2022 CCA Math Bonanza

Part of CCA Math Bonanza

Subcontests

(49)

1

Functions

f(x)

be a function such that

f(1) = 1234

f(2)=1800

f(x) = f(x-1) + 2f(x-2)-1

for all integers

x

. Evaluate the number of divisors of

\sum_{i=1}^{2022}f(i)

2022 CCA Math Bonanza Tiebreaker Round #4

1

Double angle

(2\cos^2{7.5}-\cos{75}-1)^2

can be expressed as

\frac{p}{q}

p+q

?
2022 CCA Math Bonanza Tiebreaker Round #3

1

Chinese Remainder Theorem

Determine the last three digits of

374^{2022}.

2022 CCA Math Bonanza Tiebreaker Round #2

1

Telescoping

How many positive integer factors does the following expression have?

\sum_{n=1}^{999} \log_{10} \left(\frac{n+1}{n} \right)

2022 CCA Math Bonanza Tiebreaker Round #1

1

Either circles or coordination

Five points are selected within a unit circle at random. Estimate the minimum distance between any pair of points. An estimate

E

\frac{2}{1+|log_2(A)-log_2(E)|}

A

is the actual answer.
2022 CCA Math Bonanza Lightning Round 5.4

1

What were the previous answers again?

Estimate the number of times a one-digit answer (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) has been submitted as an answer for any question by any team in the first 4 sets of this competition's lightning round. An estimate

E

\frac{2}{1+|log_2(A)-log_2(E)|}

A

is the actual answer.
2022 CCA Math Bonanza Lightning Round 5.3

1

Imagine guessing 0

Estimate the range of the submissions for this problem. Your answer must be between

[0, 1000]

E

\frac{2}{1+0.05|A-E|}

A

is the actual answer.
2022 CCA Math Bonanza Lightning Round 5.2

1

Do deer even eat grass

Alistar wants to wreak havoc on Jhin's yard, which is a 2D plane of grass. First, he selects a number

n

, randomly and uniformly from

[0,1]

, and then he eats all grass within

n

meters from where he's standing. He then moves 2 meters in a random direction, and repeats his process. He stops if any of the grass that he wants to eat (or, in other words, in his intended eating territory) is already eaten. Estimate the amount of grass Alistar is expected to eat. An estimate

E

\frac{2}{1+|A-E|}

A

is the actual answer.
2022 CCA Math Bonanza Lightning Round 5.1

1

Coloring circles

Lukas Nepomuceno draws 5 congruent circles equally spaced around a 6th, and colors all of them 1 of 3 colors. Assume that rotations and reflections of colorings are indistinguishable. How many distinct colorings are there?
2022 CCA Math Bonanza Lightning Round 4.4

1

Rock paper scissors

Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing?
2022 CCA Math Bonanza Lightning Round 4.3

1

Is this even possible...?

A rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has

20

regular triangular faces,

30

square faces, and

12

regular pentagonal faces, as shown below. How many rotational symmetries does a rhombicosidodecahedron have?
2022 CCA Math Bonanza Lightning Round 4.2

1

Forgetting about 35

There are 99 seagulls labeled 2-100 and 100 bagels labeled 1-100. Starting from Seagull 2, each Seagull

N

\frac{1}{N}

of whatever remains of each Bagel

I

N

I

. How many bagels still have more than

\frac{2}{3}

of their original size after Seagull 100 finishes eating?
2022 CCA Math Bonanza Lightning Round 4.1

1

Bird bash

A merganser mates every 7th day, a scaup mates every 11th day, and a gadwall mates every 13th day. A merganser, scaup, and gadwall all mate on Day 0. On Days N, N+1, and N+2 the merganser, scaup, and gadwall mate in some order with no two birds mating on the same day. Determine the smallest possible value of N.
2022 CCA Math Bonanza Lightning Round 3.4

1

Digits of numbers

Determine the sum of all positive integers

n<100

satisfying the following expression.

\sum_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod \;{10^{k+1})}-n \;(\bmod \;{10^k)}\right)=\prod_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod\; 10^{k+1})-n \;(\bmod\; 10^k)\right).

\textstyle\sum

\textstyle\prod

represent sum and product, respectively.
2022 CCA Math Bonanza Lightning Round 3.3

1

Circles and semicircles

In the following diagram,

AB = 1

. The radius of the circle with center

C

can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Lightning Round 3.2

1

Die rolling

Kongol rolls two fair 6-sided die. The probability that one roll is a divisor of the other can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Lightning Round 3.1

1

A "pack" of geese?

10 geese are numbered 1-10. One goose leaves the pack, and the remaining nine geese assemble in a symmetric V-shaped formation with four geese on each side. Given that the product of the geese on both halves of the "V" are the same, what is the sum of the possible values of the goose that left?
2022 CCA Math Bonanza Lightning Round 2.4

1

Bird physics

Given that the height of a greater sage grouse flying through the air is defined by the function

64x-x^2

0<x<64

, what is the first time at which the bird reaches a height of 903?
2022 CCA Math Bonanza Lightning Round 2.3

1

Pigeon Geo

ABCD

has side lengths

AB=6 \text{ miles}

BC=9\text{ miles}.

A pigeon hovers at point

P

, which is 5 miles above some randomly chosen point inside

ABCD

. Given that the expected value of

AP^2+CP^2-BP^2-DP^2

can be expressed as

\tfrac{a}{b}

ab

?
2022 CCA Math Bonanza Lightning Round 2.2

1

Don't really see how the duck part is relevant

Given that a duck found that

5-2\sqrt{3}i

is one of the roots of

-259 + 107x - 17x^2 + x^3

, what is the sum of the real parts of the other two roots?
2022 CCA Math Bonanza Lightning Round 2.1

1

Opportunity Cost

Jongol and Gongol are writing calculus questions and grading tests. They want to write 90 calculus problems and they have 120 tests to grade. Jongol can write 3 questions per minute or grade 4 tests per minute. Gongol can write 1 question per minute or grade 2 tests per minute. Evaluate the shortest possible time, in minutes, for them to complete the two tasks.
2022 CCA Math Bonanza Lightning Round 1.4

1

Calculus...?

The area of the region bound by the equations

y = 2\cos(4x)

y = \sin(2x) + 10

x=0

x = 5\pi

can be expressed as

x\pi

for some value of

x

x

.
2022 CCA Math Bonanza Lightning Round 1.3

1

Gambling is bad

Xonathan Jue goes to the casino with exactly \

1000. Each week, he has a

chance of breaking even and

2/3$ chance of losing \$500. Evaluate the expected amount of weeks before he loses everything.
2022 CCA Math Bonanza Lightning Round 1.2

1

Somewhat Symmetric Equations

a = bc

b = ca

c = a + b

c > a

a+b+c

.
2022 CCA Math Bonanza Lightning Round 1.1

1

The last whale erased

Evan, Larry, and Alex are drawing whales on the whiteboard. Evan draws 10 whales, Larry draws 15 whales, and Alex draws 20 whales. Michelle then starts randomly erasing whales one by one. The probability that she finishes erasing Larry's whales first can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Team Round #10

1

Too much geometry

Equilateral octagon

A_1A_2A_3A_4A_5A_6A_7A_8

is constructed such that

A_1A_3A_5A_7

is a square of side length

\sqrt{2}

A_2A_4A_6A_8

is a square of side length 4/3. For each vertex

A_i

of the octagon, let

B_i

be the intersection of lines

A_{i+1}A_{i+2}

A_{i-1}A_{i-2}

A_{i-8} = A_i = A_{i+8}

[B_1B_2B_3B_4B_5B_6B_7B_8]^2

.
2022 CCA Math Bonanza Team Round #9

1

Set Probability

Let n be a set of integers.

S(n)

is defined as the sum of the elements of n.

T=\{1,2,3,4,5,6,7,8,9\}

and A and B are subsets of T such that A

\cup

B=T

\cap

B=\varnothing

. The probability that

S(A)\geq4S(B)

can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Team Round #8

1

Candy feeding

A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one mood when the other baby gets one. Both babies start at the "okay" state, and a baby will start crying if they don't get a candy when they're already "upset". The probability that 10 minutes pass without either baby crying can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Team Round #7

1

Countercurrent heat exchange

A bird starts with 300 ml of blood at 100 degrees in its body, 50 ml of blood at 0 degrees in its feet. Every minute, 50 ml of blood flows from the body to the feet, and 50 ml of blood at 40% of the body temperature flows from the feet to the body. The bird feels cold once its internal body temperature (not including the feet) falls below 60 degrees. Compute how many minutes it takes for the bird to feel cold.
2022 CCA Math Bonanza Team Round #6

1

Unsorted binary search

Maggie Waggie organizes a pile of 127 calculus tests in alphabetical order, with Joccy Woccy's test being 64th in the pile. While Maggie isn't looking, Joccy walks over and randomly scrambles the entire pile of tests. When Maggie returns, she is oblivious to the fact that Joccy has tampered with the list. She uses a binary search algorithm to find Joccy's test, where she looks at the test in the middle of the pile. If the test is not Joccy's, she binary searches the top half of the list if the test appears after Joccy's name when arranged alphabetically, or the bottom half of the list otherwise. The probability that Maggie finds Joccy's test can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Team Round #5

1

Anti-King Chess

Let there exist a configuration of exactly 1 black king,

n

black chess pieces (each of which can be a pawn, knight, bishop, rook, or queen), and a white anti-king on a standard 8x8 board in which the white anti-king is not under attack, but will be if it is moved. Compute the minimal value of

n

.
*An anti-king can move to any square is not 1 square vertically, horizontally, or diagonally. It can also capture undefended pieces.
2022 CCA Math Bonanza Team Round #4

1

Packing Spheres

The smallest possible volume of a cylinder that will fit nine spheres of radius 1 can be expressed as

x\pi

for some value of

x

x

.
2022 CCA Math Bonanza Team Round #3

1

Conditional Probability?!

CCA's B building has 6 rooms on the second floor, labeled B201 to B206, as well as 8 rooms on the first floor, labeled B101 to B108. Annie is currently in room B205. Each minute, she chooses to stay or change floors with equal probability, and chooses a classroom on that floor to go to at random (she can stay in the classroom that she's already in). B104, B108, and B203 are the only rooms that have teachers who will scold her for randomly walking around during class time. The probability that she is first scolded in room B203 can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Team Round #2

1

Minimizing Exponents

a

b

c

d

be positive integers such that

77^a \cdot 637^b = 143^c \cdot 49^d

. Compute the minimal value of

a+b+c+d

.
2022 CCA Math Bonanza Team Round #1

1

Why are the last 3 all geo...

P

A

B

C

D

be points on a plane such that

PA = 9

PB = 19

PC = 9

PD = 5

\angle APB = 120^\circ

\angle BPC = 45^\circ

\angle CPD = 60^\circ

\angle DPA = 135^\circ

G_1

G_2

G_3

G_4

be the centroids of triangles

PAB

PBC

PCD

PDA

[G_1G_2G_3G_4]

can be expressed as

a\sqrt{b} + c\sqrt{d}

a+b+c+d

.
2022 CCA Math Bonanza Individual Round #15

1

Reflecting Points

ABC

be a triangle with side lengths

AB=6, AC=7,

BC=8.

H

be the orthocenter of

\triangle ABC

H'

be the reflection of

H

across the midpoint

M

BC.

\tfrac{[ABH']}{[ACH']}

can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Individual Round #14

1

Sum of lengths

A_1BC

A_1B=5

A_1C=12

BC=13

. For all natural numbers

i

B_i

be the foot of the altitude from

A_i

BC

A_{2i}

be the foot of the altitude from

B_i

A_1B

A_{2i+1}

be the foot of the altitude from

B_i

A_1C

\sum_{i=1}^{7}A_iB_i = \frac{p}{q}

p+q

.
2022 CCA Math Bonanza Individual Round #13

1

Logic AND Bashing

Find the number of 8-tuples of binary inputs

\{A, B, C, D, E, F, G, H\}

\{ (A \text{ AND } B)\text{ OR } (C \text{ AND } D)\} \text{ AND } \{ (E \text{ AND } F)\text{ OR } (G \text{ AND } H)\}

= \{ (A \text{ OR } B)\text{ AND } (C \text{ OR } D)\} \text{ OR } \{ (E \text{ OR } F)\text{ AND } (G \text{ OR } H)\}

The AND gates produce an output that is ON only if both the inputs are ON, and the OR gates produce an output that is OFF only if both inputs are OFF.
2022 CCA Math Bonanza Individual Round #12

1

Ducks swimming

A river is bounded by the lines

x=0

x=25

, with a current of 2 units/s in the positive y-direction. At

t=0

, a mallard is at

(0, 0)

, and a wigeon is at

(25, 0)

. They start swimming with a constant speed such that they meet at

(x,22)

. The mallard has a speed of 4 units/s relative to the water, and the wigeon has a speed of 3 units/s relative to the water. Find the value of

x

.
2022 CCA Math Bonanza Individual Round #11

1

Ducks running laps

\overline{AB}

be a line segment of length 2,

C_1

be the circle with diameter

\overline{AB}

C_0

be the circle with radius 2 externally tangent to

C_1

A

C_2

be the circle with radius 3 externally tangent to

C_1

B

D_1

is located at point

B

D_2

C_2

, 270 degrees counterclockwise from

B

D_0

C_0

, 270 degrees counterclockwise from

A

. At the same time, the ducks all start running counterclockwise around their corresponding circles, with each duck taking the same amount of time to complete a full lap around its circle. When the 3 ducks are first collinear, the distance between

D_0

D_2

can be expressed as

p\sqrt{q}

p+q

.
2022 CCA Math Bonanza Individual Round #10

1

p^2-1 is divisible by 8

Find the maximum value of

x

x

p^{32} - 1

p> 20

.
2022 CCA Math Bonanza Individual Round #9

1

Rubber Chicken Contest

Lason Jiu gives a problem to Sick Nong and Ayush Agrawal. Sick takes 6 minutes to solve the problem, while Ayush takes 9 minutes. Sick has a 1/3 chance of solving correctly and Ayush has a 2/3 chance of solving correctly. If they solved it incorrectly, they resume solving with the same time and accuracy. Lason gives a rubber chicken to the first person who solves it correctly. If Sick and Ayush solve the question at the same time, Lason checks Sick's work first. The probability that Ayush wins the rubber chicken can be expressed as

\frac{p}{q}

p+q

.
2022 CCA Math Bonanza Individual Round #8

1

Subset Bashing

A = \{2, 4, \ldots, 1000\},

B = \{3, 6, \ldots, 999\},

C = \{5, 10, \ldots, 1000\},

D = \{7, 14, \ldots, 994\},

E = \{11, 22, \ldots, 990\},

\textrm{and } F = \{13, 26, \ldots, 988\}.

Find the number of elements in the set

(((((A\cup B)\cap C)\cup D)\cap E)\cup F)

.
2022 CCA Math Bonanza Individual Round #7

1

Around 109.5 degrees

Let regular tetrahedron

ABCD

O

\tan^2(\angle AOB)

.
2022 CCA Math Bonanza Individual Round #6

1

Early Timesink

\Gamma_1

be a circle with radius

\frac{5}{2}

A

B

C

\Gamma_1

\overline{AB} = 3

\overline{AC} = 5

\Gamma_2

be a circle such that

\Gamma_2

AB

BC

Q

R

\Gamma_2

is also internally tangent to

\Gamma_1

P

\Gamma_2

AC

X

Y

[PXY]

can be expressed as

\frac{a\sqrt{b}}{c}

a+b+c

.
2022 CCA Math Bonanza Individual Round #5

1

Painting Shapes

Burrito Bear has a white unit square. She inscribes a circle inside of the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as

\frac{a\pi+b}{c}

a+b+c

.
2022 CCA Math Bonanza Individual Round #4

1

Only choosing large numbers

S = {1, 2, \cdots, 100}.

X

S

such that no two distinct elements in

X

multiply to an element in

X.

Find the maximum number of elements of

X

.
2022 CCA Math Bonanza Individual Round #3

1

Road Trip Arrangements

Alice, Bob, Cassie, Dave, and Ethan are going on a road trip and need to arrange themselves among a drivers seat, a passenger seat, and three distinguishable back row seats. Alice, Bob, and Cassie are not allowed to drive. Alice and Bob are also not allowed to sit in the front passenger seat. Find the number of possible seating arrangements.
2022 CCA Math Bonanza Individual Round #2

1

Kepler's 3rd Law?

Asteroids A and B have circular orbits around the same star. Asteroid A is located 400 km away from the star and takes 8000 hours to complete one full revolution. Asteroid B is located 100 km away and the speed of Asteroid B is twice the speed of Asteroid A. Find how long it takes for Asteroid B to complete one full revolution in hours.
2022 CCA Math Bonanza Individual Round #1