7
5.7
5.7
grams each. Dimes are worth
10
10
10
cents and weigh
2.3
2.3
2.3
grams each. A pile of quarters weighs
342
342
342
grams, and a pile of dimes is equal in value to the pile of quarters. What is the weight in grams of the pile of dimes? p2. Three-digit palindrome
A
B
A
‾
\overline{ABA}
A
B
A
with
A
≠
0
A \ne 0
A
=
0
has the property that the sum of the squares of its digits equals the two-digit number
A
B
‾
\overline{AB}
A
B
. What is the sum of all possible values of
A
B
A
‾
\overline{ABA}
A
B
A
? p3. Let
N
N
N
be the number
N
=
01020304...979899
N = 01020304 . . . 979899
N
=
01020304...979899
, formed by concatenating the integers from
01
01
01
to
99
99
99
inclusive, with leading zeros. How many times does the digit
7
7
7
appear in
7
×
N
7 \times N
7
×
N
? p4. Winry and Riza are each writing the integers
1
,
2
,
.
.
.
,
N
1, 2, . . . , N
1
,
2
,
...
,
N
on a chalkboard. However, Winry erases all occurrences of the digit
1
1
1
, and Riza erases all occurrences of the digit
3
3
3
. They notice that at
N
=
13
N = 13
N
=
13
, the sum of the digits written on the chalkboard on Winry’s list is the same as the sum of the digits written on the chalkboard on Riza’s list. What is the next smallest
N
N
N
where this property holds? p5. Define the Hexicab distance between two points on the Cartesian plane to be the length of the shortest path connecting the points that consists of line segments parallel to the
x
x
x
-axis, the line
y
=
3
3
x
y =\frac{\sqrt3}{3}x
y
=
3
3
x
, or the line
y
=
3
x
y =\sqrt3 x
y
=
3
x
. The region
R
R
R
consists of all points that are a Hexicab distance at most
1
1
1
from the origin. If the area of R can be expressed in the form
a
+
b
c
d
\frac{a+b\sqrt{c}}{d}
d
a
+
b
c
where
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are positive integers such that c is square-free and
g
c
d
(
a
,
b
,
d
)
=
1
gcd(a, b, d) = 1
g
c
d
(
a
,
b
,
d
)
=
1
, find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
. p6.
△
A
0
B
0
C
0
\vartriangle A_0B_0C_0
△
A
0
B
0
C
0
has
∠
A
0
=
4
0
o
\angle A_0 = 40^o
∠
A
0
=
4
0
o
,
∠
B
0
=
6
0
o
\angle B_0 = 60^o
∠
B
0
=
6
0
o
, and angle
∠
C
0
=
8
0
o
\angle C_0 = 80^o
∠
C
0
=
8
0
o
. It is rotated about vertex
A
0
A_0
A
0
clockwise by some angle
α
\alpha
α
to get
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
. Then,
△
A
1
B
1
C
1
\vartriangle A_1B_1C_1
△
A
1
B
1
C
1
is rotated clockwise about
B
1
B_1
B
1
by some angle
β
\beta
β
to get
△
A
2
B
2
C
2
\vartriangle A_2B_2C_2
△
A
2
B
2
C
2
. After that,
△
A
2
B
2
C
2
\vartriangle A_2B_2C_2
△
A
2
B
2
C
2
is rotated clockwise about vertex
C
2
C_2
C
2
by some angle
γ
\gamma
γ
to get
△
A
3
B
3
C
3
\vartriangle A_3B_3C_3
△
A
3
B
3
C
3
. If
A
0
=
A
3
A_0 = A_3
A
0
=
A
3
,
B
0
=
B
3
B_0 = B_3
B
0
=
B
3
,
C
0
=
C
3
C_0 = C_3
C
0
=
C
3
, and
0
o
<
α
≤
18
0
o
0^o < \alpha\le 180^o
0
o
<
α
≤
18
0
o
, then what is
α
\alpha
α
in degrees? p7. Ally, Bella, Aria, Barbara, Ava, Brianna, and Cindy are playing a game, with turns taken in the order listed above. Ally starts with
1
1
1
muffin, and has the choice to either “double it and give it to the next person” or “square it and give it to the next person”. Ally then passes the resulting amount to Bella, and this process continues, with each person given the same choice and passing it to the next person. This process ends with Cindy receiving C muffins. Ally, Aria, and Ava are trying to maximize
∣
2023
−
C
∣
|2023 - C|
∣2023
−
C
∣
, while Bella, Barbara, and Brianna are trying to minimize
∣
2023
−
C
∣
|2023 - C|
∣2023
−
C
∣
. If both sides play optimally, what is
∣
2023
−
C
∣
|2023 - C|
∣2023
−
C
∣
? p8. A set of integers has a coprime cycle if it can be listed in a cyclic list of the form
(
a
1
,
a
2
,
a
3
,
.
.
.
,
a
n
)
(a_1, a_2, a_3, . . . , a_n)
(
a
1
,
a
2
,
a
3
,
...
,
a
n
)
, where each element of the set appears exactly once and
g
c
d
(
a
i
,
a
i
+
1
)
=
1
gcd(a_i, a_{i+1}) = 1
g
c
d
(
a
i
,
a
i
+
1
)
=
1
for
i
=
1
,
.
.
.
,
n
i = 1, . . . , n
i
=
1
,
...
,
n
where
a
n
+
1
=
a
1
a_{n+1} = a_1
a
n
+
1
=
a
1
. For example,
(
7
,
8
,
9
,
10
,
11
,
14
,
13
,
12
)
(7, 8, 9, 10, 11, 14, 13, 12)
(
7
,
8
,
9
,
10
,
11
,
14
,
13
,
12
)
is a coprime cycle for
k
=
7
k = 7
k
=
7
, but
(
7
,
8
,
9
,
10
,
11
,
12
,
13
,
14
)
(7, 8, 9, 10, 11, 12, 13, 14)
(
7
,
8
,
9
,
10
,
11
,
12
,
13
,
14
)
is not because
g
c
d
(
14
,
7
)
≠
1
gcd(14, 7) \ne 1
g
c
d
(
14
,
7
)
=
1
. What is the smallest positive integer k such that the set
{
k
,
k
+
1
,
.
.
.
,
k
+
7
}
\{k, k + 1, . . . , k + 7\}
{
k
,
k
+
1
,
...
,
k
+
7
}
does not possess a coprime cycle? p9.
A
B
C
D
ABCD
A
BC
D
is a regular tetrahedron with side length
1
1
1
. Spheres
O
1
,
O_1,
O
1
,
O
2
O_2
O
2
,
O
3
O_3
O
3
, and
O
4
O_4
O
4
have equal radii, and are positioned inside
A
B
C
D
ABCD
A
BC
D
such that they are internally tangent to three of the faces at a time, and all four spheres intersect at a single point. If the radius of
O
1
O_1
O
1
can be expressed as
a
b
c
\frac{a \sqrt{b}}{c}
c
a
b
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are positive integers such that
b
b
b
is square-free and
g
c
d
(
a
,
c
)
=
1
gcd(a, c) = 1
g
c
d
(
a
,
c
)
=
1
, find
a
+
b
+
c
a + b + c
a
+
b
+
c
. p10. Danielle has
202
3
3
2023^3
202
3
3
boxes in which she can place non-negative integers. Each box is assigned a unique ordered triple
(
i
,
j
,
k
)
(i, j, k)
(
i
,
j
,
k
)
with
i
,
j
,
k
∈
{
1
,
2
,
3
,
⋅
⋅
⋅
,
2023
}
i, j, k \in \{1, 2, 3, · · · , 2023\}
i
,
j
,
k
∈
{
1
,
2
,
3
,⋅⋅⋅,
2023
}
. In the box with ordered triple
(
1
,
1
,
1
)
(1, 1, 1)
(
1
,
1
,
1
)
, Danielle places
a
0
a_0
a
0
. Then, for a box labeled
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, she places the smallest non-negative integer that has not been placed in any of
(
a
0
,
b
,
c
)
(a_0, b, c)
(
a
0
,
b
,
c
)
for
0
≤
a
0
<
a
0 \le a_0 < a
0
≤
a
0
<
a
,
(
a
,
b
0
,
c
)
(a, b_0, c)
(
a
,
b
0
,
c
)
for
0
≤
b
0
<
b
0 \le b_0 < b
0
≤
b
0
<
b
, or
(
a
,
b
,
c
0
)
(a, b, c_0)
(
a
,
b
,
c
0
)
for
0
≤
c
0
<
c
0 \le c_0 < c
0
≤
c
0
<
c
. Find the number Danielle places in the box labeled
(
2023
,
203
,
20
)
(2023, 203, 20)
(
2023
,
203
,
20
)
. p11. For a fixed integer
n
n
n
, let
f
(
a
,
b
,
c
,
d
)
=
(
4
a
−
6
b
+
7
c
−
3
d
+
1
)
n
f(a, b, c, d) = (4a - 6b + 7c - 3d + 1)^n
f
(
a
,
b
,
c
,
d
)
=
(
4
a
−
6
b
+
7
c
−
3
d
+
1
)
n
. The
k
k
k
th degree terms of a polynomial are the terms whose exponents sum to
k
k
k
, e.g.
a
2
b
3
c
a^2 b^3 c
a
2
b
3
c
has degree
6
6
6
. Let
A
A
A
and
B
B
B
be the sum of the coefficients of all
7
7
7
th degree terms and of all
9
9
9
th degree terms of
f
f
f
, respectively. If
B
=
224
A
,
B = 224A,
B
=
224
A
,
find the total number of positive divisors of
A
A
A
. p12. If
n
n
n
is the smallest positive integer such that
1
0
3
n
3
+
10
n
+
1
≡
−
1
10^{3n^3+10n+1 } \equiv -1
1
0
3
n
3
+
10
n
+
1
≡
−
1
(mod
14641
14641
14641
), then find the last three digits of
3
n
3^n
3
n
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.