MathDB

2

Kathy's Cards

5

red cards and

5

green cards. She shuffles the

10

cards and lays out

5

of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is

\tfrac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m + n

.

Base Ten Satisfies This Condition

Find the sum of all positive integers

b<1000

such that the base-

b

integer

36_b

is a perfect square and the base-

b

integer

27_b

is a perfect cube.

5

2

What sound does a drowning mathematician make? logloglog...

For each ordered pair of real numbers

(x,y)

satisfying

\log_2(2x+y) = \log_4(x^2+xy+7y^2)

there is a real number

K

such that

\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).

Find the product of all possible values of

K

.

Complex Number Sum and Products

Suppose that

x

,

y

, and

z

are complex numbers such that

xy = -80-320i

,

yz = 60

, and

zx = -96+24i

, where

i = \sqrt{-1}

. Then there are real numbers

a

and

b

such that

x+y+z = a+bi

. Find

a^2 + b^2

.

15

2

Dam David, back at it again with the late geo II

David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals,

A

,

B

,

C

, which can each be inscribed in a circle with radius

1

. Let

\varphi_A

denote the measure of the acute angle made by the diagonals of quadrilateral

A

, and define

\varphi_B

and

\varphi_C

similarly. Suppose that

\sin\varphi_A=\frac{2}{3}

,

\sin\varphi_B=\frac{3}{5}

, and

\sin\varphi_C=\frac{6}{7}

. All three quadrilaterals have the same area

K

, which can be written in the form

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

Inequality FE

Find the number of functions

f

from

\{0,1,2,3,4,5,6\}

to the integers such that

f(0)=0, f(6)=12

, and

|x-y| \le |f(x)-f(y)| \le 3 |x-y|

for all

x

and

y

in

\{0,1,2,3,4,5,6\}

.

13

2

Dam David, back at it again with the late geo I

Let

\triangle ABC

have side lengths

AB=30

,

BC=32

, and

AC=34

. Point

X

lies in the interior of

\overline{BC}

, and points

I_1

and

I_2

are the incenters of

\triangle ABX

and

\triangle ACX

, respectively. Find the minimum possible area of

\triangle AI_1I_2

as

X

varies along

\overline{BC}

.

Die Problem

Misha rolls a standard, fair six-sided die until she rolls

1

-

2

-

3

in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is

\tfrac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

10

2

The bash she tells you not to worry about

The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point

A

. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path

AJABCHCHIJA

, which has

10

steps. Let

n

be the number of paths with

15

steps that begin and end at point

A

. Find the remainder when

n

is divided by

1000

A

", (1-r) * dir( 90), -dir( 90)); label("

B

", (1-r) * dir(162), -dir(162)); label("

C

", (1-r) * dir(234), -dir(234)); label("

D

", (1-r) * dir(306), -dir(306)); label("

E

", (1-r) * dir(378), -dir(378)); label("

F

", (2+r) * dir(378), dir(378)); label("

G

", (2+r) * dir(306), dir(306)); label("

H

", (2+r) * dir(234), dir(234)); label("

I

", (2+r) * dir(162), dir(162)); label("

J

", (2+r) * dir( 90), dir( 90)); [/asy]

It is the Very Model of A Modern Major Function Nest

Find the number of functions

f(x)

from

\{1,2,3,4,5\}

to

\{1,2,3,4,5\}

that satisfy

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

for all

x

in

\{1,2,3,4,5\}

.

12

2

102 Combo

For every subset

T

of

U = \{ 1,2,3,\ldots,18 \}

, let

s(T)

be the sum of the elements of

T

, with

s(\emptyset)

defined to be

0

. If

T

is chosen at random among all subsets of

U

, the probability that

s(T)

is divisible by

3

is

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m

.

Sum of Areas is Equal

Let

ABCD

be a convex quadrilateral with

AB=CD=10

,

BC=14

, and

AD=2\sqrt{65}

. Assume that the diagonals of

ABCD

intersect at point

P

, and that the sum of the areas of

\triangle APB

and

\triangle CPD

equals the sum of the areas of

\triangle BPC

and

\triangle APD

. Find the area of quadrilateral

ABCD

.

8

2

Equiangular Hexagons are not Tangential

Let

ABCDEF

be an equiangular hexagon such that

AB=6, BC=8, CD=10

, and

DE=12

. Denote

d

the diameter of the largest circle that fits inside the hexagon. Find

d^2

.

Dynamic Programming by Hand Again

A frog is positioned at the origin in the coordinate plane. From the point

(x,y)

, the frog can jump to any of the points

(x+1, y), (x+2, y), (x, y+1),

or

(x, y+2)

. Find the number of distinct sequences of jumps in which the frog begins at

(0,0)

and ends at

(4,4)

.

4

2

Do u know DA? click click

In

\triangle ABC, AB = AC = 10

and

BC = 12

. Point

D

lies strictly between

A

and

B

on

\overline{AB}

and point

E

lies strictly between

A

and

C

on

\overline{AC}

so that

AD = DE = EC

. Then

AD

can be expressed in the form

\tfrac{p}{q}

, where

p

and

q

are relatively prime positive integers. Find

p + q

.

Rearranging Names: Caroline is Cornelia

In equiangular octagon

CAROLINE

,

CA = RO = LI = NE = \sqrt{2}

and

AR = OL = IN = EC = 1

. The self-intersecting octagon

CORNELIA

encloses six non-overlapping triangular regions. Let

K

be the area enclosed by

CORNELIA

, that is, that total area of the six triangular regions. Then

K=\tfrac{a}{b}

where

a

and

b

are relatively prime positive integers. Find

a + b

.

7

2

Hexagonal Prison

A right hexagonal prism has height

2

. The bases are regular hexagons with side length

1

. Any

3

of the

12

vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).

Rational Geometry

Triangle

ABC

has sides

AB=9,BC = 5\sqrt{3},

and

AC=12

. Points

A=P_0, P_1, P_2, \dots, P_{2450} = B

are on segment

\overline{AB}

with

P_k

between

P_{k-1}

and

P_{k+1}

for

k=1,2,\dots,2449

, and points

A=Q_0, Q_1, Q_2, \dots ,Q_{2450} = C

for

k=1,2,\dots,2449

. Furthermore, each segment

\overline{P_kQ_k}, k=1,2,\dots,2449

, is parallel to

\overline{BC}

. The segments cut the triangle into

2450

regions, consisting of

2449

trapezoids and

1

triangle. Each of the

2450

regions have the same area. Find the number of segments

\overline{P_kQ_k}, k=1,2 ,\dots,2450

, that have rational length.

14

2

Dynamic Programming by Hand

Let

SP_1P_2P_3EP_4P_5

be a heptagon. A frog starts jumping at vertex

S

. From any vertex of the heptagon except

E

, the frog may jump to either of the two adjacent vertices. When it reaches vertex

E

, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than

12

jumps that end at

E

.

Projective Fun

The incircle of

\omega

of

\triangle ABC

is tangent to

\overline{BC}

at

X

. Let

Y \neq X

be the other intersection of

\overline{AX}

with

\omega

. Points

P

and

Q

lie on

\overline{AB}

and

\overline{AC}

, respectively, so that

\overline{PQ}

is tangent to

\omega

at

Y

. Assume that

AP=3, PB = 4, AC=8

, and

AQ = \tfrac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

11

2

Bashtastic

Find the least positive integer

n

such that when

3^n

is written in base

143

, its two right-most digits in base

143

are

01

.

Subtract from 720 You Must

Find the number of permutations of

1,2,3,4,5,6

such that for each

k

with

1\leq k\leq 5

, at least one of the first

k

terms of the permutation is greater than

k

.

6

2

Complex Numbers are Real

Let

N

be the number of complex numbers

z

with the properties that

|z|=1

and

z^{6!}-z^{5!}

is a real number. Find the remainder when

N

is divided by

1000

.

Real Roots of a Quartic

A real number

a

is chosen randomly and uniformly from the interval

[-20, 18]

. The probability that the roots of the polynomial

x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2

are all real can be written in the form

\tfrac{m}{n}

, where

m

and

n

are relatively prime positive integers. Find

m+n

.

2

I'm basic

The number

n

can be written in base

14

as

\underline{a}

\underline{b}

\underline{c}

, can be written in base

15

as

\underline{a}

\underline{c}

\underline{b}

, and can be written in base

6

as

\underline{a}

\underline{c}

\underline{a}

\underline{c}

, where

a > 0

. Find the base-

10

representation of

n

.

Sequences Modulo 11: Indexblocked

Let

a_0 = 2

,

a_1 = 5

, and

a_2 = 8

, and for

n>2

define

a_n

recursively to be the remainder when

4(a_{n-1} + a_{n-2} + a_{n-3})

is divided by

11

. Find

a_{2018}\cdot a_{2020}\cdot a_{2022}

.

1

2

Counting is hard

Let

S

be the number of ordered pairs of integers

(a,b)

with

1 \leq a \leq 100

and

b \geq 0

such that the polynomial

x^2+ax+b

can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when

S

is divided by

1000

.

Ina, Eve, and Paul

Points

A

,

B

, and

C

lie in that order along a straight path where the distance from

A

to

C

is

1800

meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at

A

and running toward

C

, Paul starting at

B

and running toward

C

, and Eve starting at

C

and running toward

A

. When Paul meets Eve, he turns around and runs toward

A

. Paul and Ina both arrive at

B

at the same time. Find the number of meters from

A

to

B

.

9

2

Subset Formation

Find the number of four-element subsets of

\{1,2,3,4,\dots, 20\}

with the property that two distinct elements of a subset have a sum of

16

, and two distinct elements of a subset have a sum of

24

. For example,

\{3,5,13,19\}

and

\{6,10,20,18\}

are two such subsets.

Homothety vs. Shoelace

Octagon

ABCDEFGH

with side lengths

AB = CD = EF = GH = 10

and

BC= DE = FG = HA = 11

is formed by removing four

6-8-10

triangles from the corners of a

23\times 27

rectangle with side

\overline{AH}

on a short side of the rectangle, as shown. Let

J

be the midpoint of

\overline{HA}

, and partition the octagon into

7

triangles by drawing segments

\overline{JB}

,

\overline{JC}

,

\overline{JD}

,

\overline{JE}

,

\overline{JF}

, and

\overline{JG}

. Find the area of the convex polygon whose vertices are the centroids of these

7

triangles.
[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label("

A

", A, W*r); label("

B

", B, S*r); label("

C

", C, S*r); label("

D

", D, E*r); label("

E

", EE, E*r); label("

F

", F, N*r); label("

G

", G, N*r); label("

H

", H, W*r); label("

J

", J, W*r); [/asy]

2018 AIME Problems

Subcontests

Kathy's Cards

Base Ten Satisfies This Condition

What sound does a drowning mathematician make? logloglog...

Complex Number Sum and Products

Dam David, back at it again with the late geo II

Inequality FE

Dam David, back at it again with the late geo I

Die Problem

The bash she tells you not to worry about

It is the Very Model of A Modern Major Function Nest

102 Combo

Sum of Areas is Equal

Equiangular Hexagons are not Tangential

Dynamic Programming by Hand Again

Do u know DA? click click

Rearranging Names: Caroline is Cornelia

Hexagonal Prison

Rational Geometry

Dynamic Programming by Hand

Projective Fun

Bashtastic

Subtract from 720 You Must

Complex Numbers are Real

Real Roots of a Quartic

I'm basic

Sequences Modulo 11: Indexblocked

Counting is hard

Ina, Eve, and Paul

Subset Formation

Homothety vs. Shoelace

2018 AIME Problems

Subcontests

Kathy's Cards

Base Ten Satisfies This Condition

What sound does a drowning mathematician make? logloglog...

Complex Number Sum and Products

Dam David, back at it again with the late geo II

Inequality FE

Dam David, back at it again with the late geo I

Die Problem

The bash she tells you not to worry about

It is the Very Model of A Modern Major Function Nest

102 Combo

Sum of Areas is Equal

Equiangular Hexagons are not Tangential

Dynamic Programming by Hand Again

Do u know DA? *click click*

Rearranging Names: Caroline is Cornelia

Hexagonal Prison

Rational Geometry

Dynamic Programming by Hand

Projective Fun

Bashtastic

Subtract from 720 You Must

Complex Numbers are Real

Real Roots of a Quartic

I'm basic

Sequences Modulo 11: Indexblocked

Counting is hard

Ina, Eve, and Paul

Subset Formation

Homothety vs. Shoelace

Do u know DA? click click