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Girls in Math at Yale
2022 Girls in Math at Yale
2022 Girls in Math at Yale
Part of
Girls in Math at Yale
Subcontests
(20)
Mixer Round
1
Hide problems
2022 Girls in Math at Yale - Mixer Round
p1. Find the smallest positive integer
N
N
N
such that
2
N
−
1
2N -1
2
N
−
1
and
2
N
+
1
2N +1
2
N
+
1
are both composite. p2. Compute the number of ordered pairs of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
with
1
≤
a
,
b
≤
5
1 \le a, b \le 5
1
≤
a
,
b
≤
5
such that
a
b
−
a
−
b
ab - a - b
ab
−
a
−
b
is prime. p3. Given a semicircle
Ω
\Omega
Ω
with diameter
A
B
AB
A
B
, point
C
C
C
is chosen on
Ω
\Omega
Ω
such that
∠
C
A
B
=
6
0
o
\angle CAB = 60^o
∠
C
A
B
=
6
0
o
. Point
D
D
D
lies on ray
B
A
BA
B
A
such that
D
C
DC
D
C
is tangent to
Ω
\Omega
Ω
. Find
(
B
D
B
C
)
2
\left(\frac{BD}{BC} \right)^2
(
BC
B
D
)
2
. p4. Let the roots of
x
2
+
7
x
+
11
x^2 + 7x + 11
x
2
+
7
x
+
11
be
r
r
r
and
s
s
s
. If
f
(
x
)
f(x)
f
(
x
)
is the monic polynomial with roots
r
s
+
r
+
s
rs + r + s
rs
+
r
+
s
and
r
2
+
s
2
r^2 + s^2
r
2
+
s
2
, what is
f
(
3
)
f(3)
f
(
3
)
? p5. Regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
has side length
3
3
3
. Circle
ω
\omega
ω
is drawn with
A
C
AC
A
C
as its diameter.
B
C
BC
BC
is extended to intersect
ω
\omega
ω
at point
G
G
G
. If the area of triangle
B
E
G
BEG
BEG
can be expressed as
a
b
c
\frac{a\sqrt{b}}{c}
c
a
b
for positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
with
b
b
b
squarefree 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
. p6. Suppose that
x
x
x
and
y
y
y
are positive real numbers such that
log
2
x
=
log
x
y
=
log
y
256
\log_2 x = \log_x y = \log_y 256
lo
g
2
x
=
lo
g
x
y
=
lo
g
y
256
. Find
x
y
xy
x
y
. p7. Call a positive three digit integer
A
B
C
‾
\overline{ABC}
A
BC
fancy if
A
B
C
‾
=
(
A
B
‾
)
2
−
11
⋅
C
‾
\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}
A
BC
=
(
A
B
)
2
−
11
⋅
C
. Find the sum of all fancy integers. p8. Let
△
A
B
C
\vartriangle ABC
△
A
BC
be an equilateral triangle. Isosceles triangles
△
D
B
C
\vartriangle DBC
△
D
BC
,
△
E
C
A
\vartriangle ECA
△
EC
A
, and
△
F
A
B
\vartriangle FAB
△
F
A
B
, not overlapping
△
A
B
C
\vartriangle ABC
△
A
BC
, are constructed such that each has area seven times the area of
△
A
B
C
\vartriangle ABC
△
A
BC
. Compute the ratio of the area of
△
D
E
F
\vartriangle DEF
△
D
EF
to the area of
△
A
B
C
\vartriangle ABC
△
A
BC
. p9. Consider the sequence of polynomials an(x) with
a
0
(
x
)
=
0
a_0(x) = 0
a
0
(
x
)
=
0
,
a
1
(
x
)
=
1
a_1(x) = 1
a
1
(
x
)
=
1
, and
a
n
(
x
)
=
a
n
−
1
(
x
)
+
x
a
n
−
2
(
x
)
a_n(x) = a_{n-1}(x) + xa_{n-2}(x)
a
n
(
x
)
=
a
n
−
1
(
x
)
+
x
a
n
−
2
(
x
)
for all
n
≥
2
n \ge 2
n
≥
2
. Suppose that
p
k
=
a
k
(
−
1
)
⋅
a
k
(
1
)
p_k = a_k(-1) \cdot a_k(1)
p
k
=
a
k
(
−
1
)
⋅
a
k
(
1
)
for all nonnegative integers
k
k
k
. Find the number of positive integers
k
k
k
between
10
10
10
and
50
50
50
, inclusive, such that
p
k
−
2
+
p
k
−
1
=
p
k
+
1
−
p
k
+
2
p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}
p
k
−
2
+
p
k
−
1
=
p
k
+
1
−
p
k
+
2
. p10. In triangle
A
B
C
ABC
A
BC
, point
D
D
D
and
E
E
E
are on line segments
B
C
BC
BC
and
A
C
AC
A
C
, respectively, such that
A
D
AD
A
D
and
B
E
BE
BE
intersect at
H
H
H
. Suppose that
A
C
=
12
AC = 12
A
C
=
12
,
B
C
=
30
BC = 30
BC
=
30
, and
E
C
=
6
EC = 6
EC
=
6
. Triangle BEC has area 45 and triangle
A
D
C
ADC
A
D
C
has area
72
72
72
, and lines CH and AB meet at F. If
B
F
2
BF^2
B
F
2
can be expressed as
a
−
b
c
d
\frac{a-b\sqrt{c}}{d}
d
a
−
b
c
for positive integers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
with c squarefree and
g
c
d
(
a
,
b
,
d
)
=
1
gcd(a, b, d) = 1
g
c
d
(
a
,
b
,
d
)
=
1
, then find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
. p11. Find the minimum possible integer
y
y
y
such that
y
>
100
y > 100
y
>
100
and there exists a positive integer x such that
x
2
+
18
x
+
y
x^2 + 18x + y
x
2
+
18
x
+
y
is a perfect fourth power. p12. Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral such that
A
B
=
2
AB = 2
A
B
=
2
,
C
D
=
4
CD = 4
C
D
=
4
,
B
C
=
A
D
BC = AD
BC
=
A
D
, and
∠
A
D
C
+
∠
B
C
D
=
12
0
o
\angle ADC + \angle BCD = 120^o
∠
A
D
C
+
∠
BC
D
=
12
0
o
. If the sum of the maximum and minimum possible areas of quadrilateral
A
B
C
D
ABCD
A
BC
D
can be expressed as
a
b
a\sqrt{b}
a
b
for positive integers
a
a
a
,
b
b
b
with
b
b
b
squarefree, then find
a
+
b
a + b
a
+
b
. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Tiebreaker
1
Hide problems
Girls in Math at Yale 2022 Tiebreaker Round
p1. Suppose that
x
x
x
and
y
y
y
are positive real numbers such that
log
2
x
=
log
x
y
=
log
y
256
\log_2 x = \log_x y = \log_y 256
lo
g
2
x
=
lo
g
x
y
=
lo
g
y
256
. Find
x
y
xy
x
y
. p2. Let the roots of
x
2
+
7
x
+
11
x^2 + 7x + 11
x
2
+
7
x
+
11
be
r
r
r
and
s
s
s
. If f(x) is the monic polynomial with roots
r
s
+
r
+
s
rs + r + s
rs
+
r
+
s
and
r
2
+
s
2
r^2 + s^2
r
2
+
s
2
, what is
f
(
3
)
f(3)
f
(
3
)
? p3. Call a positive three digit integer
A
B
C
‾
\overline{ABC}
A
BC
fancy if
A
B
C
‾
=
(
A
B
‾
)
2
−
11
⋅
C
‾
\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}
A
BC
=
(
A
B
)
2
−
11
⋅
C
. Find the sum of all fancy integers. p4. In triangle
A
B
C
ABC
A
BC
, points
D
D
D
and
E
E
E
are on line segments
B
C
BC
BC
and
A
C
AC
A
C
, respectively, such that
A
D
AD
A
D
and
B
E
BE
BE
intersect at
H
H
H
. Suppose that
A
C
=
12
AC = 12
A
C
=
12
,
B
C
=
30
BC = 30
BC
=
30
, and
E
C
=
6
EC = 6
EC
=
6
. Triangle
B
E
C
BEC
BEC
has area
45
45
45
and triangle
A
D
C
ADC
A
D
C
has area
72
72
72
, and lines
C
H
CH
C
H
and
A
B
AB
A
B
meet at
F
F
F
. If
B
F
2
BF^2
B
F
2
can be expressed as
a
−
b
c
d
\frac{a-b\sqrt{c}}{d}
d
a
−
b
c
for positive integers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
with
c
c
c
squarefree and
g
c
d
(
a
,
b
,
d
)
=
1
gcd(a, b, d) = 1
g
c
d
(
a
,
b
,
d
)
=
1
, then find
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
. p5. Find the minimum possible integer
y
y
y
such that
y
>
100
y > 100
y
>
100
and there exists a positive integer
x
x
x
such that
x
2
+
18
x
+
y
x^2 + 18x + y
x
2
+
18
x
+
y
is a perfect fourth power. p6. Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral such that
A
B
=
2
AB = 2
A
B
=
2
,
C
D
=
4
CD = 4
C
D
=
4
,
B
C
=
A
D
BC = AD
BC
=
A
D
, and
∠
A
D
C
+
∠
B
C
D
=
12
0
o
\angle ADC + \angle BCD = 120^o
∠
A
D
C
+
∠
BC
D
=
12
0
o
. If the sum of the maximum and minimum possible areas of quadrilateral
A
B
C
D
ABCD
A
BC
D
can be expressed as
a
b
a\sqrt{b}
a
b
for positive integers
a
,
b
a, b
a
,
b
with
b
b
b
squarefree, then find
a
+
b
a + b
a
+
b
. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
R6
1
Hide problems
Girls in Math at Yale 2022 Mathathon Round 6
p16 Madelyn is being paid
$
50
\$50
$50
/hour to find useful Non-Functional Trios, where a Non-Functional Trio is defined as an ordered triple of distinct real numbers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, and a Non- Functional Trio is useful if
(
a
,
b
)
(a, b)
(
a
,
b
)
,
(
b
,
c
)
(b, c)
(
b
,
c
)
, and
(
c
,
a
)
(c, a)
(
c
,
a
)
are collinear in the Cartesian plane. Currently, she’s working on the case
a
+
b
+
c
=
2022
a+b+c = 2022
a
+
b
+
c
=
2022
. Find the number of useful Non-Functional Trios
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that
a
+
b
+
c
=
2022
a + b + c = 2022
a
+
b
+
c
=
2022
. p17 Let
p
(
x
)
=
x
2
−
k
p(x) = x^2 - k
p
(
x
)
=
x
2
−
k
, where
k
k
k
is an integer strictly less than
250
250
250
. Find the largest possible value of k such that there exist distinct integers
a
,
b
a, b
a
,
b
with
p
(
a
)
=
b
p(a) = b
p
(
a
)
=
b
and
p
(
b
)
=
a
p(b) = a
p
(
b
)
=
a
. p18 Let
A
B
C
ABC
A
BC
be a triangle with orthocenter
H
H
H
and circumcircle
Γ
\Gamma
Γ
such that
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
14
BC = 14
BC
=
14
, and
C
A
=
15
CA = 15
C
A
=
15
.
B
H
BH
B
H
and
C
H
CH
C
H
meet
Γ
\Gamma
Γ
again at points
D
D
D
and
E
E
E
, respectively, and
D
E
DE
D
E
meets
A
B
AB
A
B
and
A
C
AC
A
C
at
F
F
F
and
G
G
G
, respectively. The circumcircles of triangles
A
B
G
ABG
A
BG
and
A
C
F
ACF
A
CF
meet BC again at points
P
P
P
and
Q
Q
Q
. If
P
Q
PQ
PQ
can be expressed as
a
b
\frac{a}{b}
b
a
for positive integers
a
,
b
a, b
a
,
b
with
g
c
d
(
a
,
b
)
=
1
gcd (a, b) = 1
g
c
d
(
a
,
b
)
=
1
, find
a
+
b
a + b
a
+
b
.
R5
1
Hide problems
Girls in Math at Yale 2022 Mathathon Round 5
p13 Let
A
B
C
D
ABCD
A
BC
D
be a square. Points
E
E
E
and
F
F
F
lie outside of
A
B
C
D
ABCD
A
BC
D
such that
A
B
E
ABE
A
BE
and
C
B
F
CBF
CBF
are equilateral triangles. If
G
G
G
is the centroid of triangle
D
E
F
DEF
D
EF
, then find
∠
A
G
C
\angle AGC
∠
A
GC
, in degrees. p14 The silent reading
s
(
n
)
s(n)
s
(
n
)
of a positive integer
n
n
n
is the number obtained by dropping the zeros not at the end of the number. For example,
s
(
1070030
)
=
1730
s(1070030) = 1730
s
(
1070030
)
=
1730
. Find the largest
n
<
10000
n < 10000
n
<
10000
such that
s
(
n
)
s(n)
s
(
n
)
divides
n
n
n
and
n
≠
s
(
n
)
n\ne s(n)
n
=
s
(
n
)
. p15 Let
A
B
C
D
E
F
G
H
ABCDEFGH
A
BC
D
EFG
H
be a regular octagon with side length
12
12
12
. There exists a region
R
R
R
inside the octagon such that for each point
X
X
X
in
R
R
R
, exactly three of the rays
A
X
AX
A
X
,
B
X
BX
BX
,
C
X
CX
CX
,
D
X
DX
D
X
,
G
X
GX
GX
, and
H
X
HX
H
X
intersect segment
E
F
EF
EF
. If the area of region
R
R
R
can be expressed as
a
−
b
c
a -b\sqrt{c}
a
−
b
c
for positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
with
c
c
c
squarefree, find
a
+
b
+
c
a + b + c
a
+
b
+
c
.
R4
1
Hide problems
Girls in Math at Yale 2022 Mathathon Round 4
p10 Kathy has two positive real numbers,
a
a
a
and
b
b
b
. She mistakenly writes
log
(
a
+
b
)
=
log
(
a
)
+
log
(
b
)
,
\log (a + b) = \log (a) + \log( b),
lo
g
(
a
+
b
)
=
lo
g
(
a
)
+
lo
g
(
b
)
,
but miraculously, she finds that for her combination of
a
a
a
and
b
b
b
, the equality holds. If
a
=
2022
b
a = 2022b
a
=
2022
b
, then
b
=
p
q
b = \frac{p}{q}
b
=
q
p
, for positive integers
p
,
q
p, q
p
,
q
where
g
c
d
(
p
,
q
)
=
1
gcd(p, q) = 1
g
c
d
(
p
,
q
)
=
1
. Find
p
+
q
p + q
p
+
q
. p11 Points
X
X
X
and
Y
Y
Y
lie on sides
A
B
AB
A
B
and
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
, respectively. Ray
X
Y
→
\overrightarrow{XY}
X
Y
is extended to point
Z
Z
Z
such that
A
,
C
A, C
A
,
C
, and
Z
Z
Z
are collinear, in that order. If triangle
A
B
Z
ABZ
A
BZ
is isosceles and triangle
C
Y
Z
CYZ
C
Y
Z
is equilateral, then the possible values of
∠
Z
X
B
\angle ZXB
∠
ZXB
lie in the interval
I
=
(
a
o
,
b
o
)
I = (a^o, b^o)
I
=
(
a
o
,
b
o
)
, such that
0
≤
a
,
b
≤
360
0 \le a, b \le 360
0
≤
a
,
b
≤
360
and such that
a
a
a
is as large as possible and
b
b
b
is as small as possible. Find
a
+
b
a + b
a
+
b
. p12 Let
S
=
{
(
a
,
b
)
∣
0
≤
a
,
b
≤
3
,
a
S = \{(a, b) | 0 \le a, b \le 3, a
S
=
{(
a
,
b
)
∣0
≤
a
,
b
≤
3
,
a
and
b
b
b
are integers
}
\}
}
. In other words,
S
S
S
is the set of points in the coordinate plane with integer coordinates between
0
0
0
and
3
3
3
, inclusive. Prair selects four distinct points in
S
S
S
, for each selected point, she draws lines with slope
1
1
1
and slope
−
1
-1
−
1
passing through that point. Given that each point in
S
S
S
lies on at least one line Prair drew, how many ways could she have selected those four points?
R3
1
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Girls in Math at Yale 2022 Mathathon Round 3
p7 Cindy cuts regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
out of a sheet of paper. She folds
B
B
B
over
A
C
AC
A
C
, resulting in a pentagon. Then, she folds
A
A
A
over
C
F
CF
CF
, resulting in a quadrilateral. The area of
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is
k
k
k
times the area of the resulting folded shape. Find
k
k
k
. p8 Call a sequence
{
a
n
}
=
a
1
,
a
2
,
a
3
,
.
.
.
\{a_n\} = a_1, a_2, a_3, . . .
{
a
n
}
=
a
1
,
a
2
,
a
3
,
...
of positive integers Fib-o’nacci if it satisfies
a
n
=
a
n
−
1
+
a
n
−
2
a_n = a_{n-1}+a_{n-2}
a
n
=
a
n
−
1
+
a
n
−
2
for all
n
≥
3
n \ge 3
n
≥
3
. Suppose that
m
m
m
is the largest even positive integer such that exactly one Fib-o’nacci sequence satisfies
a
5
=
m
a_5 = m
a
5
=
m
, and suppose that
n
n
n
is the largest odd positive integer such that exactly one Fib-o’nacci sequence satisfies
a
5
=
n
a_5 = n
a
5
=
n
. Find
m
n
mn
mn
. p9 Compute the number of ways there are to pick three non-empty subsets
A
A
A
,
B
B
B
, and
C
C
C
of
{
1
,
2
,
3
,
4
,
5
,
6
}
\{1, 2, 3, 4, 5, 6\}
{
1
,
2
,
3
,
4
,
5
,
6
}
, such that
∣
A
∣
=
∣
B
∣
=
∣
C
∣
|A| = |B| = |C|
∣
A
∣
=
∣
B
∣
=
∣
C
∣
and the following property holds:
A
∩
B
∩
C
=
A
∩
B
=
B
∩
C
=
C
∩
A
.
A \cap B \cap C = A \cap B = B \cap C = C \cap A.
A
∩
B
∩
C
=
A
∩
B
=
B
∩
C
=
C
∩
A
.
R2
1
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Girls in Math at Yale 2022 Mathathon Round 2
p4 Define the sequence
a
n
{a_n}
a
n
as follows: 1)
a
1
=
−
1
a_1 = -1
a
1
=
−
1
, and 2) for all
n
≥
2
n \ge 2
n
≥
2
,
a
n
=
1
+
2
+
.
.
.
+
n
−
(
n
+
1
)
a_n = 1 + 2 + . . . + n - (n + 1)
a
n
=
1
+
2
+
...
+
n
−
(
n
+
1
)
. For example,
a
3
=
1
+
2
+
3
−
4
=
2
a_3 = 1+2+3-4 = 2
a
3
=
1
+
2
+
3
−
4
=
2
. Find the largest possible value of
k
k
k
such that
a
k
+
a
k
+
1
=
a
k
+
2
a_k+a_{k+1} = a_{k+2}
a
k
+
a
k
+
1
=
a
k
+
2
. p5 The taxicab distance between two points
(
a
,
b
)
(a, b)
(
a
,
b
)
and
(
c
,
d
)
(c, d)
(
c
,
d
)
on the coordinate plane is
∣
c
−
a
∣
+
∣
d
−
b
∣
|c-a|+|d-b|
∣
c
−
a
∣
+
∣
d
−
b
∣
. Given that the taxicab distance between points
A
A
A
and
B
B
B
is
8
8
8
and that the length of
A
B
AB
A
B
is
k
k
k
, find the minimum possible value of
k
2
k^2
k
2
. p6 For any two-digit positive integer
A
B
‾
\overline{AB}
A
B
, let
f
(
A
B
‾
)
=
A
B
‾
−
A
⋅
B
f(\overline{AB}) = \overline{AB}-A\cdot B
f
(
A
B
)
=
A
B
−
A
⋅
B
, or in other words, the result of subtracting the product of its digits from the integer itself. For example,
f
(
72
‾
)
=
72
−
7
⋅
2
=
58
f(\overline{72}) = 72-7\cdot 2 = 58
f
(
72
)
=
72
−
7
⋅
2
=
58
. Find the maximum possible
n
n
n
such that there exist distinct two-digit integers
X
Y
‾
\overline{XY}
X
Y
and
W
Z
‾
\overline{WZ}
W
Z
such that
f
(
X
Y
‾
)
=
f
(
W
Z
‾
)
=
n
f(\overline{XY} ) = f(\overline{WZ}) = n
f
(
X
Y
)
=
f
(
W
Z
)
=
n
.
R1
1
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Girls in Math at Yale 2022 Mathathon Round 1
p1 How many two-digit positive integers with distinct digits satisfy the conditions that 1) neither digit is
0
0
0
, and 2) the units digit is a multiple of the tens digit? p2 Mirabel has
47
47
47
candies to pass out to a class with
n
n
n
students, where
10
≤
n
<
20
10\le n < 20
10
≤
n
<
20
. After distributing the candy as evenly as possible, she has some candies left over. Find the smallest integer
k
k
k
such that Mirabel could have had
k
k
k
leftover candies. p3 Callie picks two distinct numbers from
{
1
,
2
,
3
,
4
,
5
}
\{1, 2, 3, 4, 5\}
{
1
,
2
,
3
,
4
,
5
}
at random. The probability that the sum of the numbers she picked is greater than the sum of the numbers she didn’t pick is
p
p
p
.
p
p
p
can be expressed as
a
b
\frac{a}{b}
b
a
for positive integers
a
,
b
a, b
a
,
b
with
g
c
d
(
a
,
b
)
=
1
gcd (a, b) = 1
g
c
d
(
a
,
b
)
=
1
. Find
a
+
b
a + b
a
+
b
.
12
1
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Girls in Math at Yale 2022 Problem 12: Computationalized Oly Geo part n
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
5
AB = 5
A
B
=
5
,
B
C
=
7
BC = 7
BC
=
7
, and
C
A
=
8
CA = 8
C
A
=
8
, and let
D
D
D
be a point on arc
B
C
^
\widehat{BC}
BC
of its circumcircle
Ω
\Omega
Ω
. Suppose that the angle bisectors of
∠
A
D
B
\angle ADB
∠
A
D
B
and
∠
A
D
C
\angle ADC
∠
A
D
C
meet
A
B
AB
A
B
and
A
C
AC
A
C
at
E
E
E
and
F
F
F
, respectively, and that
E
F
EF
EF
and
B
C
BC
BC
meet at
G
G
G
. Line
G
D
GD
G
D
meets
Ω
\Omega
Ω
at
T
T
T
. If the maximum possible value of
A
T
2
AT^2
A
T
2
can be expressed as
a
b
\frac{a}{b}
b
a
for positive integers
a
,
b
a, b
a
,
b
with
gcd
(
a
,
b
)
=
1
\gcd (a,b) = 1
g
cd
(
a
,
b
)
=
1
, find
a
+
b
a + b
a
+
b
.Proposed by Andrew Wu
11
1
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Girls in Math at Yale 2022 Problem 11: Literally 1984
Georgina calls a
992
992
992
-element subset
A
A
A
of the set
S
=
{
1
,
2
,
3
,
…
,
1984
}
S = \{1, 2, 3, \ldots , 1984\}
S
=
{
1
,
2
,
3
,
…
,
1984
}
a halfthink set if [*] the sum of the elements in
A
A
A
is equal to half of the sum of the elements in
S
S
S
, and [*] exactly one pair of elements in
A
A
A
differs by
1
1
1
.She notices that for some values of
n
n
n
, with
n
n
n
a positive integer between
1
1
1
and
1983
1983
1983
, inclusive, there are no halfthink sets containing both
n
n
n
and
n
+
1
n+1
n
+
1
. Find the last three digits of the product of all possible values of
n
n
n
.Proposed by Andrew Wu and Jason Wang(Note: wording changed from original to specify what
n
n
n
can be.)
10
1
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Girls in Math at Yale 2022 Problem 10: Dividing 15^6
How many ways are there to choose distinct positive integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
dividing
1
5
6
15^6
1
5
6
such that none of
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
or
d
d
d
divide each other? (Order does not matter.)Proposed by Miles Yamner and Andrew Wu(Note: wording changed from original to clarify)
9
1
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Girls in Math at Yale 2022 Problem 9: Monic Quadratics Strike Again
Suppose that
P
(
x
)
P(x)
P
(
x
)
is a monic quadratic polynomial satisfying
a
P
(
a
)
=
20
P
(
20
)
=
22
P
(
22
)
aP(a) = 20P(20) = 22P(22)
a
P
(
a
)
=
20
P
(
20
)
=
22
P
(
22
)
for some integer
a
≠
20
,
22
a\neq 20, 22
a
=
20
,
22
. Find the minimum possible positive value of
P
(
0
)
P(0)
P
(
0
)
.Proposed by Andrew Wu(Note: wording changed from original to specify that
a
≠
20
,
22
a \neq 20, 22
a
=
20
,
22
.)
8
1
Hide problems
Girls in Math at Yale 2022 Problem 8: Maximize [DEF]^2
Triangle
A
B
C
ABC
A
BC
has sidelengths
A
B
=
1
AB=1
A
B
=
1
,
B
C
=
3
BC=\sqrt{3}
BC
=
3
, and
A
C
=
2
AC=2
A
C
=
2
. Points
D
,
E
D,E
D
,
E
, and
F
F
F
are chosen on
A
B
,
B
C
AB, BC
A
B
,
BC
, and
A
C
AC
A
C
respectively, such that
∠
E
D
F
=
∠
D
F
A
=
9
0
∘
\angle EDF = \angle DFA = 90^{\circ}
∠
E
D
F
=
∠
D
F
A
=
9
0
∘
. Given that the maximum possible value of
[
D
E
F
]
2
[DEF]^2
[
D
EF
]
2
can be expressed as
a
b
\frac{a}{b}
b
a
for positive integers
a
,
b
a, b
a
,
b
with
gcd
(
a
,
b
)
=
1
\gcd (a, b) = 1
g
cd
(
a
,
b
)
=
1
, find
a
+
b
a + b
a
+
b
. (Here
[
D
E
F
]
[DEF]
[
D
EF
]
denotes the area of triangle
D
E
F
DEF
D
EF
.) Proposed by Vismay Sharan
7
1
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Girls in Math at Yale 2022 Problem 7: Divisible by 99
Given that six-digit positive integer
A
B
C
D
E
F
‾
\overline{ABCDEF}
A
BC
D
EF
has distinct digits
A
,
A,
A
,
B
,
B,
B
,
C
,
C,
C
,
D
,
D,
D
,
E
,
E,
E
,
F
F
F
between
1
1
1
and
8
8
8
, inclusive, and that it is divisible by
99
99
99
, find the maximum possible value of
A
B
C
D
E
F
‾
\overline{ABCDEF}
A
BC
D
EF
.Proposed by Andrew Milas
6
1
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Girls in Math at Yale 2022 Problem 6: in which Carissa doesn't get hit by a car
Carissa is crossing a very, very, very wide street, and did not properly check both ways before doing so. (Don't be like Carissa!) She initially begins walking at
2
2
2
feet per second. Suddenly, she hears a car approaching, and begins running, eventually making it safely to the other side, half a minute after she began crossing. Given that Carissa always runs
n
n
n
times as fast as she walks and that she spent
n
n
n
times as much time running as she did walking, and given that the street is
260
260
260
feet wide, find Carissa's running speed, in feet per second.Proposed by Andrew Wu
5
1
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Girls in Math at Yale 2022 Problem 5: Cat & Claire
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive integer with distinct nonzero digits,
A
B
‾
\overline{AB}
A
B
, such that
A
A
A
and
B
B
B
are both factors of
A
B
‾
\overline{AB}
A
B
." Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"Cat says, "That's cool, and my favorite number is among those four numbers! Also, the square of my number is the product of two of the other numbers among the four you mentioned!" Claire says, "Now I know your favorite number!" What is Cat's favorite number?Proposed by Andrew Wu
4
1
Hide problems
Girls in Math at Yale 2022 Problem 4: Kara Do Be Rollin' Dice
Kara rolls a six-sided die, and if on that first roll she rolls an
n
n
n
, she rolls the die
n
−
1
n-1
n
−
1
more times. She then computes that the product of all her rolls, including the first, is
8
8
8
. How many distinct sequences of rolls could Kara have rolled?Proposed by Andrew Wu
3
1
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Girls in Math at Yale 2022 Problem 3: r/collaptzse
The Collaptz function is defined as
C
(
n
)
=
{
3
n
−
1
n
odd
,
n
2
n
even
.
C(n) = \begin{cases} 3n - 1 & n\textrm{~odd}, \\ \frac{n}{2} & n\textrm{~even}.\end{cases}
C
(
n
)
=
{
3
n
−
1
2
n
n
odd
,
n
even
.
We obtain the Collaptz sequence of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of
13
13
13
begins with
13
,
38
,
19
,
56
,
28
,
⋯
13, 38, 19, 56, 28, \cdots
13
,
38
,
19
,
56
,
28
,
⋯
and so on. Find the sum of the three smallest positive integers
n
n
n
whose Collaptz sequences do not contain
1
,
1,
1
,
or in other words, do not collaptzse.Proposed by Andrew Wu and Jason Wang
2
1
Hide problems
Girls in Math at Yale 2022 Problem 2: Filling In Grid
How many ways are there to fill in a
2
×
2
2\times 2
2
×
2
square grid with the numbers
1
,
2
,
3
,
1,2,3,
1
,
2
,
3
,
and
4
4
4
such that the numbers in any two grid squares that share an edge have an absolute difference of at most
2
2
2
?Proposed by Andrew Wu
1
1
Hide problems
GIrls in Math at Yale 2022 Problem 1: Primle
Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between
10
10
10
and
99
99
99
, called the Primle. For each guess, a digit is highlighted blue if it is in the Primle, but not in the correct place. A digit is highlighted orange if it is in the Primle and is in the correct place. Finally, a digit is left unhighlighted if it is not in the Primle. If Charlotte guesses
13
13
13
and
47
47
47
and is left with the following game board, what is the Primle? \begin{array}{c} \boxed{1} \,\, \boxed{3} \$$\smallskipamount] \boxed{4}\,\, \fcolorbox{black}{blue}{\color{white}7} \end{array}Proposed by Andrew Wu and Jason Wang