2016 LMT

Part of LMT

Subcontests

(27)

3

2016 LMT Spring Guts Round p13-p24 - Lexington Mathematical Tournament

2016 \times 2016

chess board is cut into

k \ge 1

rectangle(s) with positive integer sidelengths. Let

p

be the sum of the perimeters of all

k

rectangles. Additionally, let

m

M

be the minimum and maximum possible value of

\frac{p}{k}

, respectively. Determine the ordered pair

(m,M)

.
p14. For nonnegative integers

n

f (n)

be the product of the digits of

n

\sum^{1000}_{i=1}f (i )

.
p15. How many ordered pairs of positive integers

(m,n)

have the property that

mn

2016

?
Round 6
p16. Let

a,b,c

be distinct integers such that

a +b +c = 0

. Find the minimum possible positive value of

|a^3 +b^3 +c^3|

.
p17. Find the greatest positive integer

k

11^k -2^k

is a perfect square.
p18. Find all ordered triples

(a,b,c)

a \le b \le c

of nonnegative integers such that

2a +2b +2c = ab +bc +ca

.
Round 7
p19. Let

f :N \to N

be a function such that

f ( f (n))+ f (n +1) = n +2

for all positive integers

n

f (20)+ f (16)

\vartriangle ABC

be a triangle with area

10

BC = 10

. Find the minimum possible value of

AB \cdot AC

\vartriangle ABC

be a triangle with sidelengths

AB = 19

BC = 24

C A = 23

D

be a point on minor arc

BC

of the circumcircle of

\vartriangle ABC

DB =DC

. A circle with center

D

that passes through

B

C

AC

again at a point

E \ne C

. Find the length of

AE

.
Round 8
p22. Let

m =\frac12 \sqrt{2+\sqrt{2+... \sqrt2}}

, where there are

2014

square roots. Let

f_1(x) =2x^2 -1

f_n(x) = f_1( f_{n-1}(x))

f_{2015}(m)

.
p23. How many ordered triples of integers

(a,b,c)

are there such that

0 < c \le b \le a \le 2016

a +b-c = 2016

?
p24. In cyclic quadrilateral

ABCD

\angle B AD = 120^o

\angle ABC = 150^o

CD = 8

and the area of

ABCD

6\sqrt3

. Find the perimeter of

ABCD

.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2016 LMT Spring Guts Round p1-p12 - Lexington Mathematical Tournament

Round 1
p1. Today, the date

4/9/16

has the property that it is written with three perfect squares in strictly increasing order. What is the next date with this property?
p2. What is the greatest integer less than

100

whose digit sumis equal to its greatest prime factor?
p3. In chess, a bishop can only move diagonally any number of squares. Find the number of possible squares a bishop starting in a corner of a

20\times 16

chessboard can visit in finitely many moves, including the square it stars on.
Round 2
p4. What is the fifth smallest positive integer with at least

5

distinct prime divisors?
p5. Let

\tau (n)

be the number of divisors of a positive integer

n

1

n

. Howmany positive integers

n \le 1000

are there such that

\tau (n) > 2

\tau (\tau (n)) = 2

?
p6. How many distinct quadratic polynomials

P(x)

with leading coefficient

1

exist whose roots are positive integers and whose coefficients sum to

2016

?
Round 3
p7. Find the largest prime factor of

112221

.
p8. Find all ordered pairs of positive integers

(a,b)

\frac{a^2b^2+1}{ab-1}

is an integer.
p9. Suppose

f : Z \to Z

is a function such that

f (2x)= f (1-x)+ f (1-x)

for all integers

x

. Find the value of

f (2) f (0) +f (1) f (6)

.
Round 4
p10. For any six points in the plane, what is the maximum number of isosceles triangles that have three of the points as vertices?
p11. Find the sum of all positive integers

n

\sqrt{n+ \sqrt{n -25}}

is also a positive integer.
p12. Distinct positive real numbers are written at the vertices of a regular

2016

-gon. On each diagonal and edge of the

2016

-gon, the sum of the numbers at its endpoints is written. Find the minimum number of distinct numbers that are now written, including the ones at the vertices.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here. and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2016 LMT Spring Guts Round p25-36 - Lexington Mathematical Tournament

Round 9
p25. Define a sequence

\{a_n\}_{n \ge 1}

of positive real numbers by

a_1 = 2

a^2_n -2a_n +5 =4a_{n-1}

n \ge 2

k

is a positive real number such that

a_n <k

for all positive integers

n

. Find the minimum possible value of

k

\vartriangle ABC

be a triangle with

AB = 13

BC = 14

C A = 15

. Suppose the incenter of

\vartriangle ABC

I

and the incircle is tangent to

BC

AB

D

E

, respectively. Line

\ell

passes through the midpoints of

BD

BE

X

\ell

AX \parallel BC

X I

x, y, z

be positive real numbers such that

x y + yz +zx = 20

x^2yz +x y^2z +x yz^2 = 100

. Additionally, let

s = \max (x y, yz,xz)

m = \min(x, y, z)

s

is maximal, find

m

.
Round 10
p28. Let

\omega_1

be a circle with center

O

1

that is internally tangent to a circle

\omega_2

2

T

R

\omega_1

N

be the projection of

R

TO

O

lies on segment

NT

\frac{RN}{NO} = \frac4 3

. Additionally, let

S

\omega_2

T,R,S

are collinear. Tangents are drawn from

S

\omega_1

\omega_1

P

Q

. The tangent to

\omega_1

R

PQ

Z

. Find the area of triangle

\vartriangle ZRS

m

n

be positive integers such that

k =\frac{ m^2+n^2}{mn-1}

is also a positive integer. Find the sum of all possible values of

k

f_k (x) = k \cdot \ min (x,1-x)

. Find the maximum value of

k \le 2

for which the equation

f_k ( f_k ( f_k (x))) = x

8

x

0 \le x \le 1

.
Round 11
In the following problems,

A

is the answer to Problem

31

B

is the answer to Problem

32

C

is the answer to Problem

33

. For this set, you should find the values of

A

B

C

and submit them as answers to problems

31

32

33

, respectively. Although these answers depend on each other, each problem will be scored separately.
p31. Find

A \cdot B \cdot C + \dfrac{1}{B+ \dfrac{1}{C +\dfrac{1}{B+\dfrac{1}{...}}}}

D = 7 \cdot B \cdot C

. An ant begins at the bottom of a unit circle. Every turn, the ant moves a distance of

r

units clockwise along the circle, where

r

is picked uniformly at random from the interval

\left[ \frac{\pi}{2D} , \frac{\pi}{D} \right]

. Then, the entire unit circle is rotated

\frac{\pi}{4}

radians counterclockwise. The ant wins the game if it doesn’t get crushed between the circle and the

x

-axis for the first two turns. Find the probability that the ant wins the game.
p33. Let

m

n

be the two-digit numbers consisting of the products of the digits and the sum of the digits of the integer

2016 \cdot B

, respectively. Find

\frac{n^2}{m^2 - mn}

.
Round 12
p34. There are five regular platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. For each of these solids, define its adjacency angle to be the dihedral angle formed between two adjacent faces. Estimate the sum of the adjacency angles of all five solids, in degrees. If your estimate is

E

and the correct answer is

A

, your score for this problem will be

\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).

p35. Estimate the value of

\log_{10} \left(\prod_{k|2016} k!\right),

where the product is taken over all positive divisors

k

2016

. If your estimate is

E

and the correct answer is

A

, your score for this problem will be

\max \left(0, \lceil 15 \cdot \min \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).

p36. Estimate the value of

\sqrt{2016}^{\sqrt[4]{2016}}

. If your estimate is

E

and the correct answer is

A

, your score for this problem will be

\max \left(0, \lceil 15 \cdot \min \left(\frac{\ln E}{\ln A}, 2- \frac{\ln E}{\ln A}\right) \rceil \right).

PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here and 5-8 [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

1

2016 LMT Team Round - Potpourri - Lexington Math Tournament

X,Y ,Z

be nonzero real numbers such that the quadratic function

X t^2 - Y t + Z = 0

has the unique root

t = Y

X

ABCD

AB = BC = 1

CD = AD =\sqrt2

BD =\sqrt5

AC

.
p3. Find the number of integers

n

n -2016

n^2 -2016

a

divides an integer

b

if there exists a unique integer

k

ak = b

.
p4. The points

A(-16, 256)

B(20, 400)

lie on the parabola

y = x^2

. There exists a point

C(a,a^2)

on the parabola

y = x^2

such that there exists a point

D

on the parabola

y = -x^2

ACBD

is a parallelogram. Find

a

F_0

is a unit square. To create figure

F_1

, divide each side of the square into equal fifths and add two new squares with sidelength

\frac15

to each side, with one of their sides on one of the sides of the larger square. To create figure

F_{k+1}

F_k

, repeat this same process for each open side of the smallest squares created in

F_n

A_n

F_n

\lim_{n\to \infty} A_n

. https://cdn.artofproblemsolving.com/attachments/8/9/85b764acba2a548ecc61e9ffc29aacf24b4647.png
p6. For a prime

p

S_p

be the set of nonnegative integers

n

p

for which there exists a nonnegative integer

k

2016^k -n

is divisible by

p

. Find the sum of all

p

p

does not divide the sum of the elements of

S_p

.
p7. Trapezoid

ABCD

AB \parallel CD

AD = AB = BC

\gamma

\omega

are inscribed in the trapezoid such that circle

\gamma

CD

AB

AD

\omega

CD

AB

BC

\gamma

\omega

are externally tangent to each other, find

AB

x, y, z

be real numbers such that

(x+y)^2+(y+z)^2+(z+x)^2 = 1

. Over all triples

(x, y, z)

, find the maximum possible value of

y -z

\vartriangle ABC

has sidelengths

AB = 13

BC = 14

CA = 15

P

be a point on segment

BC

\frac{BP}{CP} = 3

I_1

I_2

be the incenters of triangles

\vartriangle ABP

\vartriangle ACP

. Suppose that the circumcircle of

\vartriangle I_1PI_2

intersects segment

AP

for a second time at a point

X \ne P

. Find the length of segment

AX

1 \le i \le 9

, let Ai be the answer to problem i from this section. Let

(i_1,i_2,... ,i_9)

be a permutation of

(1, 2,... , 9)

A_{i_1} < A_{i_2} < ... < A_{i_9}

i_j

, put the number

i_j

in the box which is in the

j

th row from the top and the

j

th column from the left of the

9\times 9

grid in the bonus section of the answer sheet. Then, fill in the rest of the squares with digits

1, 2,... , 9

\bullet

3\times 3

grid contains exactly one of each digit from

1

9

\bullet

each row of the

9\times 9

grid contains exactly one of each digit from

1

9

\bullet

each column of the

9\times 9

grid contains exactly one of each digit from

1

9

.
PS. You had better use hide for answers.

1

2016 LMT Individual #25

ABCD

be a trapezoid with

AB\parallel DC

M

be the midpoint of

CD

AD\perp CD, AC\perp BM,

BC\perp BD

\frac{AB}{CD}

.
Proposed by Nathan Ramesh

1

2016 LMT Individual #24

S

be a set consisting of all positive integers less than or equal to

100

P

S

such that there do not exist two elements

x,y\in P

x=2y

. Find the maximum possible number of elements of

P

.
Proposed by Nathan Ramesh

1

2016 LMT Individual #23

Call a positive integer

n\geq 2

junk if there exist two distinct

n

digit binary strings

a_1a_2\cdots a_n

b_1b_2\cdots b_n

a_1+a_2=b_1+b_2,

a_{i-1}+a_i+a_{i+1}=b_{i-1}+b_i+b_{i+1}

2\leq i\leq n-1,

a_{n-1}+a_n=b_{n-1}+b_n

.
Find the number of junk positive integers less than or equal to

2016

.
Proposed by Nathan Ramesh

1

2016 LMT Individual #22

Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives.
Proposed by Nathan Ramesh

1

2016 LMT Individual #21

S

be the set of positive integers

n

3\cdot \varphi (n)=n,

\varphi (n)

is the number of positive integers

k\leq n

\gcd (k, n)=1

\sum_{n\in S} \, \frac{1}{n}.

Proposed by Nathan Ramesh

1

2016 LMT Individual #20

Find the number of partitions of the set

\{1,2,3,\cdots ,11,12\}

into three nonempty subsets such that no subset has two elements which differ by

1

.
Proposed by Nathan Ramesh

1

2016 LMT Individual #19

Find the shortest distance between the graphs of

y=x^2+5

x=y^2+5

.
Proposed by Nathan Ramesh

1

2016 LMT Individual #18

\triangle ABC

be a triangle with

AB=5, BC=6, CA=7

P

is a point inside

\triangle ABC

\triangle BPA\sim \triangle APC

AP

BC

X

\frac{BX}{CX}

.
Proposed by Nathan Ramesh

1

2016 LMT Individual #17

Find the minimum possible value of

\left\lfloor \dfrac{a+b}{c}\right\rfloor+2 \left\lfloor \dfrac{b+c}{a}\right\rfloor+ \left\lfloor \dfrac{c+a}{b}\right\rfloor

a,b,c

are the sidelengths of a triangle.
Proposed by Nathan Ramesh

1

2016 LMT Individual #16

N

be the number of functions

f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}

that have the property that for

1\leq x\leq 5

it is true that

f(f(x))=x

N

can be written in the form

5^a\cdot b

for positive integers

a

b

b

not divisible by

5

a+b

.
Proposed by Nathan Ramesh

2

2016 LMT Individual #15

For nonnegative integers

n

f(n)

be the number of digits of

n

that are at least

5

g(n)=3^{f(n)}

\sum_{i=1}^{1000} g(i).

Proposed by Nathan Ramesh

2016 LMT Theme #15

A round robin tournament is held with

2016

participants. Each player plays each other player once and exactly one game results in a tie. Let

W

be the sum of the squares of each team's win total and let

L

be the sum of the squares of each team's loss total. Find the maximum possible value of

W-L

.
Proposed by Matthew Weiss

2

2016 LMT Individual #14

P

Q

AC

AB

, respectively, of triangle

\triangle ABC

PB=PC

PQ\perp AB

\frac{AQ}{QB}=\frac{AP}{PB}.

\angle CBA

, in degrees.
Proposed by Nathan Ramesh

2016 LMT Theme #14

A ladder style tournament is held with

2016

participants. The players begin seeded

1,2,\cdots 2016

. Each round, the lowest remaining seeded player plays the second lowest remaining seeded player, and the loser of the game gets eliminated from the tournament. After

2015

rounds, one player remains who wins the tournament. If each player has probability of

\tfrac{1}{2}

to win any game, then the probability that the winner of the tournament began with an even seed can be expressed has

\tfrac{p}{q}

for coprime positive integers

p

q

. Find the remainder when

p

1000

.
Proposed by Nathan Ramesh

2

2016 LMT Individual #13

Find the area enclosed by the graph of

|x|+|2y|=12

.
Proposed by Nathan Ramesh

2016 LMT Theme #13

A round robin tournament is held with

2016

participants. Each player plays each other player once and no games result in ties. We say a pair of players

A

B

is a dominant pair if all other players either defeat

A

B

or are defeated by both

A

B

. Find the maximum number dominant pairs.
Proposed by Nathan Ramesh

2

2016 LMT Individual #12

Two lines intersect inside a unit square, splitting it into four regions. Find the maximum product of the areas of the four regions.
Proposed by Nathan Ramesh

2016 LMT Theme #12

A round robin tournament is held with

2016

participants. Each round, after seeing the results from the previous round, the tournament organizer chooses two players to play a game with each other that will result in a win for one of the players and a loss for the other. The tournament organizer wants each person to have a different total number of wins at the end of

k

rounds. Find the minimum possible value of

k

for which this can always be guaranteed.
Proposed by Nathan Ramesh

2

2016 LMT Individual #11

Find all ordered triples

(a,b,c)

of real numbers such that

\begin{cases} a+b=c,\\ a^2+b^2=c^2-c-6,\\ a^3+b^3 = c^3-2c^2-5c. \\ \end{cases}

Proposed by Evan Fang

2016 LMT Theme #11

A single elimination tournament is held with

2016

participants. In each round, players pair up to play games with each other. There are no ties, and if there are an odd number of players remaining before a round then one person will get a bye for the round. Find the minimum number of rounds needed to determine a winner.
Proposed by Nathan Ramesh

2

2016 LMT Individual #10

There are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings such that there is an odd number of buildings between each pair of buildings in the subset?
Proposed by Yiming Zheng

2016 LMT Theme #10

S=\{1,2,3,4,5,6\}.

Find the number of bijective functions

f:S\rightarrow S

for which there exist exactly

6

bijective functions

g:S\rightarrow S

f(g(x))=g(f(x))

x\in S

.
Proposed by Nathan Ramesh

2

2016 LMT Individual #9

An acute triangle has area

84

42

, with each side being at least

10

units long. Let

S

be the set of points that are within

5

units of some vertex of the triangle. What fraction of the area of

S

lies outside the triangle?
Proposed by Nathan Ramesh

2016 LMT Theme #9

f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}

is called good if the function

g(n)=|f(n)-n|

is injective. Furthermore, a good function

f

is called excellent if there exists another good function

f'

f(n)-f'(n)

is nonzero for exactly one value of

n

N

be the number of good functions that are not excellent. Find the remainder when

N

1000

.
Proposed by Nathan Ramesh

2

2016 LMT Individual #8

How many lattice points

P

in or on the circle

x^2+y^2=25

have the property that there exists a unique line with rational slope through

P

that divides the circle into two parts with equal areas?
Proposed by Nathan Ramesh

2016 LMT Theme #8

Consider the function

f:[0,1)\rightarrow[0,1)

f(x)=2x-\lfloor 2x\rfloor

\lfloor 2x\rfloor

is the greatest integer less than or equal to

2x

. Find the sum of all values of

x

f^{17}(x)=x.

Proposed by Matthew Weiss

2

2016 LMT Individual #7

Compute the product of the three smallest prime factors of

21!\cdot 14!+21!\cdot 21+14!\cdot 14+21\cdot 14.

Proposed by Daniel Liu

2016 LMT Theme #7

R(x)

be a function that takes a natural number as input and returns a rectangle.

R(1)

is known to have integer side lengths. Let

p(x)

be the perimeter of

R(x)

a(x)

R(x)

p(x+5)=6 p(x)

x

in the domain of

R

a(x+2)=12a(x)

x> 6

in the domain of

R

x \leq 6

a(x+1)=a(x)+2

p(16)=1296

, and let the side lengths of

R(11)

a

b

a\leq b

. Find the ordered pair

(a,b)

.
Proposed by Matthew Weiss

2

2016 LMT Individual #6

A positive integer is called cool if it can be expressed in the form

a!\cdot b!+315

a,b

are positive integers. For example,

1!\cdot 1!+315=316

is a cool number. Find the sum of all cool numbers that are also prime numbers.
Proposed by Evan Fang

2016 LMT Theme #6

How many functions

f:\{1,2,3,4\}\rightarrow \{1,2,3\}

are surjective?
Proposed by Nathan Ramesh

2

2016 LMT Individual #5

An isosceles triangle has angles of

50^\circ,x^\circ,

y^\circ

. Find the maximum possible value of

x-y

.
Proposed by Nathan Ramesh

2016 LMT Theme #5

Pixar Prison, for Pixar villains, is shaped like a 600 foot by 1000 foot rectangle with a 300 foot by 500 foot rectangle removed from it, as shown below. The warden separates the prison into three congruent polygonal sections for villains from The Incredibles, Finding Nemo, and Cars. What is the perimeter of each of these sections? [asy] draw((0,0)--(0,6)--(10,6)--(10,0)--(8,0)--(8,3)--(3,3)--(3,0)--(0,0)); label("600", (1,3.5)); label("1000", (5.5,6.5)); label("300", (4,1.5)); label("500", (5.5,3.5)); label("300", (1.5,-0.5)); [/asy]
Proposed by Peter Rowley

2

2016 LMT Individual #4

A triangle has two sides of lengths

1984

2016

. Find the maximum possible area of the triangle.
Proposed by Nathan Ramesh

2016 LMT Theme #4

A male volcano is in the shape of a hollow cone with the point side up, but with everything above a height of 6 meters removed. The resulting shape has a bottom radius of 10 meters and a top radius of 7 meters, with a height of 6 meters. He sat above his bay, watching all the couples play. His lava grew and grew until he was half full of lava. Then, he erupted, lowering the height of the lava to 2 meters. What fraction of the lava remained in the volcano?
Proposed by Matthew Weiss

2

2016 LMT Individual #3

The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.
Proposed by Clive Chan

2016 LMT Theme #3

Geri plays chess against himself. White has a 5% chance of winning, Black has a 5% chance of winning, and there is a 90% chance of a draw. What is the expected number of games Geri will have to play against himself for one of the colors to win four times?
Proposed by Matthew Weiss

2

2016 LMT Individual #2

Mike rides a bike for

30

minutes, traveling

8

miles. He started riding at

20

miles per hour, but by the end of his journey he was only traveling at

10

miles per hour. What was his average speed, in miles per hour?
Proposed by Nathan Ramesh

2016 LMT Theme #2

Mater is confused and starts going around the track in the wrong direction. He can go around 7 times in an hour. Lightning and Chick start in the same place at Mater and at the same time, both going the correct direction. Lightning can go around 91 times per hour, while Chick can go around 84 times per hour. When Lightning passes Chick for the third time, how many times will he have passed Mater (if Lightning is passing Mater just as he passes Chick for the third time, count this as passing Mater)?
Proposed by Matthew Weiss

2

2016 LMT Individual #1

Find the ordered triple of natural numbers

(x,y,z)

x \le y \le z

x^x+y^y+z^z = 3382.

Proposed by Evan Fang

2016 LMT Theme #1

Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?
Proposed by Matthew Weiss