MathDB

1

2018 CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

(0, 0)

to

(a, b)

, where

a

and

b

are positive real numbers with

a < b

. The robots move at the same constant speed. However, the first robot can only travel in directions parallel to the lines

x = 0

or

y = 0

, while the second robot can only travel in directions parallel to the lines

y = x

or

y = -x

. Both robots take the shortest possible path to

(a, b)

and arrive at the same time. Find the ratio

\frac{a}{b}

.
p2. Suppose

x + \frac{1}{x} + y + \frac{1}{y} = 12

and

x^2 + \frac{1}{x^2} + y^2 + \frac{1}{y^2} = 70

. Compute

x^3 + \frac{1}{x^3} + y^3 + \frac{1}{y^3}

.
p3. Find the largest non-negative integer

a

such that

2^a

divides

3^{2^{2018}}+ 3.

p4. Suppose

z

and

w

are complex numbers, and

|z| = |w| = z \overline{w}+\overline{z}w = 1

. Find the largest possible value of

Re(z + w)

, the real part of

z + w

.
p5. Two people,

A

and

B

, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles.

A

and

B

each take turns making moves, with

A

making the first move. The last player able to make a move wins. Suppose that the three piles have

10

,

x

, and

30

matches. Find the largest value of

x

for which

A

does not have a winning strategy.
p6. Let

A_1A_2A_3A_4A_5A_6

be a regular hexagon with side length

1

. For

n = 1

,

...

,

6

, let

B_n

be a point on the segment

A_nA_{n+1}

chosen at random (where indices are taken mod

6

, so

A_7 = A_1

). Find the expected area of the hexagon

B_1B_2B_3B_4B_5B_6

.
p7. A termite sits at the point

(0, 0, 0)

, at the center of the octahedron

|x| + |y| + |z| \le 5

. The termite can only move a unit distance in either direction parallel to one of the

x

,

y

, or

z

axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of

5

steps, can the termite use to reach the surface of the octahedron?
p8. Let

P(x) = x^{4037} - 3 - 8 \cdot \sum^{2018}_{n=1}3^{n-1}x^n

Find the number of roots

z

of

P(x)

with

|z| > 1

, counting multiplicity.
p9. How many times does

01101

appear as a not necessarily contiguous substring of

0101010101010101

? (Stated another way, how many ways can we choose digits from the second string, such that when read in order, these digits read

01101

?)
p10. A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example,

28

is a perfect number because

1 + 2 + 4 + 7 + 14 = 28

. Let

n_i

denote the ith smallest perfect number. Define

f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}

(where

\sum_{i|n_x}

means we sum over all positive integers

i

that are divisors of

n_x

). Compute

f(2)

, given there are at least

50

perfect numbers.
p11. Let

O

be a circle with chord

AB

. The perpendicular bisector to

AB

is drawn, intersecting

O

at points

C

and

D

, and intersecting

AB

at the midpoint

E

. Finally, a circle

O'

with diameter

ED

is drawn, and intersects the chord

AD

at the point

F

. Given

EC = 12

, and

EF = 7

, compute the radius of

O

.
p12. Suppose

r

,

s

,

t

are the roots of the polynomial

x^3 - 2x + 3

. Find

\frac{1}{r^3 - 2}+\frac{1}{s^3 - 2}+\frac{1}{t^3 - 2}.

p13. Let

a_1

,

a_2

,...,

a_{14}

be points chosen independently at random from the interval

[0, 1]

. For

k = 1

,

2

,

...

,

7

, let

I_k

be the closed interval lying between

a_{2k-1}

and

a_{2k}

(from the smaller to the larger). What is the probability that the intersection of

I_1

,

I_2

,

...

,

I_7

is nonempty?
p14. Consider all triangles

\vartriangle ABC

with area

144\sqrt3

such that

\frac{\sin A \sin B \sin C}{ \sin A + \sin B + \sin C}=\frac14.

Over all such triangles

ABC

, what is the smallest possible perimeter?
p15. Let

N

be the number of sequences

(x_1,x_2,..., x_{2018})

of elements of

\{1, 2,..., 2019\}

, not necessarily distinct, such that

x_1 + x_2 + ...+ x_{2018}

is divisible by

2018

. Find the last three digits of

N

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

10

1

2018 Fall Team #10

Let

ABC

be a triangle such that

AB = 13

,

BC = 14

,

AC = 15

. Let

M

be the midpoint of

BC

and define

P \ne B

to be a point on the circumcircle of

ABC

such that

BP \perp PM

. Furthermore, let

H

be the orthocenter of

ABM

and define

Q

to be the intersection of

BP

and

AC

. If

R

is a point on

HQ

such that

RB \perp BC

, find the length of

RB

.

9

1

2018 Fall Team #9

Say that a function

f : \{1, 2, . . . , 1001\} \to Z

is almost polynomial if there is a polynomial

p(x) = a_{200}x^{200} +... + a_1x + a_0

such that each an is an integer with

|a_n| \le 201

, and such that

|f(x) - p(x)| \le 1

for all

x \in \{1, 2, . . . , 1001\}

. Let

N

be the number of almost polynomial functions. Compute the remainder upon dividing

N

by

199

.

8

1

2018 Fall Team #8

Find the largest positive integer

n

that cannot be written as

n = 20a + 28b + 35c

for nonnegative integers

a, b

, and

c

.

7

1

2018 Fall Team #7

For a positive number

n

, let

g(n)

be the product of all

1 \le k \le n

such that gcd

(k, n) =1

, and say that

n > 1

is reckless if

n

is odd and

g(n) \equiv -1

(mod

n

). Find the number of reckless numbers less than

50

.

6

1

2018 Fall Team #6

Karina has a polynomial

p_1(x) = x^2 + x + k

, where

k

is an integer. Noticing that

p_1

has integer roots, she forms a new polynomial

p_2(x) = x^2 + a_1x + b_1

, where

a_1

and

b_1

are the roots of

p_1

and

a_1 \ge b_1

. The polynomial

p_2

also has integer roots, so she forms a new polynomial

p_3(x) = x^2 + a_2x + b_2

, where

a_2

and

b_2

are the roots of

p_2

and

a_2 \ge b_2

. She continues this process until she reaches

p_7(x)

and finds that it does not have integer roots. What is the largest possible value of

k

?

5

2

2018 Fall Team #5

Let

\vartriangle ABC

be a right triangle such that

AB = 3

,

BC = 4

,

AC = 5

. Let point

D

be on

AC

such that the incircles of

\vartriangle ABD

and

\vartriangle BCD

are mutually tangent. Find the length of

BD

.

2018 CHMMC Tiebreaker 5 - roots of p(x) = 2x^5 - 3x^3 + 2x -7

Let

a,b, c, d,e

be the roots of

p(x) = 2x^5 - 3x^3 + 2x -7

. Find the value of

(a^3 - 1)(b^3 - 1)(c^3 - 1)(d^3 - 1)(e^3 - 1).

4

2

2018 Fall Team #4

If Percy rolls a fair six-sided die until he rolls a

5

, what is his expected number of rolls, given that all of his rolls are prime?

2018 CHMMC Tiebreaker 4 - f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4

Find the sum of the real roots of

f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4

.

3

2

2018 Fall Team #3

Let

p

be the third-smallest prime number greater than

5

such that:

\bullet

2p + 1

is prime, and

\bullet

5^p \not\equiv 1

(mod

2p + 1

). Find

p

.

2018 CHMMC Tiebreaker 3 - sum ( 1/(n^2 + 3n) - 1/(n^2 + 3n + 2) )

Compute

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

2

2018 Fall Team #2

Compute the sum

\sum^{200}_{n=1}\frac{1}{n(n+1)(n+2)}

.

2018 CHMMC Tiebreaker 2 - cat is tied to one corner of base of a tower

A cat is tied to one corner of the base of a tower. The base forms an equilateral triangle of side length

4

m, and the cat is tied with a leash of length

8

m. Let

A

be the area of the region accessible to the cat. If we write

A = \frac{m}{n} k - \sqrt{\ell}

, where

m,n, k, \ell

are positive integers such that

m

and

n

are relatively prime, and

\ell

is squarefree, what is the value of

m + n + k + \ell

?

1

2

2018 Fall Team #1

Anita plays the following single-player game: She is given a circle in the plane. The center of this circle and some point on the circle are designated “known points”. Now she makes a series of moves, each of which takes one of the following forms: (i) She draws a line (infinite in both directions) between two “known points”; or (ii) She draws a circle whose center is a “known point” and which intersects another “known point”. Once she makes a move, all intersections between her new line/circle and existing lines/circles become “known points”, unless the new/line circle is identical to an existing one. In other words, Anita is making a ruler-and-compass construction, starting from a circle. What is the smallest number of moves that Anita can use to construct a drawing containing an equilateral triangle inscribed in the original circle?

2018 CHMMC Tiebreaker 1 - infinitely many lily numbered pads

A large pond contains infinitely many lily pads labelled

1

,

2

,

3

,

...

, placed in a line, where for each

k

, lily pad

k + 1

is one unit to the right of lily pad

k

. A frog starts at lily pad

100

. Each minute, if the frog is at lily pad

n

, it hops to lily pad

n + 1

with probability

\frac{n-1}{n}

, and hops all the way back to lily pad

1

with probability

\frac{1}{n}

. Let

N

be the position of the frog after

1000

minutes. What is the expected value of

N

?

2018 CHMMC (Fall)

Subcontests

2018 CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

2018 Fall Team #10

2018 Fall Team #9

2018 Fall Team #8

2018 Fall Team #7

2018 Fall Team #6

2018 Fall Team #5

2018 CHMMC Tiebreaker 5 - roots of p(x) = 2x^5 - 3x^3 + 2x -7

2018 Fall Team #4

2018 CHMMC Tiebreaker 4 - f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4

2018 Fall Team #3

2018 CHMMC Tiebreaker 3 - sum ( 1/(n^2 + 3n) - 1/(n^2 + 3n + 2) )

2018 Fall Team #2

2018 CHMMC Tiebreaker 2 - cat is tied to one corner of base of a tower

2018 Fall Team #1

2018 CHMMC Tiebreaker 1 - infinitely many lily numbered pads