MathDB

1

2021-22 Winter CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

p1. Fleming has a list of 8 mutually distinct integers between

90

to

99

, inclusive. Suppose that the list has median

94

, and that it contains an even number of odd integers. If Fleming reads the numbers in the list from smallest to largest, then determine the sixth number he reads.
p2. Find the number of ordered pairs

(x,y)

of three digit base-

10

positive integers such that

x-y

is a positive integer, and there are no borrows in the subtraction

x-y

. For example, the subtraction on the left has a borrow at the tens digit but not at the units digit, whereas the subtraction on the right has no borrows.
\begin{tabular}{ccccc} & 4 & 7 & 2 \\ - & 1 & 9 & 1\\ \hline & 2 & 8 & 1 \\ \end{tabular}\,\,\, \,\,\, \begin{tabular}{ccccc} & 3 & 7 & 9 \\ - & 2 & 6 & 3\\ \hline & 1 & 1 & 6 \\ \end{tabular}
p3. Evaluate

1 \cdot 2 \cdot 3-2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5- 4 \cdot 5 \cdot 6+ ... +2017 \cdot 2018 \cdot 2019 -2018 \cdot 2019 \cdot 2020+1010 \cdot 2019 \cdot 2021

p4. Find the number of ordered pairs of integers

(a,b)

such that

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

is an integer.
p5. Lin Lin has a

4\times 4

chessboard in which every square is initially empty. Every minute, she chooses a random square

C

on the chessboard, and places a pawn in

C

if it is empty. Then, regardless of whether

C

was previously empty or not, she then immediately places pawns in all empty squares a king’s move away from

C

. The expected number of minutes before the entire chessboard is occupied with pawns equals

\frac{m}{n}

for relatively prime positive integers

m

,

n

. Find

m+n

. A king’s move, in chess, is one square in any direction on the chessboard: horizontally, vertically, or diagonally.
p6. Let

P(x) = x^5-3x^4+2x^3-6x^2+7x+3

and

a_1,...,a_5

be the roots of

P(x)

. Compute

\sum^5_{k=1}(a^3_k -4a^2_k +a_k +6).

p7. Rectangle

AXCY

with a longer length of

11

and square

ABCD

share the same diagonal

\overline{AC}

. Assume

B

,

X

lie on the same side of

\overline{AC}

such that triangle

BXC

and square

ABCD

are non-overlapping. The maximum area of

BXC

across all such configurations equals

\frac{m}{n}

for relatively prime positive integers

m

,

n

. Compute

m+n

.
p8. Earl the electron is currently at

(0,0)

on the Cartesian plane and trying to reach his house at point

(4,4)

. Each second, he can do one of three actions: move one unit to the right, move one unit up, or teleport to the point that is the reflection of its current position across the line

y=x

. Earl cannot teleport in two consecutive seconds, and he stops taking actions once he reaches his house. Earl visits a chronologically ordered sequence of distinct points

(0,0)

,

...

,

(4,4)

due to his choice of actions. This is called an Earl-path. How many possible such Earl-paths are there?
p9. Let

P(x)

be a degree-

2022

polynomial with leading coefficient

1

and roots

\cos \left( \frac{2\pi k}{2023} \right)

for

k = 1

,

...

,

2022

(note

P(x)

may have repeated roots). If

P(1) =\frac{m}{n}

where

m

and

n

are relatively prime positive integers, then find the remainder when

m+n

is divided by

100

.
p10. A randomly shuffled standard deck of cards has

52

cards,

13

of each of the four suits. There are

4

Aces and

4

Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the first King and before the first Ace is

\frac{m}{n}

where

m

and

n

are relatively prime positive integers. Find

m+n

.
p11. The following picture shows a beam of light (dashed line) reflecting off a mirror (solid line). The angle of incidence is marked by the shaded angle; the angle of reflection is marked by the unshaded angle. https://cdn.artofproblemsolving.com/attachments/9/d/d58086e5cdef12fbc27d0053532bea76cc50fd.png The sides of a unit square

ABCD

are magically distorted mirrors such that whenever a light beam hits any of the mirrors, the measure of the angle of incidence between the light beam and the mirror is a positive real constant

q

degrees greater than the measure of the angle of reflection between the light beam and the mirror. A light beam emanating from

A

strikes

\overline{CD}

at

W_1

such that

2DW_1 =CW_1

, reflects off of

\overline{CD}

and then strikes

\overline{BC}

at

W_2

such that

2CW_2 = BW_2

, reflects off of

\overline{BC}

, etc. To this end, denote

W_i

the

i

-th point at which the light beam strikes

ABCD

. As

i

grows large, the area of

W_iW_{i+1}W_{i+2}W_{i+3}

approaches

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. Compute

m+n

.
p12. For any positive integer

m

, define

\phi (m)

the number of positive integers

k \le m

such that

k

and

m

are relatively prime. Find the smallest positive integer

N

such that

\sqrt{ \phi (n) }\ge 22

for any integer

n \ge N

.
p13. Let

n

be a fixed positive integer, and let

\{a_k\}

and

\{b_k\}

be sequences defined recursively by

a_1 = b_1 = n^{-1}

a_j = j(n- j+1)a_{j-1}\,\,\, , \,\,\, j > 1

b_j = nj^2b_{j-1}+a_j\,\,\, , \,\,\, j > 1

When

n = 2021

, then

a_{2021} +b_{2021} = m \cdot 2017^2

for some positive integer

m

. Find the remainder when

m

is divided by

2017

.
p14. Consider the quadratic polynomial

g(x) = x^2 +x+1020100

. A positive odd integer

n

is called

g

-friendly if and only if there exists an integer

m

such that

n

divides

2 \cdot g(m)+2021

. Find the number of

g

-friendly positive odd integers less than

100

.
p15. Let

ABC

be a triangle with

AB < AC

, inscribed in a circle with radius

1

and center

O

. Let

H

be the intersection of the altitudes of

ABC

. Let lines

\overline{OH}

,

\overline{BC}

intersect at

T

. Suppose there is a circle passing through

B

,

H

,

O

,

C

. Given

\cos (\angle ABC-\angle BCA) = \frac{11}{32}

, then

TO = \frac{m\sqrt{p}}{n}

for relatively prime positive integers

m

,

n

and squarefree positive integer

p

. Find

m+n+ p

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

6

2

2021-22 Winter Team #6

There is a unique degree-

10

monic polynomial with integer coefficients

f(x)

such that

f \left( \sum^9_{j=0}\sqrt[10]{2021^j}\right)= 0.

Find the remainder when

f(1)

is divided by

1000

.

CHMMC 2021-21 Proof 6

Let

ABC

be an acute triangle with orthocenter

H

. A point

L \ne A

lies on the plane of

ABC

such that

\overline{HL} \perp \overline{AL}

and

LB : LC = AB : AC

. Suppose

M_1 \ne B

lies on

\overline{BL}

such that

\overline{HM_1} \perp \overline{BM_1}

and

M_2 \ne C

lies on

\overline{CL}

such that

\overline{HM_2} \perp \overline{CM_2}

. Prove that

\overline{M_1M_2}

bisects

\overline{AL}

.

7

1

2021-22 Winter Team #7

Let

ABC

be a triangle with

AB = 5

,

BC = 6

, and

CA = 7

. Denote

\Gamma

the incircle of

ABC

, let

I

be the center of

\Gamma

. The circumcircle of

BIC

intersects

\Gamma

at

X_1

and

X_2

. The circumcircle of

CIA

intersects

\Gamma

at

Y_1

and

Y_2

. The circumcircle of

AIB

intersects

\Gamma

at

Z_1

and

Z_2

. The area of the triangle determined by

\overline{X_1X_2}

,

\overline{Y_1Y_2}

, and

\overline{Z_1Z_2}

equals

\frac{m \sqrt{p}}{n}

for positive integers

m, n

, and

p

, where

m

and

n

are relatively prime and

p

is squarefree. Compute

m+n+ p

.

10

1

2021-22 Winter Team #10

In triangle

ABC

, let

O

be the circumcenter. The incircle of

ABC

is tangent to

\overline{BC}

,

\overline{CA},

and

\overline{AB}

at points

D, E

, and

F

, respectively. Let

G

be the centroid of triangle

DEF

. Suppose the inradius and circumradius of

ABC

is

3

and

8

, respectively. Over all such triangles

ABC

, pick one that maximizes the area of triangle

AGO

. If we write

AG^2 =\frac{m}{n}

for relatively prime positive integers

m

and

n

, then find

m

.

9

1

2021-22 Winter Team #9

Find the largest prime divisor of

\sum^{30}_{n=3} {{n \choose 3} \choose 2}

8

1

2021-22 Winter Team #8

Depei is imprisoned by an evil wizard and is coerced to play the following game. Every turn, Depei flips a fair coin. Then, the following events occur in this order:

\bullet

The wizard computes the difference between the total number of heads and the total number of tails Depei has flipped. If that number is greater than or equal to

4

or less than or equal to

-3

, then Depei is vaporized by the wizard.

\bullet

The wizard determines if Depei has flipped at least

10

heads or at least

10

tails. If so, then the wizard releases Depei from the prison. The probability that Depei is released by the evil wizard equals

\frac{m}{2^k}

, where

m, k

are positive integers. Compute

m+k

.

5

2

2021-22 Winter Team #5

How many cubics in the form

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

for integers

a,d

have roots that are all non-negative integers?

CHMMC 2021-21 Proof 5

Find all functions

f : R \to R

such that

f(f(x) + f(y)^2) = f(x)^2 +y^2f(y)^3.

Here

R

denotes the usual real numbers.

4

2

2021-22 Winter Team #4

How many ordered triples

(a, b, c)

of integers

1 \le a, b, c \le 31

are there such that the remainder of

ab+bc+ca

divided by

31

equals

8

?

CHMMC 2021-22 Proof 4

Show that for any three positive integers

a,m,n

such that

m

divides

n

, there exists an integer

k

such that

gcd(a,m) = gcd(a+km,n)

.

3

2

2021-22 Winter Team #3

Suppose

a, b, c

are complex numbers with

a + b + c = 0

,

a^2 + b^2 + c^2 = 0

, and

|a|,|b|,|c| \le 5

. Suppose further at least one of

a, b, c

have real and imaginary parts that are both integers. Find the number of possibilities for such ordered triples

(a, b, c)

.

CHMMC 2021-21 Proof 3

Let

F(x_1,..., x_n)

be a polynomial with real coefficients in

n > 1

“indeterminate” variables

x_1,..., x_n

. We say that

F

is

n

-alternating if for all integers

1 \le i < j \le n

,

F(x_1,..., x_i,..., x_j,..., x_n) = - F(x_1,..., x_j,..., x_i,..., x_n),

i.e. swapping the order of indeterminates

x_i, x_j

flips the sign of the polynomial. For example,

x^2_1x_2 - x^2_2x_1

is

2

-alternating, whereas

x_1x_2x_3 +2x_2x_3

is not

3

-alternating.
Note: two polynomials

P(x_1,..., x_n)

and

Q(x_1,..., x_n)

are considered equal if and only if each monomial constituent

ax^{k_1}_1... x^{k_n}_n

of

P

appears in

Q

with the same coefficient

a

, and vice versa. This is equivalent to saying that

P(x_1,..., x_n) = 0

if and only if every possible monomial constituent of

P

has coefficient

0

.
(1) Compute a

3

-alternating polynomial of degree

3

. (2) Prove that the degree of any nonzero

n

-alternating polynomial is at least

{n \choose 2}

.

2

2021-22 Winter Team #2

A prefrosh is participating in Caltech’s “Rotation.” They must rank Caltech’s

8

houses, which are Avery, Page, Lloyd, Venerable, Ricketts, Blacker, Dabney, and Fleming, each a distinct integer rating from

1

to

8

inclusive. The conditions are that the rating

x

they give to Fleming is at most the average rating

y

given to Ricketts, Blacker, and Dabney, which is in turn at most the average rating

z

given to Avery, Page, Lloyd, and Venerable. Moreover

x, y, z

are all integers. How many such rankings can the prefrosh provide?

CHMMC 2021-22 Proof 2

For any positive integer

n

, let

p(n)

be the product of its digits in base-

10

representation. Find the maximum possible value of

\frac{p(n)}{n}

over all integers

n \ge 10

.

1

2

2021-22 Winter Team #1

Let

ABC

be a right triangle with hypotenuse

\overline{AC}

and circumcenter

O

. Point

E

lies on

\overline{AB}

such that

AE = 9

,

EB = 3

, point

F

lies on

\overline{BC}

such that

BF = 6

,

FC = 2

. Now suppose

W, X, Y

, and

Z

are the midpoints of

\overline{EB}

,

\overline{BF}

,

\overline{FO}

, and

\overline{OE}

, respectively. Compute the area of quadrilateral

W XY Z

.

CHMMC 2021-22 Proof 1 (a-b)(b-c) + (b-c)(c-a) + (c-a)(a-b) = 0

Find all ordered triples

(a,b, c)

of real numbers such that

(a-b)(b-c) + (b-c)(c-a) + (c-a)(a-b) = 0.

2021 CHMMC Winter (2021-22)

Subcontests

2021-22 Winter CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

2021-22 Winter Team #6

CHMMC 2021-21 Proof 6

2021-22 Winter Team #7

2021-22 Winter Team #10

2021-22 Winter Team #9

2021-22 Winter Team #8

2021-22 Winter Team #5

CHMMC 2021-21 Proof 5

2021-22 Winter Team #4

CHMMC 2021-22 Proof 4

2021-22 Winter Team #3

CHMMC 2021-21 Proof 3

2021-22 Winter Team #2

CHMMC 2021-22 Proof 2

2021-22 Winter Team #1

CHMMC 2021-22 Proof 1 (a-b)(b-c) + (b-c)(c-a) + (c-a)(a-b) = 0