2023 MOAA

Part of Math Open At Andover problems

Subcontests

(25)

1

2023 MOAA Tiebreakers

TB1. Two not necessarily distinct positive integers

a,b

are randomly chosen from the set

\{1,2,\ldots, 20\}

. Find the expected value of the number of distinct prime factors of

ab

.
Proposed by Harry Kim
TB2. Square

ABCD

has side length

15

E

F

AD

BC

respectively such that

AE = 5

BF = 5

. Find the area of intersection between triangles

\triangle{AFC}

\triangle{BED}

.
Proposed by Andy Xu
TB3. If

x

y

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

\frac{x}{y}+\frac{y}{x} = 3

xy

.
Proposed by Harry Kim and Andy Xu

1

2023 MOAA Gunga P24

\omega

is inscribed in acute triangle

ABC

I

denote the center of

\omega

D,E,F

be the points of tangency of

\omega

BC, CA, AB

respectively. Let

M

be the midpoint of

BC

P

be the intersection of the line through

I

perpendicular to

AM

EF

AP=9

EC=2EA

BD=3

. Find the sum of all possible perimeters of

\triangle ABC

.
Proposed by Harry Kim

1

2023 MOAA Gunga P23

For every positive integer

n

f(n) = \frac{n^4+n^3+n^2-n+1}{n^6-1}

\sum_{n = 2}^{20} f(n) = \frac{a}{b}

for relatively prime positive integers

a

b

, find the sum of the prime factors of

b

.
Proposed by Harry Kim

1

2023 MOAA Gunga P22

Harry the knight is positioned at the origin of the Cartesian plane. In a "knight hop", Harry can move from the point

(i,j)

to a point with integer coordinates that is a distance of

\sqrt{5}

(i,j)

. What is the number of ways that Harry can return to the origin after 6 knight hops?
Proposed by Harry Kim

1

2023 MOAA Gunga P21

In obtuse triangle

ABC

\angle B > 90^\circ

H

O

be its orthocenter and circumcenter respectively. Let

D

be the foot of the altitude from

A

HC

E

be the foot of the altitude from

B

AC

O,E,D

lie on a line. If

OC=8

OE=4

, find the area of triangle

HAB

.
Proposed by Harry Kim

1

2023 MOAA Gunga P20

Big Bad Brandon is assigning groups of his Big Bad Burglars to attack 7 different towers. Each Burglar can only belong to one attack group and Brandon takes over a tower if the number of Burglars attacking the tower strictly exceeds the number of knights guarding it. He knows there the total number of knights guarding the towers is 99 but does not know the exact number of knights guarding each tower. What is the minimum number of Burglars that Brandon needs to guarantee he can take over at least 4 of the 7 towers?
Proposed by Eric Wang

1

2023 MOAA Gunga P19

Compute the remainder when

\binom{205}{101}

101 \times 103

.
Proposed by Brandon Xu

1

2023 MOAA Gunga P18

\triangle{ABC}

is isosceles with

AB = AC

. Let the incircle of

\triangle{ABC}

BC

AC

D

E

respectively. Let

F \neq A

be the point such that

DF = DA

EF = EA

AF = 8

and the circumradius of

\triangle{AED}

5

, find the area of

\triangle{ABC}

.
Proposed by Anthony Yang and Andy Xu

1

2023 MOAA Gunga P17

Call a polynomial with real roots n-local if the greatest difference between any pair of its roots is

n

f(x)=x^2+ax+b

be a 1-local polynomial with distinct roots such that

a

b

are non-zero integers. If

f(f(x))

is a 23-local polynomial, find the sum of the roots of

f(x)

.
Proposed by Anthony Yang

1

2023 MOAA Gunga P16

Compute the sum

\frac{\varphi(50!)}{\varphi(49!)}+ \frac{\varphi(51!)}{\varphi(50!)} + \dots + \frac{\varphi(100!)}{\varphi(99!)}

\varphi(n)

returns the number of positive integers less than

n

that are relatively prime to

n

.
Proposed by Andy Xu

2

2023 MOAA Team P15

ABC

has circumcircle

\omega

D

be the foot of the altitude from

A

BC

AD

\omega

E \neq A

M

be the midpoint of

AD

\angle{BMC} = 90^\circ

AB = 9

AE = 10

\triangle{ABC}

can be expressed in the form

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

a,b,c

are positive integers and

b

is square-free. Find

a+b+c

.
Proposed by Andy Xu

2023 MOAA Gunga P15

ABC

AB = 5

BC = 7

CA = 8

M

be the midpoint of

BC

P

Q

AB

AC

respectively such that

MP \perp AB

MQ \perp AC

H

is the orthocenter of

\triangle{APQ}

then the area of

\triangle{HPM}

can be expressed in the form

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

a

c

are relatively prime positive integers and

b

is square-free. Find

a+b+c

.
Proposed by Harry Kim

2

2023 MOAA Team P14

For a positive integer

n

f(n)

denote the number of positive integers

a\leq n

\gcd(a,n) = \gcd(a+1,n) = 1

. Find the sum of all

n

f(n)=15

.
Proposed by Harry Kim

2023 MOAA Gunga P14

N

be the number of ordered triples of 3 positive integers

(a,b,c)

6a

10b

15c

are all perfect squares and

abc = 210^{210}

. Find the number of divisors of

N

.
Proposed by Andy Xu

2

2023 MOAA Team P13

If real numbers

x

y

z

x^2-yz = 1

y^2-xz = 4

|x+y+z|

is minimized, then

z^2-xy

can be expressed in the form

\sqrt{a}-b

a

b

are positive integers. Find

a+b

.
Proposed by Andy Xu

2023 MOAA Gunga P13

\alpha

\beta

\gamma

be the roots of the polynomial

2023x^3-2023x^2-1

\frac{1}{\alpha^3}+\frac{1}{\beta^3}+\frac{1}{\gamma^3}

.
Proposed by Andy Xu

2

2023 MOAA Team P12

N

be the number of

105

-digit positive integers that contain the digit 1 an odd number of times. Find the remainder when

N

1000

.
Proposed by Harry Kim

2023 MOAA Gunga P12

Andy is planning to flip a fair coin 10 times. Among the 10 flips, Valencia randomly chooses one flip to exchange Andy's fair coin with her special coin which lands on heads with a probability of

\frac{1}{4}

. If the coin is exchanged in a certain flip, then that flip, along with all following flips will be performed with the special coin. The expected number of heads Andy flips can be expressed as

\frac{m}{n}

m

n

are positive integers. Find

m+n

.
Proposed by Andy Xu

2

2023 MOAA Team P11

Let the quadratic

P(x)=x^2+5x+1

. Two distinct real numbers

a,b

P(a+b)=ab

P(ab)=a+b

Find the sum of all possible values of

a^2

.
Proposed by Harry Kim

2023 MOAA Gunga P11

s(n)

denote the sum of the digits of

n

p(n)

be the product of the digits of

n

. Find the smallest integer

k

s(k)+p(k)=49

s(k+1)+p(k+1)=68

.
Proposed by Anthony Yang

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