MathDB

1

2021 Girls in Math at Yale - Mixer Round

(a, b, c)

satisfying

\bullet

a, b, c

are are single-digit positive integers, and

\bullet

a \cdot b + c = a + b \cdot c

.
p2. In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form an increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to

94

. How many possible combinations of test scores could they have had? (Test scores at Greendale range between

0

and

100

, inclusive.)
p3. Suppose that

a + \frac{1}{b} = 2

and

b +\frac{1}{a} = 3

. If

\frac{a}{b} + \frac{b}{a}

can be expressed as

\frac{p}{q}

in simplest terms, find

p + q

.
p4. Suppose that

A

and

B

are digits between

1

and

9

such that

0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B_1B_1B_1...}) + 1

Find the sum of all possible values of

10A + B

.
p5. Let

ABC

be an isosceles right triangle with

m\angle ABC = 90^o

. Let

D

and

E

lie on segments

AC

and

BC

, respectively, such that triangles

\vartriangle ADB

and

\vartriangle CDE

are similar and

DE = EB

. If

\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}

with

a, b

positive integers and a squarefree, then find

a + b

.
p6. Five bowling pins

P_1

,

P_2

,...,

P_5

are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is a b where a and b are relatively prime, find

a + b

. (Pins

P_i

and

P_j

are adjacent if and only if

|i -j| = 1

.)
p7. Let triangle

ABC

have side lengths

AB = 10

,

BC = 24

, and

AC = 26

. Let

I

be the incenter of

ABC

. If the maximum possible distance between

I

and a point on the circumcircle of

ABC

can be expressed as

a +\sqrt{b}

for integers

a

and

b

with

b

squarefree, find

a + b

. (The incenter of any triangle

XY Z

is the intersection of the angle bisectors of

\angle Y XZ

,

\angle XZY

, and

\angle ZY X

.)
p8. How many terms in the expansion of

(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})

have coefficients equal to

1011

?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Tiebreaker

1

Girls in Math at Yale 2021 Tiebreaker Round

p1. In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form a strictly increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to

94

. How many possible combinations of test scores could they have had? (Test scores at Greendale range between

0

and

100

, inclusive.)
p2. Suppose that

A

and

B

are digits between

1

and

9

such that

0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B1B1B1...}) + 1

Find the sum of all possible values of

10A + B

.
p3. Let

ABC

be an isosceles right triangle with

m\angle ABC = 90^o

. Let

D

and

E

lie on segments

\overline{AC}

and

\overline{BC}

, respectively, such that triangles

\vartriangle ADB

and

\vartriangle CDE

are similar and

DE =EB

. If

\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}

with

a

,

b

positive integers and

a

squarefree, then find

a + b

.
p4. Five bowling pins

P_1, P_2, ..., P_5

are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is

\frac{a}{b}

where

a

and

b

are relatively prime, find

a + b

. (Pins

P_i

and

P_j

are adjacent if and only if

|i - j| = 1

.)
p5. How many terms in the expansion of

(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})

have coeffcients equal to

1011

?
p6. Suppose

f(x)

is a monic quadratic polynomial with distinct nonzero roots

p

and

q

, and suppose

g(x)

is a monic quadratic polynomial with roots

p + \frac{1}{q}

and

q + \frac{1}{p}

. If we are given that

g(-1) = 1

and

f(0)\ne -1

, then there exists some real number

r

that must be a root of

f(x)

. Find

r

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

R6

1

GiM at Yale 2021 Mathathon R6: oops no flavortext

16. Suppose trapezoid

JANE

is inscribed in a circle of radius

25

such that the center of the circle lies inside the trapezoid. If the two bases of

JANE

have side lengths

14

and

30

and the average of the lengths of the two legs is

\sqrt{m}

, what is

m

?
17. What is the radius of the circle tangent to the

x

-axis, the line

y=\sqrt{3}x

, and the circle

(x-10\sqrt{3})^2+(y-10)^2=25

?
18. Find the smallest positive integer

n

such that

3n^3-9n^2+5n-15

is divisible by

121

but not

2

.

R5

1

GiM at Yale 2021 Mathathon R5: INTO THE UNKNOWNNNNNNNNNNNNNNNNN

13. The triangle with vertices

(0,0), (a,b)

, and

(a,-b)

has area

10

. Find the sum of all possible positive integer values of

a

, given that

b

is a positive integer.
14. Elsa is venturing into the unknown. She stands on

(0,0)

in the coordinate plane, and each second, she moves to one of the four lattice points nearest her, chosen at random and with equal probability. If she ever moves to a lattice point she has stood on before, she has ventured back into the known, and thus stops venturing into the unknown from then on. After four seconds have passed, the probability that Elsa is still venturing into the unknown can be expressed as

\frac{a}{b}

in simplest terms. Find

a+b

.
(A lattice point is a point with integer coordinates.)
15. Let

ABCD

be a square with side length

4

. Points

X, Y,

and

Z

, distinct from points

A, B, C,

and

D

, are selected on sides

AD, AB,

and

CD

, respectively, such that

XY = 3, XZ = 4

, and

\angle YXZ = 90^{\circ}

. If

AX = \frac{a}{b}

in simplest terms, then find

a + b

.

R4

1

GiM at Yale 2021 Mathathon R4: oops all combo/when does the game stop?!

10. Prair picks a three-digit palindrome

n

at random. If the probability that

2n

is also a palindrome can be expressed as

\frac{p}{q}

in simplest terms, find

p + q

. (A palindrome is a number that reads the same forwards as backwards; for example,

161

and

2992

are palindromes, but

342

is not.)
11. If two distinct integers are picked randomly between

1

and

50

inclusive, the probability that their sum is divisible by

7

can be expressed as

\frac{p}{q}

, where

p

and

q

are relatively prime positive integers. Find

p+q

.
12. Ali is playing a game involving rolling standard, fair six-sided dice. She calls two consecutive die rolls such that the first is less than the second a "rocket." If, however, she ever rolls two consecutive die rolls such that the second is less than the first, the game stops. If the probability that Ali gets five rockets is

\frac{p}{q}

, where

p

and

q

are relatively prime positive integers, find

p+q

.

R3

1

GiM at Yale 2021 Mathathon R3: 25exists/originallykara/consecutives!

7. Peggy picks three positive integers between

1

and

25

, inclusive, and tells us the following information about those numbers:
[*] Exactly one of them is a multiple of

2

; [*] Exactly one of them is a multiple of

3

; [*] Exactly one of them is a multiple of

5

; [*] Exactly one of them is a multiple of

7

; [*] Exactly one of them is a multiple of

11

.
What is the maximum possible sum of the integers that Peggy picked?
8. What is the largest positive integer

k

such that

2^k

divides

2^{4^8}+8^{2^4}+4^{8^2}

?
9. Find the smallest integer

n

such that

n

is the sum of

7

consecutive positive integers and the sum of

12

consecutive positive integers.

R2

1

GiM at Yale 2021 Mathathon R2: we only know "max" and "min"

4. Suppose that

\overline{A2021B}

is a six-digit integer divisible by

9

. Find the maximum possible value of

A \cdot B

.
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose that

a

and

b

are positive integers such that

\frac{a}{b-20}

and

\frac{b+21}{a}

are positive integers. Find the maximum possible value of

a + b

.

R1

1

GiM at Yale 2021 Mathathon R1: sillyalg/hextriangles/sirenaprimes

1. If

5x+3y-z=4

,

x=y

, and

z=4

, find

x+y+z

.
2. How many ways are there to pick three distinct vertices of a regular hexagon such that the triangle with those three points as its vertices shares exactly one side with the hexagon?
3. Sirena picks five distinct positive primes,

p_1 < p_2 < p_3 < p_4 < p_5

, and finds that they sum to

192

. If the product

p_1p_2p_3p_4p_5

is as large as possible, what is

p_1 - p_2 + p_3 - p_4 + p_5

?

12

1

Girls in Math at Yale 2021 Problem 12: Internally Tangent Circles

Let

\Gamma_1

and

\Gamma_2

be externally tangent circles with radii lengths

2

and

6

, respectively, and suppose that they are tangent to and lie on the same side of line

\ell

. Points

A

and

B

are selected on

\ell

such that

\Gamma_1

and

\Gamma_2

are internally tangent to the circle with diameter

AB

. If

AB = a + b\sqrt{c}

for positive integers

a, b, c

with

c

squarefree, then find

a + b + c

.
Proposed by Andrew Wu, Deyuan Li, and Andrew Milas

11

1

Girls in Math at Yale 2021 Problem 11: Min Plus Max Volumes

A right rectangular prism has integer side lengths

a

,

b

, and

c

. If

\text{lcm}(a,b)=72

,

\text{lcm}(a,c)=24

, and

\text{lcm}(b,c)=18

, what is the sum of the minimum and maximum possible volumes of the prism?
Proposed by Deyuan Li and Andrew Milas

10

1

Girls in Math at Yale 2021 Problem 10: 20^21

Suppose that

a_1, a_2, a_3, \ldots

is an infinite geometric sequence such that for all

i \ge 1

,

a_i

is a positive integer. Suppose furthermore that

a_{20} + a_{21} = 20^{21}

. If the minimum possible value of

a_1

can be expressed as

2^a 5^b

for positive integers

a

and

b

, find

a + b

.
Proposed by Andrew Wu

9

1

Girls in Math at Yale 2021 Problem 9: Ali Can't Pronounce Stuff

Ali defines a pronunciation of any sequence of English letters to be a partition of those letters into substrings such that each substring contains at least one vowel. For example,

\text{A } \vert \text{ THEN } \vert \text{ A}

,

\text{ATH } \vert \text{ E } \vert \text{ NA}

,

\text{ATHENA}

, and

\text{AT } \vert \text{ HEN } \vert \text{ A}

are all pronunciations of the sequence

\text{ATHENA}

. How many distinct pronunciations does

\text{YALEMATHCOMP}

have? (Y is not a vowel.)
Proposed by Andrew Wu, with significant inspiration from ali cy

8

1

Girls in Math at Yale 2021 Problem 8: ABBA

Let

A

and

B

be digits between

0

and

9

, and suppose that the product of the two-digit numbers

\overline{AB}

and

\overline{BA}

is equal to

k

. Given that

k+1

is a multiple of

101

, find

k

.
Proposed by Andrew Wu

7

1

Girls in Math at Yale 2021 Problem 7: Circles!!!!!!!

Suppose two circles

\Omega_1

and

\Omega_2

with centers

O_1

and

O_2

have radii

3

and

4

, respectively. Suppose that points

A

and

B

lie on circles

\Omega_1

and

\Omega_2

, respectively, such that segments

AB

and

O_1O_2

intersect and that

AB

is tangent to

\Omega_1

and

\Omega_2

. If

O_1O_2=25

, find the area of quadrilateral

O_1AO_2B

. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(12cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -12.81977592804657, xmax = 32.13023014338037, ymin = -14.185056097058798, ymax = 12.56855801985179; /* image dimensions */
/* draw figures */ draw(circle((-3.4277328104418046,-1.4524996726688195), 3), linewidth(1.2)); draw(circle((21.572267189558197,-1.4524996726688195), 4), linewidth(1.2)); draw((-2.5877328104418034,1.4275003273311748)--(20.452267189558192,-5.2924996726687885), linewidth(1.2)); /* dots and labels */ dot((-3.4277328104418046,-1.4524996726688195),linewidth(3pt) + dotstyle); label("

O_1

", (-4.252707018231291,-1.545940604327141), N * labelscalefactor); dot((21.572267189558197,-1.4524996726688195),linewidth(3pt) + dotstyle); label("

O_2

", (21.704189347819636,-1.250863978037686), NE * labelscalefactor); dot((-2.5877328104418034,1.4275003273311748),linewidth(3pt) + dotstyle); label("

A

", (-2.3937351324858342,1.6999022848568643), NE * labelscalefactor); dot((20.452267189558192,-5.2924996726687885),linewidth(3pt) + dotstyle); label("

B

", (20.671421155806545,-4.9885012443707835), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]
Proposed by Deyuan Li and Andrew Milas

6

1

Girls in Math at Yale 2021 Problem 6: Kara Rolls Dice

Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
[*] She rolled a

6

exactly three times; [*] The product of her first three rolls is the same as the product of her last three rolls.
How many distinct sequences of six rolls could Kara have rolled?
Proposed by Andrew Wu

5

1

Girls in Math at Yale 2021 Problem 5: I Mislabeled Vertices Once

Let

ATHEM

be a convex pentagon with

AT = 14

,

TH = MA = 20

,

HE = EM = 15

, and

\angle THE = \angle EMA = 90^{\circ}

. Find the area of

ATHEM

.
Proposed by Andrew Wu

4

1

Girls in Math at Yale 2021 Problem 4: 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 that is the product of three distinct prime numbers!"
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! My favorite number is not among those four numbers, though."
Claire says, "Now I know your favorite number!"
What is Cat's favorite number?
Proposed by Andrew Wu and Andrew Milas

3

1

Girls in Math at Yale 2021 Problem 3: Boring Sequence

Suppose that

a_1 = 1,

a_2 = 2

, and for any

n \ge 3

,

a_n = a_1 + a_2 + \cdots + a_{n-1}

. Find

\frac{a_{2021}}{a_{2020}}

.
Proposed by Andrew Wu

2

1

Girls in Math at Yale 2021 Problem 2: berry good!

A box of strawberries, containing

12

strawberries total, costs

\$ 2

. A box of blueberries, containing

48

blueberries total, costs

\$ 3

. Suppose that for

\$ 12

, Sareen can either buy

m

strawberries total or

n

blueberries total. Find

n - m

.
Proposed by Andrew Wu

1

Girls in Math at Yale 2021 Problem 1: E X P O N E N T

Given that

2x + 7y = 3

, find

2^{6x + 21y - 4}

.
Proposed by Deyuan Li

2021 Girls in Math at Yale

Subcontests

2021 Girls in Math at Yale - Mixer Round

Girls in Math at Yale 2021 Tiebreaker Round

GiM at Yale 2021 Mathathon R6: oops no flavortext

GiM at Yale 2021 Mathathon R5: INTO THE UNKNOWNNNNNNNNNNNNNNNNN

GiM at Yale 2021 Mathathon R4: oops all combo/when does the game stop?!

GiM at Yale 2021 Mathathon R3: 25exists/originallykara/consecutives!

GiM at Yale 2021 Mathathon R2: we only know &quot;max&quot; and &quot;min&quot;

GiM at Yale 2021 Mathathon R1: sillyalg/hextriangles/sirenaprimes

Girls in Math at Yale 2021 Problem 12: Internally Tangent Circles

Girls in Math at Yale 2021 Problem 11: Min Plus Max Volumes

Girls in Math at Yale 2021 Problem 10: 20^21

Girls in Math at Yale 2021 Problem 9: Ali Can't Pronounce Stuff

Girls in Math at Yale 2021 Problem 8: ABBA

Girls in Math at Yale 2021 Problem 7: Circles!!!!!!!

Girls in Math at Yale 2021 Problem 6: Kara Rolls Dice

Girls in Math at Yale 2021 Problem 5: I Mislabeled Vertices Once

Girls in Math at Yale 2021 Problem 4: Cat &amp; Claire

Girls in Math at Yale 2021 Problem 3: Boring Sequence

Girls in Math at Yale 2021 Problem 2: berry good!

Girls in Math at Yale 2021 Problem 1: E X P O N E N T

GiM at Yale 2021 Mathathon R2: we only know "max" and "min"

Girls in Math at Yale 2021 Problem 4: Cat & Claire