2019 CHMMC (Fall)

Part of CHMMC problems

Subcontests

(11)

1

2019 CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

p1. Consider a cube with side length

2

. Take any one of its vertices and consider the three midpoints of the three edges emanating from that vertex. What is the distance from that vertex to the plane formed by those three midpoints?
p2. Digits

H

M

C

satisfy the following relations where

\overline{ABC}

denotes the number whose digits in base

10

A

B

C

\overline{H}\times \overline{H} = \overline{M}\times \overline{C} + 1

\overline{HH}\times \overline{H} = \overline{MC}\times \overline{C} + 1

\overline{HHH}\times \overline{H} = \overline{MCC}\times \overline{C} + 1

\overline{HMC}

.
p3. Two players play the following game on a table with fair two-sided coins. The first player starts with one, two, or three coins on the table, each with equal probability. On each turn, the player flips all the coins on the table and counts how many coins land heads up. If this number is odd, a coin is removed from the table. If this number is even, a coin is added to the table. A player wins when he/she removes the last coin on the table. Suppose the game ends. What is the probability that the first player wins?
p4. Cyclic quadrilateral

[BLUE]

\angle E

R

be a point not in

[BLUE]

[BLUR] =[BLUE]

\angle ELB = 45^o

\overline{EU} = \overline{UR}

\angle RUE

.
p5. There are two tracks in the

x, y

plane, defined by the equations

y =\sqrt{3 - x^2}\,\,\, \text{and} \,\,\,y =\sqrt{4- x^2}

A baton of length

1

has one end attached to each track and is allowed to move freely, but no end may be picked up or go past the end of either track. What is the maximum area the baton can sweep out?
p6. For integers

1 \le a \le 2

1 \le b \le 10

1 \le c \le 12

1 \le d \le 18

f(a, b, c, d)

be the unique integer between

0

8150

inclusive that leaves a remainder of a when divided by

3

, a remainder of

b

when divided by

11

, a remainder of

c

when divided by

13

, and a remainder of

d

when divided by

19

\sum_{a+b+c+d=23}f(a, b, c, d).

\cos ( \theta)

\sum^{\infty}_{n=0} \frac{ \cos (n\theta)}{3^n} = 1.

p8. How many solutions does this equation

\left(\frac{a+b}{2}\right)^2=\left(\frac{b+c}{2019}\right)^2

have in positive integers

a, b, c

that are all less than

2019^2

?
p9. Consider a square grid with vertices labeled

1, 2, 3, 4

clockwise in that order. Fred the frog is jumping between vertices, with the following rules: he starts at the vertex label

1

, and at any given vertex he jumps to the vertex diagonally across from him with probability

\frac12

and the vertices adjacent to him each with probability

\frac14

2019

jumps, suppose the probability that the sum of the labels on the last two vertices he has visited is

3

can be written as

2^{-m} -2^{-n}

for positive integers

m,n

m + n

.
p10. The base ten numeral system uses digits

0-9

and each place value corresponds to a power of

10

2019 = 2 \cdot 10^3 + 0 \cdot 10^2 + 1 \cdot 10^1 + 9 \cdot 10^0.

\phi =\frac{1 +\sqrt5}{2}

. We can define a similar numeral system, base , where we only use digits

0

1

, and each place value corresponds to a power of . For example, 11.01 = 1 \cdot \phi^1 + 1 \cdot \phi^0 + 0 \cdot \phi^{-1} + 1 \cdot \phi^{-2} Note that base representations are not unique, because, for example, 100_{\phi} = 11_{\phi}. Compute the base

\phi

representation of

7

with the fewest number of

1

ABC

be a triangle with

\angle BAC = 60^o

and with circumradius

1

G

be its centroid and

D

be the foot of the perpendicular from

A

BC

AG =\frac{\sqrt6}{3}

AD

f(a, b)

be a function with the following properties for all positive integers

a \ne b

f(1, 2) = f(2, 1)

f(a, b) + f(b, a) = 0

f(a + b, b) = f(b, a) + b

\sum^{2019}_{i=1} f(4^i - 1, 2^i) + f(4^i + 1, 2^i)

p13. You and your friends have been tasked with building a cardboard castle in the two-dimensional Cartesian plane. The castle is built by the following rules: 1. There is a tower of height

2^n

at the origin. 2. From towers of height

2^i \ge 2

, a wall of length

2^{i-1}

can be constructed between the aforementioned tower and a new tower of height

2^{i-1}

. Walls must be parallel to a coordinate axis, and each tower must be connected to at least one other tower by a wall. If one unit of tower height costs

\$9

and one unit of wall length costs

\$3

n = 1000

, how many distinct costs are there of castles that satisfy the above constraints? Two castles are distinct if there exists a tower or wall that is in one castle but not in the other.
p14. For

n

(a_1, a_2, ..., a_n)

0 \le a_i < n

i = 1, 2,..., n

a_1 \ne 0

(\overline{a_1a_2 ... a_n})_n

to be the number with digits

a_1

a_2

...

a_n

written in base

n

S_n = \{(a_1, a_2, a_3,..., a_n)| \,\,\, (n + 1)| (\overline{a_1a_2 ... a_n})_n, a_1 \ge 1\}

n

-tuples such that

(\overline{a_1a_2 ... a_n})_n

is divisible by

n + 1

n > 1

n

|S_n| + 2019

P

be the set of polynomials with degree

2019

with leading coefficient

1

and non-leading coefficients from the set

C = \{-1, 0, 1\}

. For example, the function

f = x^{2019} - x^{42} + 1

P

, but the functions

f = x^{2020}

f = -x^{2019}

f = x^{2019} + 2x^{21}

P

.
Define a swap on a polynomial

f

to be changing a term

ax^n

bx^n

b \in C

and there are no terms with degree smaller than

n

with coefficients equal to

a

b

. For example, a swap from

x^{2019} + x^{17} - x^{15} + x^{10}

x^{2019} + x^{17} - x^{15} - x^{10}

would be valid, but the following swaps would not be valid:

x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019}

x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019} + x^3 + x^2

x^{2019} + x^2 + x + 1 \,\,\, \text{to} \,\,\, x^{2019} - x^2 - x - 1

B

be the set of polynomials in

P

where all non-leading terms have the same coefficient. There are

p

polynomials that can be reached from each element of

B

s

swaps, and there exist

0

polynomials that can be reached from each element of

B

s

p \cdot s

, expressing your answer as a prime factorization.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

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2019 Fall Team #8

Consider an infinite sequence of reals

x_1, x_2, x_3, ...

x_1 = 1

x_2 =\frac{2\sqrt3}{3}

and with the recursive relationship

n^2 (x_n - x_{n-1} - x_{n-2}) - n(3x_n + 2x_{n-1} + x_{n-2}) + (x_nx_{n-1}x_{n-2} + 2x_n) = 0.

x_{2019}

1

2019 Fall Team #10

n

players are playing a game. Each player has

n

tokens. Every turn, two players with at least one token are randomly selected. The player with less tokens gives one token to the player with more tokens. If both players have the same number of tokens, a coin flip decides which player receives a token and which player gives a token. The game ends when one player has all the tokens. If

n = 2019

, suppose the maximum number of turns the game could take to end can be written as

\frac{1}{d} (a \cdot 2019^3 + b \cdot 2019^2 + c \cdot 2019)

a, b, c, d

\frac{abc}{d}

1

2019 Fall Team #9

Consider a rectangle with length

6

4

. A rectangle with length

3

1

is placed inside the larger rectangle such that it is distance

1

from the bottom and leftmost sides of the larger rectangle. We randomly select one point from each side of the larger rectangle, and connect these

4

points to form a quadrilateral. What is the probability that the smaller rectangle is strictly contained within that quadrilateral?

1

2019 Fall Team #7

S

be the set of all positive integers

n

satisfying the following two conditions:

\bullet

n

is relatively prime to all positive integers less than or equal to

\frac{n}{6}

\bullet

2^n \equiv 4

n

What is the sum of all numbers in

S

1

2019 Fall Team #6

\prod^{2019}_{i=1} (2^{2^i}- 2^{2^{i-1}} + 1).

1

2019 Fall Team #5

A tournament has

5

players and is in round-robin format (each player plays each other exactly once). Each game has a

\frac13

chance of player

A

\frac13

chance of player

B

\frac13

chance of ending in a draw. The probability that at least one player draws all of their games can be written in simplest form as

\frac{m}{3^n}

m, n

are positive integers. Find

m + n

1

2019 Fall Team #4

\vartriangle ABC

be a triangle such that the area

[ABC] = 10

\tan (\angle ABC) = 5

. If the smallest possible value of

(\overline{AC})^2

can be expressed as

-a + b\sqrt{c}

for positive integers

a, b, c

a + b + c

1

2019 Fall Team #3

A frog is jumping between lattice points on the coordinate plane in the following way: On each jump, the frog randomly goes to one of the

8

closest lattice points to it, such that the frog never goes in the same direction on consecutive jumps. If the frog starts at

(20, 19)

(20, 20)

, then what is the expected value of the frog’s position after it jumps for an infinitely long time?

1

2019 Fall Team #2

Alex, Bob, Charlie, Daniel, and Ethan are five classmates. Some pairs of them are friends. How many possible ways are there for them to be friends such that everyone has at least one friend, and such that there is exactly one loop of friends among the five classmates? Note: friendship is two-way, so if person x is friends with person y then person y is friends with person

x

1

2019 Fall Team #1

ABC

be an equilateral triangle of side length

6

D, E

F

AB

BC

AC

respectively such that

AD = BE = CF = 2

O

be the circumcircle of

DEF

, that is, the circle that passes through points

D, E

F

. What is the area of the region inside triangle

ABC

but outside circle

O