2022 LMT Spring

Part of LMT

Subcontests

(12)

3

2022 LMT Spring Guts Round p16-p27- Lexington Mathematical Tournament

Round 6
p16. Given that

x

y

are positive real numbers such that

x^3+y = 20

, the maximum possible value of

x + y

can be written as

\frac{a\sqrt{b}}{c}

a

b

c

d

are positive integers such that

gcd(a,c) = 1

b

is square-free. Find

a +b +c +d

\vartriangle DRK

DR = 13

DK = 14

RK = 15

E

be the intersection of the altitudes of

\vartriangle DRK

. Find the value of

\lfloor DE +RE +KE \rfloor

.
p18. Subaru the frog lives on lily pad

1

. There is a line of lily pads, numbered

2

3

4

5

6

7

. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either

1

2

greater, chosen at random from valid possibilities. There are alligators on lily pads

2

5

. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number

1

. Find the expected number of jumps it takes Subaru to reach pad

7

.
Round 7
This set has problems whose answers depend on one another.
p19. Let

B

be the answer to Problem

20

C

be the answer to Problem

21

f (x) = x^3-Bx-C = (x-r )(x-s)(x-t )

r

s

t

are complex numbers, find the value of

r^2+s^2+t^2

A

be the answer to Problem

19

C

be the answer to Problem

21

\omega_1

\omega_2

X

Y

P \ne Y

be the point on

\omega_1

PY

\omega_2

, and let point

Q \ne Y

be the point on

\omega_2

QY

\omega_1

PX = A

QX =C

XY

A

be the answer to Problem

19

B

be the answer to Problem

20

. Given that the positive difference between the number of positive integer factors of

A^B

and the number of positive integer factors of

B^A

D

, and given that the answer to this problem is an odd prime, find

\frac{D}{B}-40

.
Round 8
p22. Let

v_p (n)

p

and positive integer

n

output the greatest nonnegative integer

x

p^x

n

\sum^{50}_{i=1}\sum^{i}_{p=1} { v_p (i )+1 \choose 2},

where the inner summation only sums over primes

p

1

i

a

b

c

be positive real solutions to the following equations.

\frac{2b^2 +2c^2 -a^2}{4}= 25

\frac{2c^2 +2a^2 -b^2}{4}= 49

\frac{2a^2 +2b^2 -c^2}{4}= 64

The area of a triangle with side lengths

a

b

c

can be written as

\frac{x\sqrt{y}}{z}

x

z

are relatively prime positive integers and

y

is square-free. Find

x + y +z

.
p24. Alan, Jiji, Ina, Ryan, and Gavin want to meet up. However, none of them know when to go, so they each pick a random

1

hour period from

5

11

AM to meet up at Alan’s house. Find the probability that there exists a time when all of them are at the house at one time.
Round 9
p25. Let

n

be the number of registered participantsin this

LMT

. Estimate the number of digits of

\left[ {n \choose 2} \right]

10

. If your answer is

A

and the correct answer is

C

, then your score will be

\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.

\gamma

be theminimum value of

x^x

over all real numbers

x

\lfloor 10000\gamma \rfloor

. If your answer is

A

and the correct answer is

C

, then your score will be

\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.

E = \log_{13} 1+log_{13}2+log_{13}3+...+log_{13}513513.

\lfloor E \rfloor

. If your answer is

A

and the correct answer is

C

, your score will be

\left \lfloor \max \left( 0,20 - \left| \ln \left( \frac{A}{C}\right) \cdot 5 \right|\right| \right \rfloor.

PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167127p28823220]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2022 LMT Spring Guts Round p1-p15- Lexington Mathematical Tournament

Round 1
p1. A box contains

1

ball labelledW,

1

E

1

L

1

C

1

O

8

M

1

last ball labelled

E

. One ball is randomly drawn from the box. The probability that the ball is labelled

E

\frac{1}{a}

a

G +E +N = 7

G +E +O = 15

N +T = 22.

Find the value of

T +O

.
p3. The area of

\vartriangle LMT

22

MT = 4

and that there is a right angle at

M

, find the length of

LM

.
Round 2
p4. Kevin chooses a positive

2

-digit integer, then adds

6

times its unit digit and subtracts

3

times its tens digit from itself. Find the greatest common factor of all possible resulting numbers.
p5. Find the maximum possible number of times circle

D

can intersect pentagon

GRASS'

over all possible choices of points

G

R

A

S

S'

.
p6. Find the sum of the digits of the integer solution to

(\log_2 x) \cdot (\log_4 \sqrt{x}) = 36

.
Round 3
p7. Given that

x

y

are positive real numbers such that

x^2 + y = 20

, the maximum possible value of

x + y

can be written as

\frac{a}{b}

a

b

are relatively prime positive integers. Find

a +b

\vartriangle DRK

DR = 13

DK = 14

RK = 15

E

be the point such that

ED = ER = EK

. Find the value of

\lfloor DE +RE +KE \rfloor

.
p9. Subaru the frog lives on lily pad

1

. There is a line of lily pads, numbered

2

3

4

5

6

7

. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either

1

2

greater, chosen at random from valid possibilities. There are alligators on lily pads

2

5

. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number

1

. Find how many times Subaru is expected to die before he reaches pad

7

.
Round 4
p10. Find the sum of the following series:

\sum^{\infty}_{i=1} = \frac{\sum^i_{j=1} j}{2^i}=\frac{1}{2^1}+\frac{1+2}{2^2}+\frac{1+2+3}{2^3}+\frac{1+2+3+4}{2^4}+...

\phi (x)

be the number of positive integers less than or equal to

x

that are relatively prime to

x

. Find the sum of all

x

\phi (\phi(x)) = x -3

1

is relatively prime to every positive integer.
p12. On a piece of paper, Kevin draws a circle. Then, he draws two perpendicular lines. Finally, he draws two perpendicular rays originating from the same point (an

L

shape). What is the maximum number of sections into which the lines and rays can split the circle?
Round 5
p13. In quadrilateral

ABCD

\angle A = 90^o

\angle C = 60^o

\angle ABD = 25^o

\angle BDC = 5^o

AB = 4\sqrt3

, the area of quadrilateral

ABCD

can be written as

a\sqrt{b}

10a +b

.
p14. The value of

\sum^6_{n=2} \left( \frac{n^4 +1}{n^4 -1}\right) -2 \sum^6_{n=2}\left(\frac{n^3 -n^2+n}{n^4 -1}\right)

can be written as

\frac{m}{n}

m

n

are relatively prime positive integers. Find

100m+n

.
p15. Positive real numbers

x

y

satisfy the following

2

x^{1+x^{1+x^{1+...}}}= 8

\sqrt[24]{y +\sqrt[24]{y + \sqrt[24]{y +...}}} = x

Find the value of

\lfloor y \rfloor

.
PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167130p28823260]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2022 LMT Spring Speed Round - Lexington Mathematical Tournament

p1. Aidan walks into a skyscraper’s first floor lobby and takes the elevator up

50

floors. After exiting the elevator, he takes the stairs up another

10

floors, then takes the elevator down

30

floors. Find the floor number Aidan is currently on.
p2. Jeff flips a fair coin twice and Kaylee rolls a standard

6

-sided die. The probability that Jeff flips

2

heads and Kaylee rolls a

4

P

\frac{1}{P}

.
p3. Given that

a\odot b = a + \frac{a}{b}

(4\odot 2)\odot 3

.
p4. The following star is created by gluing together twelve equilateral triangles each of side length

3

. Find the outer perimeter of the star. https://cdn.artofproblemsolving.com/attachments/e/6/ad63edbf93c5b7d4c7e5d68da2b4632099d180.png
p5. In Lexington High School’sMath Team, there are

40

20

of whom do science bowl and

22

of whom who do LexMACS. What is the least possible number of students who do both science bowl and LexMACS?
p6. What is the least positive integer multiple of

3

whose digits consist of only

0

1

s? The number does not need to have both digits.
p7. Two fair

6

-sided dice are rolled. The probability that the product of the numbers rolled is at least

30

can be written as

\frac{a}{b}

a

b

are relatively prime positive integers. Find

a +b

.
p8. At the LHSMath Team Store,

5

1

\$13

5

1

\$17

. Find how much

15

15

hoodies cost, in dollars.
p9. Eric wants to eat ice cream. He can choose between

3

options of spherical ice cream scoops. The first option consists of

4

scoops each with a radius of

3

inches, which costs a total of

\$3

. The second option consists of a scoop with radius

4

inches, which costs a total of

\$2

. The third option consists of

5

scoops each with diameter

2

inches, which costs a total of

\$1

. The greatest possible ratio of volume to cost of ice cream Eric can buy is nπ cubic inches per dollar. Find

3n

.
p10. Jen claims that she has lived during at least part of each of five decades. What is the least possible age that Jen could be? (Assume that age is always rounded down to the nearest integer.)
p11. A positive integer

n

is called Maisylike if and only if

n

has fewer factors than

n -1

. Find the sum of the values of

n

that are Maisylike, between

2

10

, inclusive.
p12. When Ginny goes to the nearby boba shop, there is a

30\%

chance that the employee gets her drink order wrong. If the drink she receives is not the one she ordered, there is a

60\%

chance that she will drink it anyways. Given that Ginny drank a milk tea today, the probability she ordered it can be written as

\frac{a}{b}

a

b

are relatively prime positive integers. Find the value of

a +b

.
p13. Alex selects an integer

m

1

100

, inclusive. He notices there are the same number of multiples of

5

as multiples of

7

m+9

, inclusive. Find how many numbers Alex could have picked.
p14. In LMTown there are only rainy and sunny days. If it rains one day there’s a

30\%

chance that it will rain the next day. If it’s sunny one day there’s a

90\%

chance it will be sunny the next day. Over n days, as n approaches infinity, the percentage of rainy days approaches

k\%

10k

.
p15. A bag of letters contains

3

3

3

T’s. Aidan picks three letters at random from the bag with replacement, and Andrew picks three letters at random fromthe bag without replacement. Given that the probability that both Aidan and Andrew pick one each of L, M, and T can be written as

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

\omega

is inscribed in a square with side length

2

. In each corner tangent to

2

of the square’s sides and externally tangent to

\omega

is another circle. The radius of each of the smaller

4

circles can be written as

(a -\sqrt{b})

a

b

are positive integers. Find

a +b

. https://cdn.artofproblemsolving.com/attachments/d/a/c76a780ac857f745067a8d6c4433efdace2dbb.png
p17. In the nonexistent land of Lexingtopia, there are

10

days in the year, and the Lexingtopian Math Society has

5

members. The probability that no two of the LexingtopianMath Society’s members share the same birthday can be written as

\frac{a}{b}

a

b

are relatively prime positive integers. Find

a +b

D(n)

be the number of diagonals in a regular

n

\sum^{26}_{n=3} D(n).

p19. Given a square

A_0B_0C_0D_0

as shown below with side length

1

, for all nonnegative integers

n

, construct points

A_{n+1}

B_{n+1}

C_{n+1}

D_{n+1}

A_nB_n

B_nC_n

C_nD_n

D_nA_n

, respectively, such that

\frac{A_n A_{n+1}}{A_{n+1}B_n}=\frac{B_nB_{n+1}}{B_{n+1}C_n} =\frac{C_nC_{n+1}}{C_{n+1}D_n}=\frac{D_nD_{n+1}}{D_{n+1}A_n} =\frac34.

https://cdn.artofproblemsolving.com/attachments/6/a/56a435787db0efba7ab38e8401cf7b06cd059a.png The sum of the series

\sum^{\infty}_{i=0} [A_iB_iC_iD_i ] = [A_0B_0C_0D_0]+[A_1B_1C_1D_1]+[A_2B_2C_2D_2]...

[P]

denotes the area of polygon

P

can be written as

\frac{a}{b}

a

b

are relatively prime positive integers. Find

a +b

m

n

be two real numbers such that

\frac{2}{n}+m = 9

\frac{2}{m}+n = 1

Find the sum of all possible values of

m

plus the sumof all possible values of

n

\sigma (x)

denote the sum of the positive divisors of

x

. Find the smallest prime

p

\sigma (p!) \ge 20 \cdot \sigma ([p -1]!).

\vartriangle ABC

be an isosceles triangle with

AB = AC

M

be the midpoint of side

\overline{AB}

. Suppose there exists a point X on the circle passing through points

A

M

C

BMCX

is a parallelogram and

M

X

are on opposite sides of line

BC

\overline{AX}

\overline{BC}

intersect at a point

Y

BY = 8

AY^2

.
p23. Kevin chooses

2

integers between

1

100

, inclusive. Every minute, Corey can choose a set of numbers and Kevin will tell him how many of the

2

chosen integers are in the set. How many minutes does Corey need until he is certain of Kevin’s

2

chosen numbers?
p24. Evaluate

1^{-1} \cdot 2^{-1} +2^{-1} \cdot 3^{-1} +3^{-1} \cdot 4^{-1} +...+(2015)^{-1} \cdot (2016)^{-1} \,\,\, (mod \,\,\,2017).

p25. In scalene

\vartriangle ABC

, construct point

D

on the opposite side of

AC

B

\angle ABD = \angle DBC = \angle BC A

AD =DC

I

be the incenter of

\vartriangle ABC

BC = 64

AD = 225

BI

. https://cdn.artofproblemsolving.com/attachments/b/1/5852dd3eaace79c9da0fd518cfdcd5dc13aecf.png
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

1

2022 LMT Spring Accuracy Problem 11 - Tiebreaker

L

be the number of times the letter

L

appeared on the Speed Round,

M

be the number of times the letter

M

appeared on the Speed Round, and

T

be the number of times the letter

T

appeared on the Speed Round. Find the value of

LMT

2

2022 Spring LMT Team 10

In a country with

5

distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities

\{A,B,C\}

such that there are roads between

A

B

B

C

C

A

. How many road systems between the five cities are possible?

2022 LMT Spring Accuracy Problem 10

In a room, there are

100

light switches, labeled with the positive integers

{1,2, . . . ,100}

. They’re all initially turned off. On the

i

1 \le i \le 100

, Bob flips every light switch with label number

k

i

\frac{k}{i}

times. Find the sum of the labels of the light switches that are turned on at the end of the

100

2

2022 Spring LMT Team 9

r_1, r_2, ..., r_{2021}

be the not necessarily real and not necessarily distinct roots of

x^{2022} + 2021x = 2022

S_i = r_i^{2021}+2022r_i

1 \le i \le 2021

\left|\sum^{2021}_{i=1} S_i \right| = |S_1 +S_2 +...+S_{2021}|

2022 LMT Spring Accuracy Problem 9

A rook is randomly placed on an otherwise empty

8 \times 8

chessboard. Owen makes moves with the rook by randomly choosing

1

14

possible moves. Find the expected value of the number of moves it takes Owen to move the rook to the top left square. Note that a rook can move any number of squares either in the horizontal or vertical direction each move.

2

2022 Spring LMT Team 8

53

37,984,318,966,591,152,105,649,545,470,741,788,308,402,068,827,142,719

can be expressed as

n^21

n

is a positive integer. Find

n

2022 LMT Spring Accuracy Problem 8

A ray originating at point

P

intersects a circle with center

O

A

B

PB > PA

\overline{OP}

intersects the circle at point

C

PA = 31

PC = 17

\angle PBO = 60^o

, find the radius of the circle.

2

2022 Spring LMT Team 7

Kevin has a square piece of paper with creases drawn to split the paper in half in both directions, and then each of the four small formed squares diagonal creases drawn, as shown below. https://cdn.artofproblemsolving.com/attachments/2/2/70d6c54e86856af3a977265a8054fd9b0444b0.png Find the sum of the corresponding numerical values of figures below that Kevin can create by folding the above piece of paper along the creases. (The figures are to scale.) Kevin cannot cut the paper or rip it in any way. https://cdn.artofproblemsolving.com/attachments/a/c/e0e62a743c00d35b9e6e2f702106016b9e7872.png

2022 LMT Spring Accuracy Problem 7

A teacher wishes to separate her

12

students into groups. Yesterday, the teacher put the students into

4

3

. Today, the teacher decides to put the students into

4

3

again. However, she doesn’t want any pair of students to be in the same group on both days. Find how many ways she could formthe groups today.

2

2022 Spring LMT Team 6

y

f_y (x)

f_y (0) = y

f_y (1) = y +12

f_y (2) = 3y^2

f_y (3) = 2y +4

y

f_y(4)

can be expressed in the form

ay^2 +by +c

a,b,c

are integers. Find

a +b +c

2022 LMT Spring Accuracy Problem 6

Jacob likes to watchMickeyMouse Clubhouse! One day, he decides to create his own MickeyMouse head shown below, with two circles

\omega_1

\omega_2

\omega

O_1

O_2

O

, respectively. Let

\omega_1

\omega

P_1

Q_1

\omega_2

\omega

P_2

Q_2

P_1

O_2

Q_1

P_2

O_1

Q_2

P_1

P_2

O_1O_2

O_1P_1 = P_1P_2 = P_2O_2 = 2

Q_1O_1 \parallel Q_2O_2

\omega

can be written as

n \pi

n

. https://cdn.artofproblemsolving.com/attachments/6/d/d98a05ee2218e80fd84d299d47201669736d99.png

2

2022 Spring LMT Team 5

\sum^{2020}_{n=1} \gcd (n^3 -2n^2 +2021,n^2 -3n +3).

2022 LMT Spring Accuracy Problem 5

5

identical blue marbles and

5

identical green marbles. In how many ways can

5

marbles from the bag be arranged in a row if each blue marble must be adjacent to at least

1

2

2022 Spring LMT Team 4

Jeff has a deck of

12

4

L

4

M

4

T

s. Armaan randomly draws three cards without replacement. The probability that he takes

3

L

s can be written as

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m +n

2022 LMT Spring Accuracy Problem 4

Kevin runs uphill at a speed that is

4

meters per second slower than his speed when he runs downhill. Kevin takes a total of

80

seconds to run up and down a hill on one path. Given that the path is

300

meters long (he travels

600

meters total), find how long Kevin takes to run up the hill in seconds.

2

2022 Spring LMT Team 3 . x^2 + \frac{144}{x^2} = 25

Let the four real solutions to the equation

x^2 + \frac{144}{x^2} = 25

r_1, r_2, r_3

r_4

|r_1| +|r_2| +|r_3| +|r_4|

2022 LMT Spring Accuracy Problem 3

Find the difference between the greatest and least values of

lcm (a,b,c)

a

b

c

are distinct positive integers between

1

10

2

2022 Spring LMT Team 2

Five people are standing in a straight line, and the distance between any two people is a unique positive integer number of units. Find the least possible distance between the leftmost and rightmost people in the line in units.

2022 LMT Spring Accuracy Problem 2

a \spadesuit b = \frac{a^2-b^2}{2b-2a}

3 \spadesuit x = -10

x

2

2022 Spring LMT Team 1

Derek and Jacob have a cake in the shape a rectangle with dimensions

14

9

inches. They make a deal to split it: Derek takes home the portion of the cake that is less than one inch from the border, while Jacob takes home the remainder of the cake. Let

D : J

be the ratio of the amount of cake Derek took to the amount of cake Jacob took, where

D

J

are relatively prime positive integers. Find

D + J

2022 LMT Spring Accuracy Problem 1

Kevin colors a ninja star on a piece of graph paper where each small square has area

1

square inch. Find the area of the region colored, in square inches. https://cdn.artofproblemsolving.com/attachments/3/3/86f0ae7465e99d3e4bd3a816201383b98dc429.png