2023 CUBRMC

Part of CUBRMC

Subcontests

(11)

1

2023 CUBRMC Individual Round = Cornell University Big Red Math Competition

p1. Find the largest

4

digit integer that is divisible by

2

5

3

.
p2. The diagram below shows the eight vertices of a regular octagon of side length

2

. These vertices are connected to form a path consisting of four crossing line segments and four arcs of degree measure

270^o

. Compute the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/0/0/eec34d8d2439b48bb5cca583462c289287f7d0.png
p3. Consider the numbers formed by writing full copies of

2023

next to each other, like so:

2023202320232023...

How many copies of

2023

are next to each other in the smallest multiple of

11

that can be written in this way?
p4. A positive integer

n

10

n = a_1a_2 ...a_k

is called powerful if the digits

a_i

are nonzero for all

1 \le i \le k

n = a^{a_1}_1 + a^{a_2}_2 +...+ a^{a_k}_k .

What is the unique four-digit positive integer that is powerful?
p5. Six

(6)

chess players, whose names are Alice, Bob, Crystal, Daniel, Esmeralda, and Felix, are sitting in a circle to discuss future content pieces for a show. However, due to fights they’ve had, Bob can’t sit beside Alice or Crystal, and Esmeralda can’t sit beside Felix. Determine the amount of arrangements the chess players can sit in. Two arrangements are the same if they only differ by a rotation.
p6. Given that the infinite sum

\frac{1}{1^4} +\frac{1}{2^4} +\frac{1}{3^4} +...

\frac{\pi^4}{90}

, compute the value of

\dfrac{\dfrac{1}{1^4} +\dfrac{1}{2^4} +\dfrac{1}{3^4} +...}{\dfrac{1}{1^4} +\dfrac{1}{3^4} +\dfrac{1}{5^4} +...}

ABC

is equilateral. There are

3

distinct points,

X

Y

Z

\vartriangle ABC

that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio

1 : 1 : 2

in some order. Find the ratio of the area of

\vartriangle ABC

\vartriangle XY Z

.
p8. For a fixed prime

p

, a finite non-empty set

S = \{s_1,..., s_k\}

p

-admissible if there exists an integer

n

for which the product

(s_1 + n)(s_2 + n) ... (s_k + n)

is not divisible by

p

\{4, 6, 8\}

2

-admissible since

(4+1)(6+1)(8+1) = 315

is not divisible by

2

. Find the size of the largest subset of

\{1, 2,... , 360\}

that is two-,three-, and five-admissible.
p9. Kwu keeps score while repeatedly rolling a fair

6

-sided die. On his first roll he records the number on the top of the die. For each roll, if the number was prime, the following roll is tripled and added to the score, and if the number was composite, the following roll is doubled and added to the score. Once Kwu rolls a

1

, he stops rolling. For example, if the first roll is

1

, he gets a score of

1

, and if he rolls the sequence

(3, 4, 1)

, he gets a score of

3 + 3 \cdot 4 + 2 \cdot 1 = 17

. What is his expected score?
p10. Let

\{a_1, a_2, a_3, ...\}

be a geometric sequence with

a_1 = 4

a_{2023} = \frac14

f(x) = \frac{1}{7(1+x^2)}

f(a_1) + f(a_2) + ... + f(a_{2023}).

S

be the set of quadratics

x^2 + ax + b

a

b

real, that are factors of

x^{14} - 1

f(x)

be the sum of the quadratics in

S

f(11)

.
p12. Find the largest integer

0 < n < 100

n^2 + 2n

4(n- 1)! + n + 4

\omega

be a unit circle with center

O

OQ

P

is a point on the radius

OQ

Q

such that there exists a unique chord of

\omega

P

whose midpoint when rotated

120^o

counterclockwise about

Q

\omega

OP

.
p14. A sequence of real numbers

\{a_i\}

n \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + ... + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n

for each integer

n \ge 1

. Find the value of

a_{2023}

\vartriangle ABC

\angle ABC = 90^o

I

be its incenter. Let line

BI

AC

D

CI

AB

E

ID = IE = 1

BI

.
p16. For a positive integer

n

S_n

be the set of permutations of the first

n

positive integers. If

p = (a_1, ..., a_n) \in S_n

, then define the bijective function

\sigma_p : \{1,..., n\} \to \{1, ..., n\}

\sigma_p (i) = a_i

for all integers

1 \le i \le n

. For any two permutations

p, q \in S_n

p

q

are friends if there exists a third permutation

r \in S_n

such that for all integers

1 \le i \le n

\sigma_p(\sigma_r (i)) = \sigma_r(\sigma_q(i)).

Find the number of friends, including itself, that the permutation

(4, 5, 6, 7, 8, 9, 10, 2, 3, 1)

S_{10}

.
PS. You had better use hide for answers.

1

2023 CUBRMC Team #9 n|1+\sum^{n-1}_{k=1}k^{2k}

Find the sum of all integers

n

1 < n < 30

n

1+\sum^{n-1}_{k=1}k^{2k}.

1

2023 CUBRMC Team #8 probability r is strictly closer to multiple of 58 than 37

r

is real number sampled at random with uniform probability, find the probability that

r

is strictly closer to a multiple of

58

than it is to a multiple of

37

1

2023 CUBRMC Team #7 min a if a^4 + 2a^2b + 2ab + b^2 = 960

Among all ordered pairs of real numbers

(a, b)

a^4 + 2a^2b + 2ab + b^2 = 960

, find the smallest possible value for

a

1

2023 CUBRMC Team #6 sum of all positive divisors of 40081

Find the sum of all positive divisors of

40081

1

2023 CUBRMC Team #5 f(x)=2x^2 - 3x + r

The quadratic polynomial

f(x)

has the expansion

2x^2 - 3x + r

. What is the largest real value of

r

for which the ranges of the functions

f(x)

f(f(x))

are the same set?

1

2023 CUBRMC Team #10 min(DP,EP),

ABC

have side lengths

AB = 19

BC = 180

AC = 181

, and angle measure

\angle ABC = 90^o

. Let the midpoints of

AB

BC

M

N

respectively. The circle centered at

M

and passing through point

C

intersects with the circle centered at the

N

and passing through point

A

D

E

DE

AC

P

(DP,EP)

2

2023 CUBRMC Team #4 four share meals

Alice, Bob, Carol, and David decide that they will share meals and that one of them will cook each night. Because David enjoys cooking, he will cook on

4

days of the week, while Alice, Bob, and Carol each pick a day of the week to cook on. If Alice, Bob, and Carol each choose the day they cook uniformly at random so as to avoid overlap, what is the probability that David does not cook on three consecutive days? For example, Monday, Tuesday and Wednesday are considered as three consecutive days, so are Saturday, Sunday and Monday.

2023 CUBRMC Proof #4 ain't no sunshine without geo (orthodiagonal quad)

ABCD

\Omega

be on the same plane, and

AA'

BB'

CC'

DD'

\Omega

WXY Z

be a convex quadrilateral with side lengths

WX = AA'

XY = BB'

Y Z = CC'

ZW = DD'

WXY Z

has an inscribed circle, prove that the diagonals

WY

XZ

are perpendicular to each other.

2

2023 CUBRMC Proof #3 m-n is prime and mn perfect square

Find all positive integer pairs

(m, n)

m- n

is a positive prime number and

mn

is a perfect square. Justify your answer.

2023 CUBRMC Team #3 two buckets each have four balls

Two buckets each have four balls; two red balls and two white balls in the first, and two red balls and two blue balls in the second. At first, a bucket is selected, then a ball in the bucket is selected, with both buckets and balls inside the selected bucket having equal probability of being chosen. Then, without replacement of the first ball, the process is repeated once more. Determine the probability that the first ball drawn being red if the second ball drawn was blue.

2

2023 CUBRMC Proof #2 3n + 1 teams in the MLS

This season, there are

3n + 1

teams in the MLS (Major League Soccer). As of now, each team has played exactly

n -1

matches. Prove that there exist

4

teams such that none of the

4

teams have faced each other.

2023 CUBRMC Team #2 area of concave decagon

The concave decagon shown below is embedded in the Cartesian coordinate plane such that all of its vertices have integer coordinates. Two opposite edges have length

5

, whereas the remaining eight edges have length

\sqrt{10}

. Every pair of opposite edges is parallel. The sides of the decagon do not intersect each other, and the decagon has vertical and horizontal axes of symmetry. Find the area of the decagon. https://cdn.artofproblemsolving.com/attachments/1/5/daa4ab3d71af4b3274cd222f9a091eea3be705.png

2

2023 CUBRMC Proof # 1 \sqrt{(z + x)(z + y)} - z >=\sqrt{xy}

x, y, z

be positive real numbers. Prove that

\sqrt{(z + x)(z + y)} - z \ge \sqrt{xy}.

2023 CUBRMC Team #1 n-(sum of digits of n)

Ben starts with an integer greater than

9

and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive

1

-digit integer. Find all possible

1

-digit integers Ben can end with from this process.