2021 CMIMC

Part of CMIMC Problems

Subcontests

(32)

1

2021 Team P15

Adam has a circle of radius

1

centered at the origin.
- First, he draws

6

segments from the origin to the boundary of the circle, which splits the upper (positive

y

) semicircle into

7

equal pieces. - Next, starting from each point where a segment hit the circle, he draws an altitude to the

x

-axis. - Finally, starting from each point where an altitude hit the

x

-axis, he draws a segment directly away from the bottommost point of the circle

(0,-1)

, stopping when he reaches the boundary of the circle. What is the product of the lengths of all

18

segments Adam drew?
https://cdn.discordapp.com/attachments/813077401265242143/816190774257516594/circle2.png
Proposed by Adam Bertelli

1

2021 Team P14

S

be the set of lattice points

(x,y) \in \mathbb{Z}^2

-10\leq x,y \leq 10

. Let the point

(0,0)

O

. Let Scotty the Dog's position be point

P

, where initially

P=(0,1)

. At every second, consider all pairs of points

C,D \in S

such that neither

C

D

OP

, and the area of quadrilateral

OCPD

(with the points going clockwise in that order) is

1

. Scotty finds the pair

C,D

maximizing the sum of the

y

C

D

, and randomly jumps to one of them, setting that as the new point

P

50

such moves, Scotty ends up at point

(1, 1)

. Find the probability that he never returned to the point

(0,1)

50

moves.
Proposed by David Tang

1

2021 Team P13

p=3\cdot 10^{10}+1

be a prime and let

p_n

denote the probability that

p\mid (k^k-1)

k

chosen uniformly from

\{1,2,\cdots,n\}

p_n\cdot p

converges to a value

L

n

goes to infinity, what is

L

?
Proposed by Vijay Srinivasan

1

2021 Team P12

\triangle ABC

be a triangle, and let

l

be the line passing through its incenter and centroid. Assume that

B

C

lie on the same side of

l

, and that the distance from

B

l

is twice the distance from

C

l

. Suppose also that the length

BA

is twice that of

CA

\triangle ABC

has integer side lengths and is as small as possible, what is

AB^2+BC^2+CA^2

?
Proposed by Thomas Lam

1

2021 Team P11

The set of all points

(x,y)

in the plane satisfying

x<y

x^3-y^3>x^2-y^2

A

. What is the value of

A

?
Proposed by Adam Bertelli

1

2021 Team P10

How many functions

f:\{1,2,3, \ldots, 7\} \rightarrow \{1,2,3, \ldots, 7\}

are there such that the set

\mathcal{F} = \{f(i) : i\in\{1,\ldots, 7\}\}

has cardinality four, while the set

\mathcal{G} = \{f(f(f(i))) : i\in\{1,\ldots, 7\}\}

consists of a single element?
Proposed by Sam Delatore

1

2021 Team P9

ABC

be a triangle with circumcenter

O

. Additionally,

\angle BAC=20^\circ

\angle BCA = 70^\circ

D, E

be points on side

AC

BO

\angle ABD

BE

\angle CBD

P

Q

are points on line

BC

DP

EQ

are perpendicular to

AC

\angle PAQ

?
Proposed by Daniel Li

1

2021 Team P8

Determine the number of functions

f

from the integers to

\{1,2,\cdots,15\}

f(x)=f(x+15)

f(x+f(y))=f(x-f(y))

x,y

.
Proposed by Vijay Srinivasan

1

2021 Team P7

P

Q

be fixed points in the Euclidean plane. Consider another point

O_0

O_{i+1}

as the center of the unique circle passing through

O_i

P

Q

O_i,P,Q

are never collinear.) How many possible positions of

O_0

O_{2021}=O_{0}

?
Proposed by Fei Peng

1

2021 Team P6

P(x), Q(x),

R(x)

be three monic quadratic polynomials with only real roots, satisfying

P(Q(x))=(x-1)(x-3)(x-5)(x-7)

Q(R(x))=(x-2)(x-4)(x-6)(x-8)

for all real numbers

x.

P(0)+Q(0)+R(0)?

Proposed by Kyle Lee

1

2021 Team P5

N

be the fifth largest number that can be created by combining

2021

1

's using addition, multiplication, and exponentiation, in any order (parentheses are allowed). If

f(x)=\log_2(x)

k

is the least positive integer such that

f^k(N)

is not a power of

2

, what is the value of

f^k(N)

f^k(N)=f(f(\cdots(f(N))))

f

k

times.)
Proposed by Adam Bertelli

1

2021 Team P4

How many four-digit positive integers

\overline{a_1a_2a_3a_4}

have only nonzero digits and have the property that

|a_i-a_j| \neq 1

1 \leq i<j \leq 4?

Proposed by Kyle Lee

2

2021 Team P3

\sum_{i=0}^{\infty}\frac{7^i}{(7^i+1)(7^i+7)}

Proposed by Connor Gordon

2021 TCS Problem 3

There is a tiger (which is treated as a point) in the plane that is trying to escape. It starts at the origin at time

t = 0

, and moves continuously at some speed

k

. At every positive integer time

t

, you can place one closed unit disk anywhere in the plane, so long as the disk does not intersect the tiger's current position. The tiger is not allowed to move into any previously placed disks (i.e. the disks block the tiger from moving). Note that when you place the disks, you can "see" the tiger (i.e. know where the tiger currently is).
Your goal is to prevent the tiger from escaping to infinity. In other words, you must show there is some radius

R(k)

such that, using your algorithm, it is impossible for a tiger with speed

k

to reach a distance larger than

R(k)

from the origin (where it started).
Find an algorithm for placing disks such that there exists some finite real

R(k)

such that the tiger will never be a distance more than

R(k)

away from the origin.
An algorithm that can trap a tiger of speed

k

will be awarded:
1 pt for

k<0.05

k=0.05

k=0.2

k=0.3

k=1

k=2

k=2.3

k=2.6

k=2.9

k=3.9

2

2021 Team P2

p_1, p_2, p_3, p_4, p_5, p_6

be distinct primes greater than

5

. Find the minimum possible value of

p_1 + p_2 + p_3 + p_4 + p_5 + p_6 - 6\min\left(p_1, p_2, p_3, p_4, p_5, p_6\right)

Proposed by Oliver Hayman

2021 TCS Problem 2

You are initially given the number

n=1

. Each turn, you may choose any positive divisor

d\mid n

n

d+1

. For instance, on the first turn, you must select

d=1

n=1\cdot(1+1)=2

as your new value of

n

. On the next turn, you can select either

d=1

2

n=2\cdot(1+1)=4

n=2\cdot(2+1)=6

, respectively, and so on.
Find an algorithm that, in at most

k

steps, results in

n

being divisible by the number

2021^{2021^{2021}} - 1

.
An algorithm that completes in at most

k

steps will be awarded:
1 pt for

k>2021^{2021^{2021}}

k=2021^{2021^{2021}}

k=10^{10^4}

k=10^{10}

k=10^5

k=6\cdot10^4

k=5\cdot10^4

2

2021 Team P1

Given a trapezoid with bases

AB

CD

, there exists a point

E

CD

such that drawing the segments

AE

BE

partitions the trapezoid into

3

similar isosceles triangles, each with long side twice the short side. What is the sum of all possible values of

\frac{CD}{AB}

?
Proposed by Adam Bertelli

2021 TCS Problem 1

n^2

indistinguishable pieces on an

n\times n

chessboard, where

n=2^{90}\approx 1.27\times10^{27}

. Of these pieces,

n

of them are slightly lighter than usual, while the rest are all the same standard weight, but you are unable to discern this simply by feeling the pieces.\\
However, beneath each row and column of the chessboard, you have set up a scale, which, when turned on, will tell you only whether the average weight of all the pieces on that row or column is the standard weight, or lighter than standard. On a given step, you are allowed to rearrange every piece on the chessboard, and then turn on all the scales simultaneously, and record their outputs, before turning them all off again. (Note that you can only turn on the scales if all

n^2

pieces are placed in different squares on the board.)
Find an algorithm that, in at most

k

steps, will always allow you to rearrange the pieces in such a way that every row and column measures lighter than standard on the final step.
An algorithm that completes in at most

k

steps will be awarded:
1 pt for

k>10^{55}

k=10^{55}

k=10^{30}

k=10^{28}

k=10^{20}

k=10^5

k=2021

k=500

2

2021 Geo Div 1 P3 (Div 2 P7)

\triangle ABC

be a triangle with

AB=10

AC=16,

I

be the intersection of the internal angle bisectors of

\triangle ABC.

Suppose the tangents to the circumcircle of

\triangle BIC

B

C

intersect at a point

P

PA=8.

Compute the length of

{BC}.

Proposed by Kyle Lee

2021 Combo Div 1 P3 (Div 2 P7)

How many permutations of the string

0123456

are there such that no contiguous substrings of lengths

1<\ell<7

have a sum of digits divisible by

7

?
Proposed by Srinivasan Sathiamurthy

2

2021 Geo Div 1 P2 (Div 2 P6)

In convex quadrilateral

ABCD

\angle ADC = 90^\circ + \angle BAC

AB = BC = 17

CD = 16

, what is the maximum possible area of the quadrilateral?
Proposed by Thomas Lam

2021 Combo Div 1 P2 (Div 2 P6)

Adam is playing Minesweeper on a

9\times9

grid of squares, where exactly

\frac13

(or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent squares are mines.
First, he clicks the square directly above the center square, which shows the number

4

. Next, he clicks the square directly below the center square, which shows the number

1

. What is the probability that the center square is a mine?
Proposed by Adam Bertelli

2

2021 Geo Div 2 P5

(0,0)

, chilling, when she sees a spider located at

(1,0)

! Emily runs a continuous path to her home, located at

(\sqrt{2}+2,0)

, such that she is always moving away from the spider and toward her home. That is, her distance from the spider always increases whereas her distance to her home always decreases. What is the area of the set of all points that Emily could have visited on her run home?
Proposed by Thomas Lam

2021 Combo Div 2 P5

Bill Gates and Jeff Bezos are playing a game. Each turn, a coin is flipped, and if Bill and Jeff have

m,n>0

dollars, respectively, the winner of the coin toss will take

\min{(m,n)}

from the loser. Given that Bill starts with

20

dollars and Jeff starts with

21

dollars, what is the probability that Bill ends up with all of the money?
Proposed by Daniel Li

1

2021 Geo Div 2 P3

Consider trapezoid

[ABCD]

AB\parallel CD

AB = 5

CD = 9

\angle C = 15^\circ

\angle D = 75^\circ

M_1

be the midpoint of

AB

M_2

be the midpoint of

CD

. What is the distance

M_1M_2

?
Proposed by Daniel Li

1

2021 Geo Div 1 P1 (Div 2 P2)

A

B

C

lie on a line, in that order, with

AB=8

BC=2

B

20^\circ

counter-clockwise about

A

B'

, tracing out an arc

R_1

C

is then rotated

20^\circ

clockwise about

A

C'

, tracing out an arc

R_2

. What is the area of the region bounded by arc

R_1

B'C

R_2

C'B

?
Proposed by Thomas Lam

3

2021 Alg/NT Div 1 P8

There are integers

v,w,x,y,z

and real numbers

0\le \theta < \theta' \le \pi

\cos 3\theta = \cos 3\theta' = v^{-1}, \qquad w+x\cos \theta + y\cos 2\theta = z\cos \theta'.

z\ne 0

v

is positive, find the sum of the

4

smallest possible values of

v

.
Proposed by Vijay Srinivasan

2021 Geo Div 1 P8

ABC

be a triangle with

AB < AC

\omega

be a circle through

A

tangent to both the

B

-excircle and the

C

\omega

intersect lines

AB, AC

X,Y

respectively and

X,Y

lie outside of segments

AB, AC

O

be the center of

\omega

OI_C, OI_B

BC

J,K

respectively. Suppose

KJ = 4

KO = 16

OJ = 13

\frac{[KI_BI_C]}{[JI_BI_C]}

.
Proposed by Grant Yu

2021 Combo Div 1 P8

An augmentation on a graph

G

is defined as doing the following:
- Take some set

D

G

, and duplicate each vertex

v_i \in D

to create a new vertex

v_i'

.
- If there's an edge between a pair of vertices

v_i, v_j \in D

, create an edge between vertices

v_i'

v_j'

. If there's an edge between a pair of vertices

v_i \in D

v_j \notin D

, you can choose to create an edge between

v_i'

v_j

but do not have to.
A graph is called reachable from

G

if it can be created through some sequence of augmentations on

G

H

n

vertices and satisfies that both

H

and the complement of

H

are reachable from a complete graph of

2021

vertices. If the maximum and minimum values of

n

M

m

M+m

.
Proposed by Oliver Hayman

3

2021 Alg/NT Div 1 P7

As a gift, Dilhan was given the number

n=1^1\cdot2^2\cdots2021^{2021}

, and each day, he has been dividing

n

2021!

exactly once. One day, when he did this, he discovered that, for the first time,

n

was no longer an integer, but instead a reduced fraction of the form

\frac{a}b

. What is the sum of all distinct prime factors of

b

?
Proposed by Adam Bertelli

2021 Geo Div 1 P7

Convex pentagon

ABCDE

\overline{BC}=17

\overline{AB}=2\overline{CD}

\angle E=90^\circ

. Additionally,

\overline{BD}-\overline{CD}=\overline{AC}

\overline{BD}+\overline{CD}=25

M

N

be the midpoints of

BC

AD

respectively. Ray

EA

is extended out to point

P

, and a line parallel to

AD

is drawn through

P

, intersecting line

EM

Q

G

be the midpoint of

AQ

N

G

EM

PM

respectively, and the perimeter of

\triangle QBC

42

, find the length of

\overline{EM}

.
Proposed by Adam Bertelli

2021 Combo Div 1 P7

How many non-decreasing tuples of integers

(a_1, a_2, \dots, a_{16})

are there such that

0 \leq a_i \leq 16

i

, and the sum of all

a_i

is even?
Proposed by Nancy Kuang

3

2021 Alg/NT Div 1 P6

Find the remainder when

\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor

10^4

.
Proposed by Vijay Srinivasan

2021 Geo Div 1 P6

\omega

\Gamma

O_1

O_2

and having radii

42

54

respectively, intersect at points

X,Y

\angle O_1XO_2 = 105^{\circ}

A

B

\omega

\Gamma

respectively such that

\angle O_1XA = \angle AXB = \angle BXO_2

Y

lies on both minor arcs

XA

XB

P

to be the intersection of

AO_2

BO_1

XP

AB

C

. What is the value of

\frac{AC}{BC}

?
Proposed by Puhua Cheng

2021 Combo Div 1 P6

Alice and Bob each flip

20

fair coins. Given that Alice flipped at least as many heads as Bob, what is the expected number of heads that Alice flipped?
Proposed by Adam Bertelli

3

2021 Alg/NT Div 1 P5

f

is a degree 42 polynomial such that for all integers

0\le i\le 42

,

f(i)+f(43+i)+f(2\cdot43+i)+\cdots+f(46\cdot43+i)=(-2)^i

f(2021)-f(0)

.
Proposed by Adam Bertelli

2021 Geo Div 1 P5

\gamma_1, \gamma_2, \gamma_3

be three circles with radii

3, 4, 9,

respectively, such that

\gamma_1

\gamma_2

are externally tangent at

C,

\gamma_3

is internally tangent to

\gamma_1

\gamma_2

A

B,

respectively. Suppose the tangents to

\gamma_3

A

B

X.

The line through

X

C

\gamma_3

P

Q.

Compute the length of

PQ.

Proposed by Kyle Lee

2021 Combo Div 1 P5

There are exactly 7 possible tetrominos (groups of 4 connected squares in a grid):
https://cdn.discordapp.com/attachments/813077401265242143/816189385859006474/tetris.png
Daniel has a

2 \times 20210

rectangle and wants to tile the interior with tetrominos without overlaps, pieces sticking out, or extra pieces left over. Note that you are allowed to rotate tetrominos but not reflect them.
For how many multisets of tetrominos (ie. an ordered tuple of how many of each tile he has) is it possible to exactly tile his

2\times20210

rectangle?
Proposed by Dilhan Salgado

3

2021 Alg/NT Div 1 P4 (Div 2 P8)

f(x) = \frac{x^2}8

. Starting at the point

(7,3)

, what is the length of the shortest path that touches the graph of

f

x

-axis?
Proposed by Sam Delatore

2021 Geo Div 1 P4 (Div 2 P8)

ABCDEF

be an equilateral heaxagon such that

\triangle ACE \cong \triangle DFB

AC = 7

CE=8

EA=9

, what is the side length of this hexagon?
Proposed by Thomas Lam

2021 Combo Div 1 P4 (Div 2 P8)

Suppose you have a

6

sided dice with

3

faces colored red,

2

faces colored blue, and

1

face colored green. You roll this dice

20

times and record the color that shows up on top. What is the expected value of the product of the number of red faces, blue faces, and green faces?
Proposed by Daniel Li

1

2021 Alg/NT Div 2 P7

For each positive integer

n,

\sigma(n)

denote the sum of the positive integer divisors of

n.

How many positive integers

n \leq 2021

\sigma(3n) \geq \sigma(n)+\sigma(2n)?

Proposed by Kyle Lee

1

2021 Alg/NT Div 1 P3 (Div 2 P6)

a

b

be complex numbers such that

(a+1)(b+1)=2

(a^2+1)(b^2+1)=32.

Compute the sum of all possible values of

(a^4+1)(b^4+1).

Proposed by Kyle Lee

1

2021 Alg/NT Div 1 P2 (Div 2 P5)

Suppose there are

160

n

1

st pigeon flies to the

1

2

nd pigeon flies to the

4

th hole, and so on, such that the

i

th pigeon flies to the

(i^2\text{ mod }n)

k\text{ mod }n

is the remainder when

k

n

. What is minimum

n

such that there is at most one pigeon per hole?
Proposed by Christina Yao

3

2021 Alg/NT Div 2 P4

101

st smallest integer which can represented in the form

3^a+3^b+3^c

a,b,

c

are integers?
Proposed by Dilhan Salgado

2021 Geo Div 2 P4

2\sqrt5

4\sqrt5

rectangle is rotated by an angle

\theta

about one of its diagonals. If the total volume swept out by the rotating rectangle is

62\pi

, find the measure of

\theta

in degrees.
Proposed by Connor Gordon

2021 Combo Div 2 P4

Vijay has a stash of different size stones: in particular, he has

2021

types of stones, with sizes from

0

2020

2r+1

r

.
Vijay starts randomly (and without replacement) taking out stones from his stash and laying them out in a line. Vijay notices that the first stone of size

2020

comes before the first stone of size

2019

, the first stone of size

2019

is before the first stone of size

2018

, and so on. What is the probability of this happening?
Express your answer in terms of only basic arithmetic operations (division, exponentiation, etc.) and the factorial function.
Proposed by Misha Ivkov

2

2021 Alg/NT Div 1 P1 (Div 2 P3)

How many multiples of

12

12!

and have exactly

12

divisors?
Proposed by Adam Bertelli

2021 Combo Div 1 P1 (Div 2 P3)

Adam has a box with

15

pool balls in it, numbered from

1

15

, and picks out

5

of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to

1

. How many selections of

5

balls could he have drawn from the box?
Proposed by Adam Bertelli

2

2021 Alg/NT Div 2 P2

a,b

are positive real numbers such that

a+a^2 = 1

b^2+b^4=1

a^2+b^2

.
Proposed by Thomas Lam

2021 Combo Div 2 P2

Dilhan has objects of

3

A

B

C

6

f_{A,B},f_{A,C},f_{B,A},f_{B,C},f_{C,A},f_{C,B}

f_{X,Y}

takes in an object of type

X

and outputs an object of type

Y

. Dilhan wants to compose his

6

functions, without repeats, such that the resulting expression is well-typed, meaning an object can be taken in by the first function, and the resulting output can then be taken in by the second function, and so on. In how many orders can he compose his

6

functions, satisfying this constraint?
Proposed by Adam Bertelli

3

2021 Alg/NT Div 2 P1

Find the unique 3 digit number

N=\underline{A}

\underline{B}

\underline{C},

(A, B, C)

are all nonzero, with the property that the product

P=\underline{A}

\underline{B}

\underline{C}

\times

\underline{A}

\underline{B}

\times

\underline{A}

is divisible by

1000

.
Proposed by Kyle Lee

2021 Geo Div 2 P1

ABC

has a right angle at

A

AB=20

AC=21

\omega_A

\omega_B

\omega_C

are centered at

A

B

C

respectively and pass through the midpoint

M

\overline{BC}

\omega_A

\omega_B

X\neq M

\omega_A

\omega_C

Y\neq M

XY

.
Proposed by Connor Gordon

2021 Combo Div 2 P1

9

9

chessboard with

9

kings (which can move to any of

8

adjacent squares) in the bottom row. What is the minimum number of moves, if two pieces cannot occupy the same square at the same time, to move all the kings into an

X

5\times5

region where there are

5

kings along each diagonal of the

X

, as shown below)?
\begin{tabular}{ c c c c c } O & & & & O \\ & O & & O & \\ & & O & & \\ & O & & O & \\ O & & & & O \\ \end{tabular}
Proposed by David Tang