2021 LMT Spring

Part of LMT

Subcontests

(47)

3

2021 LMT Spring Guts Round p13-p24- Lexington Mathematical Tournament

Round 5
p13. Pieck the Frog hops on Pascal’s Triangle, where she starts at the number

1

at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after

7

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

.
p14. Maisy chooses a random set

(x, y)

x^2 + y^2 -26x -10y \le 482.

The probability that

y>0

can be expressed as

\frac{A\pi -B\sqrt{C}}{D \pi}

A+B +C +D

.
[color=#f00]Due to the problem having a typo, all teams who inputted answers received points
p15.

6

points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments).
Round 6
p16. Find the number of

3

3

grids such that each square in the grid is colored white or black and no two black squares share an edge.
p17. Let

ABC

be a triangle with side lengths

AB = 20

BC = 25

AC = 15

D

be the point on BC such that

CD = 4

E

be the foot of the altitude from

A

BC

F

be the intersection of

AE

with the circle of radius

7

A

F

is outside of triangle

ABC

DF

can be expressed as

\sqrt{m}

m

is a positive integer. Find

m

.
p18. Bill and Frank were arrested under suspicion for committing a crime and face the classic Prisoner’s Dilemma. They are both given the choice whether to rat out the other and walk away, leaving their partner to face a

9

year prison sentence. Given that neither of them talk, they both face a

3

year sentence. If both of them talk, they both will serve a

6

year sentence. Both Bill and Frank talk or do not talk with the same probabilities. Given the probability that at least one of them talks is

\frac{11}{36}

, find the expected duration of Bill’s sentence in months.
Round 7
p19. Rectangle

ABCD

E

\overline{CD}

F

is the intersection of

\overline{AC}

\overline{BE}

. Given that the area of

\vartriangle AFB

175

and the area of

\vartriangle CFE

28

, find the area of

ADEF

.
p20. Real numbers

x, y

z

satisfy the system of equations

5x+ 13y -z = 100,

25x^2 +169y^2 -z2 +130x y= 16000,

80x +208y-2z = 2020.

Find the value of

x yz

.
[color=#f00]Due to the problem having infinitely many solutions, all teams who inputted answers received points.
p21. Bob is standing at the number

1

on the number line. If Bob is standing at the number

n

, he can move to

n +1

n +2

n +4

. In howmany different ways can he move to the number

10

?
Round 8
p22. A sequence

a_1,a_2,a_3, ...

of positive integers is defined such that

a_1 = 4

, and for each integer

k \ge 2

2(a_{k-1} +a_k +a_{k+1}) = a_ka_{k-1} +8.

a_6 = 488

a_2 +a_3 +a_4 +a_5

\overline{PQ}

is a diameter of circle

\omega

1

O

A

be a point such that

AP

\omega

\gamma

be a circle with diameter

AP

A'

AQ

hits the circle with diameter

AP

A''

AO

hits the circle with diameter

OP

A'A''

PQ

R

. Given that the value of the length

RA'

is is always less than

k

k

is minimized, find the greatest integer less than or equal to

1000k

.
p24. You have cards numbered

1,2,3, ... ,100

all in a line, in that order. You may swap any two adjacent cards at any time. Given that you make

{100 \choose 2}

total swaps, where you swap each distinct pair of cards exactly once, and do not do any swaps simultaneously, find the total number of distinct possible final orderings of the cards.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2021 LMT Spring Guts Round p1-p12 - Lexington Mathematical Tournament

Round 1
p1. How many ways are there to arrange the letters in the word

NEVERLAND

2

N

’s are adjacent and the two

E

’s are adjacent? Assume that letters that appear the same are not distinct.
p2. In rectangle

ABCD

E

F

AB

CD

, respectively such that

DE = EF = FB

\angle CDE = 45^o

AB + AD

AB

AD

are relatively prime positive integers.
p3. Maisy Airlines sees

n

takeoffs per day. Find the minimum value of

n

such that theremust exist two planes that take off within aminute of each other.
Round 2
p4. Nick is mixing two solutions. He has

100

mL of a solution that is

30\%

X

400

mL of a solution that is

10\%

X

. If he combines the two, what percent

X

is the final solution?
p5. Find the number of ordered pairs

(a,b)

a

b

are positive integers, such that

\frac{1}{a}+\frac{2}{b}=\frac{1}{12}.

25

balls are arranged in a

5

5

square. Four of the balls are randomly removed from the square. Given that the probability that the square can be rotated

180^o

and still maintain the same configuration can be expressed as

\frac{m}{n}

m

n

are relatively prime, find

m+n

.
Round 3
p7. Maisy the ant is on corner

A

13\times 13\times 13

box. She needs to get to the opposite corner called

B

. Maisy can only walk along the surface of the cube and takes the path that covers the least distance. Let

C

D

be the possible points where she turns on her path. Find

AC^2 + AD^2 +BC^2 +BD^2 - AB^2 -CD^2

.
p8. Maisyton has recently built

5

intersections. Some intersections will get a park and some of those that get a park will also get a chess school. Find how many different ways this can happen.
p9. Let

f (x) = 2x -1

. Find the value of

x

| f ( f ( f ( f ( f (x)))))-2020|

.
Round 4
p10. Triangle

ABC

is isosceles, with

AB = BC > AC

. Let the angle bisector of

\angle A

\overline{BC}

D

, and let the altitude from

A

\overline{BC}

E

\angle A = \angle C= x^o

, then the measure of

\angle DAE

can be expressed as

(ax -b)^o

, for some constants

a

b

ab

.
p11. Maisy randomly chooses

4

w

x

y

z

w, x, y, z \in \{1,2,3, ... ,2019,2020\}

. Given that the probability that

w^2 + x^2 + y^2 + z^2

is not divisible by

4

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

.
p12. Evaluate

-\log_4 \left(\log_2 \left(\sqrt{\sqrt{\sqrt{...\sqrt{16}}}} \right)\right),

where there are

100

square root signs.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2021 LMT Spring Guts Round p25-p36- Lexington Mathematical Tournament

Round 9
p25. Let

a

b

c

be positive numbers with

a +b +c = 4

a,b,c \le 2

M =\frac{a^3 +5a}{4a^2 +2}+\frac{b^3 +5b}{4b^2 +2}+\frac{c^3 +5c}{4c^2 +2},

then find the maximum possible value of

\lfloor 100M \rfloor

\vartriangle ABC

AB = 15

AC = 16

BC = 17

E

F

are chosen on sides

AC

AB

, respectively, such that

CE = 1

BF = 3

D

is chosen on side

BC

, and let the circumcircles of

\vartriangle BFD

\vartriangle CED

intersect at point

P \ne D

\angle PEF = 30^o

, the length of segment

PF

can be expressed as

\frac{m}{n}

m+n

.
p27. Arnold and Barnold are playing a game with a pile of sticks with Arnold starting first. Each turn, a player can either remove

7

13

sticks. If there are fewer than

7

sticks at the start of a player’s turn, then they lose. Both players play optimally. Find the largest number of sticks under

200

where Barnold has a winning strategy
Round 10
p28. Let

a

b

c

be positive real numbers such that

\log_2(a)-2 = \log_3(b) =\log_5(c)

a +b = c

a +b +c

?
p29. Two points,

P(x, y)

Q(-x, y)

are selected on parabola

y = x^2

x > 0

and the triangle formed by points

P

Q

, and the origin has equal area and perimeter. Find

y

5

families are attending a wedding.

2

families consist of

4

2

families consist of

3

1

family consists of

2

people. A very long row of

25

chairs is set up for the families to sit in. Given that all members of the same family sit next to each other, let the number of ways all the people can sit in the chairs such that no two members of different families sit next to each other be

n

. Find the number of factors of

n

.
Round 11
p31. Let polynomial

P(x) = x^3 +ax^2 +bx +c

have (not neccessarily real) roots

r_1

r_2

r_3

2ab = a^3 -20 = 6c -21

, then the value of

|r^3_1+r^3_2+r^3_3|

can be written as

\frac{m}{n}

m

n

are relatively prime positive integers. Find the value of

m+n

.
p32. In acute

\vartriangle ABC

H

I

O

G

be the orthocenter, incenter, circumcenter, and centroid of

\vartriangle ABC

, respectively. Suppose that there exists a circle

\omega

passing through

B

I

H

C

, the circumradius of

\vartriangle ABC

312

OG = 80

H'

, distinct from

H

, be the point on

\omega

\overline{HH'}

is a diameter of

\omega

. Given that lines

H'O

BC

meet at a point

P

, find the length

OP

.

p33. Find the number of ordered quadruples

(x, y, z,w)

0 \le x, y, z,w \le 1000

are integers and

x!+ y! =2^z \cdot w!

0! = 1

).
Round 12
p34. Let

Z

be the product of all the answers from the teams for this question. Estimate the number of digits of

Z

. If your estimate is

E

and the answer is

A

, your score for this problem will be

\max \left( 0, \lceil 15- |A-E| \rceil \right).

Your answer must be a positive integer.
p35. Let

N

be number of ordered pairs of positive integers

(x, y)

3x^2 -y^2 = 2

x < 2^{75}

N

. If your estimate is

E

and the answer is

A

, your score for this problem will be

\max \left( 0, \lceil 15- 2|A-E| \rceil \right).

30

points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). If your estimate is

E

and the answer is

A

, your score for this problem will be

\max \left( 0, \left \lceil 15- \ln \frac{A}{E} \right \rceil \right).

PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here and 5-8 [url=https://artofproblemsolving.com/community/c3h3166476p28814111]here.. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

1

2021 Spring LMT Division B Problem 19

Kevin is at the point

(19,12)

. He wants to walk to a point on the ellipse

9x^2 + 25y^2 = 8100

, and then walk to

(-24, 0)

. Find the shortest length that he has to walk.
Proposed by Kevin Zhao

1

2021 Spring LMT Division B Problem 16

Bob plants two saplings. Each day, each sapling has a

1/3

chance of instantly turning into a tree. Given that the expected number of days it takes both trees to grow is

m/n

m

n

are relatively prime positive integers, find

m +n

.
Proposed by Powell Zhang

1

2021 Spring LMT Division B Problem 14

In the expansion of

(2x +3y)^{20}

, find the number of coefficients divisible by

144

.
Proposed by Hannah Shen

1

2021 Spring LMT Division B Problem 13

4

\overline{a b c d}

unnoticeable if

a +c = b +d

\overline{a b c d} +\overline{c d a b}

is a multiple of

7

. Find the number of unnoticeable numbers.
Note:

a

b

c

d

are nonzero distinct digits.
Proposed by Aditya Rao

1

2021 Spring LMT Division B Problem 10

f (x)

be a function mapping real numbers to real numbers. Given that

f (f (x)) =\frac{1}{3x}

f (2) =\frac19

f\left(\frac{1}{6}\right)

.
Proposed by Zachary Perry

1

2021 Spring LMT Division B Problem 9

Convex pentagon

PQRST

PQ = T P = 5

QR = RS = ST = 6

\angle QRS = \angle RST = 90^o

. Given that points

U

V

exist such that

RU =UV = VS = 2

, find the area of pentagon

PQUVT

.
Proposed by Kira Tang

1

2021 Spring LMT Division B Problem 8

Find the number of arithmetic sequences

a_1,a_2,a_3

of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence.
Proposed by Sammy Charney

1

2021 Spring LMT Division B Problem 7

x

y

are positive real numbers such that

\frac{5}{x}=\frac{y}{13}=\frac{x}{y}

, find the value of

x^3 + y^3

.
Proposed by Ephram Chun

1

2021 Spring LMT Division B Problem 6

Maisy is at the origin of the coordinate plane. On her first step, she moves

1

unit up. On her second step, she moves

1

unit to the right. On her third step, she moves

2

units up. On her fourth step, she moves

2

units to the right. She repeats this pattern with each odd-numbered step being

1

unit more than the previous step. Given that the point that Maisy lands on after her

21

st step can be written in the form

(x, y)

, find the value of

x + y

.
Proposed by Audrey Chun

1

2021 Spring LMT Division B Problem 5

Find the number of ways there are to permute the elements of the set

\{1,2,3,4,5,6,7,8,9\}

such that no two adjacent numbers are both even or both odd.
Proposed by Ephram Chun

1

2021 Spring LMT Division B Problem 4

S

contains exactly

36

elements in the form of

2^m \cdot 5^n

0 \le m,n \le 5

. Two distinct elements of

S

are randomly chosen. Given that the probability that their product is divisible by

10^7

a/b

a

b

are relatively prime positive integers, find

a +b

.
Proposed by Ada Tsui

1

2021 Spring LMT Division B Problem 3

Aidan rolls a pair of fair, six sided dice. Let

n

be the probability that the product of the two numbers at the top is prime. Given that

n

can be written as

a/b

a

b

are relatively prime positive integers, find

a +b

.
Proposed by Aidan Duncan

1

2021 Spring LMT Division B Problem 2

Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length

2

.
Proposed by Alex Li

1

2021 Spring LMT Division B Problem 1

Given that the expression

\frac{20^{21}}{20^{20}} +\frac{20^{20}}{20^{21}}

can be written in the form

m/n

m

n

are relatively prime positive integers, find

m +n

.
Proposed by Ada Tsui

1

2021 Spring LMT Division B Problem 28

Maisy and Jeff are playing a game with a deck of cards with

4

0

4

1

4

2

’s, all the way up to

4

9

’s. You cannot tell apart cards of the same number. After shuffling the deck, Maisy and Jeff each take

4

cards, make the largest

4

-digit integer they can, and then compare. The person with the larger

4

-digit integer wins. Jeff goes first and draws the cards

2,0,2,1

from the deck. Find the number of hands Maisy can draw to beat that, if the order in which she draws the cards matters.
Proposed by Richard Chen

1

2021 Spring LMT Division A Problem 30

Ryan Murphy is playing poker. He is dealt a hand of

5

cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g.

3,4,5,6,7

9,10,J,Q,K

3

of a kind (at least

3

cards of the same rank; e.g.

5, 5, 5, 7, 7

5, 5, 5, 7,K

m/n

m

n

are relatively prime positive integers, find

m +n

.
Proposed by Aditya Rao

1

2021 Spring LMT Division A Problem 29, B Problem 30

6

people playing the card game Tractor, all

54

3

decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be

X

. Find the largest integer

n

n

X

is rational.
Proposed by Sammy Charney
Due to the problem having infinitely many solutions, all teams who inputted answers received points.

1

2021 Spring LMT Division A Problem 28, B Problem 29

Addison and Emerson are playing a card game with three rounds. Addison has the cards

1, 3

5

, and Emerson has the cards

2, 4

6

. In advance of the game, both designate each one of their cards to be played for either round one, two, or three. Cards cannot be played for multiple rounds. In each round, both show each other their designated card for that round, and the person with the higher-numbered card wins the round. The person who wins the most rounds wins the game. Let

m/n

be the probability that Emerson wins, where

m

n

are relatively prime positive integers. Find

m +n

.
Proposed by Ada Tsui

1

2021 Spring LMT Division A Problem 27

Chandler the Octopus is at a tentacle party! At this party, there is

1

2

2

3

3

4

tentacles, all the way up to

14

15

tentacles. Each tentacle is distinguishable from all other tentacles. For some

2\le m < n \le 15

, a creature with m tentacles “meets” a creature with n tentacles; “meeting” another creature consists of shaking exactly 1 tentacle with each other. Find the number of ways there are to pick distinct

m < n

2

15

, inclusive, and then to pick a creature with

m

tentacles to “meet” a selected creature with

n

tentacles.
Proposed by Armaan Tipirneni, Richard Chen, and Denise the Octopus

1

2021 Spring LMT Division A Problem 26, B Problem 27

Chandler the Octopus along with his friends Maisy the Bear and Jeff the Frog are solving LMT problems. It takes Maisy

3

minutes to solve a problem, Chandler

4

minutes to solve a problem and Jeff

5

minutes to solve a problem. They start at

12:00

pm, and Chandler has a dentist appointment from

12:10

12:30

, after which he comes back and continues solving LMT problems. The time it will take for them to finish solving

50

LMT problems, in hours, is

m/n

m

n

are relatively prime positive integers. Find

m +n

.
Note: they may collaborate on problems.
Proposed by Aditya Rao

1

2021 Spring LMT Division A Problem 25, B Problem 26

Chandler the Octopus is making a concoction to create the perfect ink. He adds

1.2

grams of melanin,

4.2

grams of enzymes, and

6.6

grams of polysaccharides. But Chandler accidentally added n grams of an extra ingredient to the concoction, Chemical

X

, to create glue. Given that Chemical

X

contains none of the three aforementioned ingredients, and the percentages of melanin, enzymes, and polysaccharides in the final concoction are all integers, find the sum of all possible positive integer values of

n

.
Proposed by Taiki Aiba

1

2021 Spring LMT Division B Problem 21

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Take five good haikus Scramble their lines randomly What are the chances That you end up with Five completely good haikus (With five, seven, five)? Your answer will be m over n where m,n Are numbers such that m,n positive Integers where gcd Of m,n is 1. Take this answer and Add the numerator and Denominator.
Proposed by Jeff Lin

1

2021 Spring LMT Division A Problem 24

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Using the four words “Hi”, “hey”, “hello”, and “haiku”, How many haikus Can somebody make? (Repetition is allowed, Order does matter.)
Proposed by Jeff Lin

1

2021 Spring LMT Division A Problem 23, B Problem 24

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
A group of haikus Some have one syllable less Sixteen in total. The group of haikus Some have one syllable more Eighteen in total. What is the largest Total count of syllables That the group can’t have? (For instance, a group Sixteen, seventeen, eighteen Fifty-one total.) (Also, you can have No sixteen, no eighteen Syllable haikus)
Proposed by Jeff Lin

1

2021 Spring LMT Division A Problem 22, B Problem 23

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Ada has been told To write down five haikus plus Two more every hour. Such that she needs to Write down five in the first hour Seven, nine, so on. Ada has so far Forty haikus and writes down Seven every hour. At which hour after She begins will she not have Enough haikus done?
Proposed by Ada Tsui

1

2021 Spring LMT Division A Problem 21, B Problem 22

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
In how many ways Can you add three integers Summing seventeen? Order matters here. For example, eight, three, six Is not eight, six, three. All nonnegative, Do not need to be distinct. What is your answer?
Proposed by Derek Gao

1

2021 LMT Spring Division A Problem 1

LMT

\overline{MA}

as an altitude. Given that

MA = 16

MT = 20

LT = 25

, find the length of the altitude from

L

\overline{MT}

.
Proposed by Kevin Zhao

1

2021 LMT Spring Division A Problem 2

f(x)

has the property that

f(x) = -\frac{1}{f(x-1)}.

f(0)=-\frac{1}{21},

find the value of

f(2021).

Proposed by Ada Tsui

1

2021 LMT Spring Division A Problem 3

Find the greatest possible sum of integers

a

b

\frac{2021!}{20^a\cdot 21^b}

is a positive integer.
Proposed by Aidan Duncan

1

2021 LMT Spring Division A Problem 4

Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

.
Proposed by Alex Li

1

2021 LMT Spring Division A Problem 5

ABCD

E

F

\overline{BC}

\overline{AD}

, respectively. Two congruent semicircles are drawn with centers

E

F

such that they both lie entirely on or inside the rectangle, the semicircle with center

E

C

, and the semicircle with center

F

A

AB=8

CE=5

, and the semicircles are tangent, find the length

BC

.
Proposed by Ada Tsui

1

2021 LMT Spring Division A Problem 6

Given that the expected amount of

1

s in a randomly selected

2021

-digit number is

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

.
Proposed by Hannah Shen

1

2021 LMT Spring Division A Problem 7

A geometric sequence consists of

11

terms. The arithmetic mean of the first

6

63

, and the arithmetic mean of the last

6

2016

7

th term in the sequence.
Proposed by Powell Zhang

1

2021 LMT Spring Division A Problem 8

\triangle{ABC}

has interior point

O

AO = \sqrt{52}

BO = 3

CO = 5

\angle{ABC}=120^{\circ}

, find the length

AB

.
Proposed by Powell Zhang

1

2021 LMT Spring Division A Problem 9

Find the sum of all positive integers

n

7<n < 100

1573_{n}

6

factors when written in base

10

.
Proposed by Aidan Duncan

1

2021 LMT Spring Division A Problem 10

Pieck the Frog hops on Pascal's Triangle, where she starts at the number

1

at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after

7

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

.
Proposed by Steven Yu

1

2021 LMT Spring Division A Problem 11 / Division B Problem 17

\triangle ABC

\angle BAC = 60^{\circ}

and circumcircle

\omega

, the angle bisector of

\angle BAC

intersects side

\overline{BC}

D

AD

is extended past

D

A'

E

F

be the feet of the perpendiculars of

A'

AB

AC

, respectively. Suppose that

\omega

is tangent to line

EF

P

E

F

\tfrac{EP}{FP} = \tfrac{1}{2}

EF=6

\triangle ABC

can be written as

\tfrac{m\sqrt{n}}{p}

m

p

are relatively prime positive integers, and

n

is a positive integer not divisible by the square of any prime. Find

m+n+p

.
Proposed by Taiki Aiba

1

2021 LMT Spring Division A Problem 12 / Division B Problem 18

23

balls on a table, all of which are either red or blue, such that the probability that there are

n

23-n

blue balls on the table (

1 \le n \le 22

) is proportional to

n

. (e.g. the probability that there are

2

21

blue balls is twice the probability that there are

1

22

blue balls.) Given that the probability that the red balls and blue balls can be arranged in a line such that there is a blue ball on each end, no two red balls are next to each other, and an equal number of blue balls can be placed between each pair of adjacent red balls is

\frac{a}{b}

a

b

are relatively prime positive integers, find

a+b

. Note: There can be any nonzero number of consecutive blue balls at the ends of the line.
Proposed by Ada Tsui

1

2021 LMT Spring Division A Problem 13

In a round-robin tournament, where any two players play each other exactly once, the fact holds that among every three students

A

B

C

, one of the students beats the other two. Given that there are six players in the tournament and Aidan beats Zach but loses to Andrew, find how many ways there are for the tournament to play out. Note: The order in which the matches take place does not matter.
Proposed by Kevin Zhao

1

2021 LMT Spring Division A Problem 14

Alex, Bob, and Chris are driving cars down a road at distinct constant rates. All people are driving a positive integer number of miles per hour. All of their cars are

15

feet long. It takes Alex

1

second longer to completely pass Chris than it takes Bob to completely pass Chris. The passing time is defined as the time where their cars overlap. Find the smallest possible sum of their speeds, in miles per hour.
Proposed by Sammy Charney

1

2021 LMT Spring Division A Problem 15 / Division B Problem 20

Andy and Eddie play a game in which they continuously flip a fair coin. They stop flipping when either they flip tails, heads, and tails consecutively in that order, or they flip three tails in a row. Then, if there has been an odd number of flips, Andy wins, and otherwise Eddie wins. Given that the probability that Andy wins is

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

.
Proposed by Anderw Zhao and Zachary Perry

1

2021 LMT Spring Division A Problem 16

Find the number of ordered pairs

(a,b)

of positive integers less than or equal to

20

such that \gcd(a,b)>1 \text{and} \frac{1}{\gcd(a,b)}+\frac{a+b}{\text{lcm}(a,b)} \geq 1.
Proposed by Zachary Perry

1

2021 LMT Spring Division A Problem 17

Given that the value of

\sum_{k=1}^{2021} \frac{1}{1^2+2^2+3^2+\cdots+k^2}+\sum_{k=1}^{1010} \frac{6}{2k^2-k}+\sum_{k=1011}^{2021} \frac{24}{2k+1}

can be expressed as

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

.
Proposed by Aidan Duncan

1

2021 LMT Spring Division A Problem 18

X

Y

are on a parabola of the form

y=\frac{x^2}{a^2}

A

(x, y) = (0, a)

XY

A

and hits the line

y=-a

B

\omega

be the circle passing through

(0, -a)

A

B

P

\omega

PA = 8

X

A

B

AX=2

XB=10

PX \cdot PY

.
Proposed by Kevin Zhao

1

2021 LMT Spring Division A Problem 19

S

be the sum of all possible values of

a \cdot c

a^3+3ab^2-72ab+432a=4c^3

a

b

c

are positive integers,

a+b > 11

a > b-13

c \le 1000

. Find the sum of all distinct prime factors of

S

.
Proposed by Kevin Zhao

1

2021 LMT Spring Division A Problem 20

\Omega

be a circle with center

O

\omega_1

\omega_2

be circles with centers

O_1

O_2

, respectively, internally tangent to

\Omega

A

B

, respectively, such that

O_1

\overline{OA}

O_2

\overline{OB}

\omega_1

. There exists a point

P

AB

P

\omega_1

\omega_2

. Let the external tangent of

\omega_1

\omega_2

on the same side of line

AB

O

\omega_1

X

\omega_2

Y

, and let lines

AX

BY

N

O_1X = 81

O_2Y = 18

NX \cdot NA

can be written as

a\sqrt{b} + c

a

b

c

are positive integers, and

b

is not divisible by the square of a prime. Find

a+b+c

.
Proposed by Kevin Zhao