2018 PUMaC Live Round

Part of 2018 Princeton University Math Competition

Subcontests

(27)

1

2018 PUMaC Live Round Miscellaneous 3

x,y\in\mathbb{Z}

y^4+4y^3+28y+8x^3+6y^2+32x+1=(x^2-y^2)(x^2+y^2+24).

Find the sum of all possible values of

|xy|

1

2018 PUMaC Live Round Estimation 3

Andrew starts with the

2018

-tuple of binary digits

(0,0,\dots,0)

. On each turn, he randomly chooses one index (between

1

2018

) and flips the digit at that index (makes it

1

0

and vice versa). What is the smallest

k

such that, after

k

steps, the expected number of ones in the sequence is greater than

1008?

You must give your answer as a nonnegative integer. If your answer is

A

and the correct answer is

C

, then your score will be

\max\{\lfloor18.5-\tfrac{|A-C|^{1.8}}{40}\rfloor,0\}.

1

2018 PUMaC Live Round Calculus 3

\mathcal{R}(f(x))

denote the number of distinct real roots of

f(x)

\sum_{a=1}^{1009}\sum_{b=1010}^{2018}\mathcal{R}(x^{2018}-ax^{2016}+b).

1

2018 PUMaC Live Round 8.3

a

b

are positive integers such that

3\sqrt{2+\sqrt{2+\sqrt{3}}}=a\cos\frac{\pi}{b}

a+b

1

2018 PUMaC Live Round 8.2

ABC

AB=10

\angle{A}=75^{\circ}

\angle{B}=60^{\circ}

\angle C = 45^{\circ}

I_A

be the center of the excircle opposite

A

D

E

be the circumcenters of triangle

BCI_A

ACI_A

respectively. If

O

is the circumcenter of triangle

ABC

, then the area of triangle

EOD

can be written as

\tfrac{a\sqrt{b}}{c}

for square-free

b

a,c

. Find the value of

a+b+c

1

2018 PUMaC Live Round 8.1

a

b

c

be such that the coefficient of the

x^ay^bz^c

term in the expansion of

(x+2y+3z)^{100}

is maximal (no other term has a strictly larger coefficient). Find the sum of all possible values of

1,000,000a+1,000b+c

1

2018 PUMaC Live Round 7.3

ABCD

has right angles at

B

D

AB<BC

E\in AB

F\in AD

AE=4

EF=7

FA=5

AB=8

X

ABCD

while satisfying

XE-XF=1

XE+XF+2XA=23

XA

may be written as

\tfrac{a-b\sqrt{c}}{d}

a,b,c,d

positive integers with

\gcd(a^2,b^2,c,d^2)=1

c

squarefree. Find

a+b+c+d

1

2018 PUMaC Live Round 7.2

Compute the smallest positive integer

n

that is a multiple of

29

with the property that for every positive integer that is relatively prime to

n

k^{n}\equiv 1\pmod{n}.

1

2018 PUMaC Live Round 7.1

Find the number of nonzero terms of the polynomial

P(x)

x^{2018}+x^{2017}+x^{2016}+x^{999}+1=(x^4+x^3+x^2+x+1)P(x).

1

2018 PUMaC Live Round Miscellaneous 2

What is the sum of the possible values for the complex number

a

such that the coefficient of the

x^5

term in the power series expansion of

\tfrac{x^3+ax^2+3x-4}{2x^2+ax+2}

1?

1

2018 PUMaC Live Round Estimation 2

How many perfect squares have the digits

1

9

each exactly once when written in base

10

?
You must give your answer as a nonnegative integer. If your answer is

A

and the correct answer is

C

, your score will be

\lfloor12.5\cdot\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor.

1

2018 PUMaC Live Round Calculus 2

Three friends are trying to meet for lunch at a cafe. Each friend will arrive independently at random between

1\!:\!00

2\!:\!00

pm. Each friend will only wait for

5

minutes by themselves before leaving. However, if another friend arrives within those

5

minutes, the pair will wait

15

minutes from the time the second friend arrives. If the probability that the three friends meet for lunch can be expressed in simplest form as

\tfrac{m}{n}

m+n

1

2018 PUMaC Live Round 5.3

k

be the largest integer such that

2^k

\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}\right)^2\right)\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}^2\right)\right).

k

1

2018 PUMaC Live Round 5.2

x^2

\tan^{-1}(x)+\tan^{-1}(3x)=\frac{\pi}{6}

0<x<\frac{\pi}{6}

1

2018 PUMaC Live Round 5.1

w

h

be positive integers and define

N(w,h)

to be the number of ways of arranging

wh

people of distinct heights for a photoshoot in such a way that they form

w

h

people, with the people of each column sorted by height (i.e. shortest at the front, tallest at the back). Find the largest value of

N(w,h)

1008

1

2018 PUMaC Live Round 4.3

0\leq a,b,c,d\leq 10

. For how many ordered quadruples

(a,b,c,d)

ad-bc

11?

1

2018 PUMaC Live Round 4.2

Some number of regular polygons meet at a point on the plane such that the polygons' interiors do not overlap, but the polygons fully surround the point (i.e. a sufficiently small circle centered at the point would be contained in the union of the polygons). What is the largest possible number of sides in any of the polygons?

1

2018 PUMaC Live Round 4.1

400000001

can be written as

p\cdot q

p

q

are prime numbers. Find the sum of the prime factors of

p+q-1

1

2018 PUMaC Live Round Miscellaneous 1

Consider all cubic polynomials

f(x)

f(2018)=2018

f

y

-axis at height

2018

, the coefficients of

f

2018

f(2019)>(2018)

.
We define the infinimum of a set

S

as follows. Let

L

be the set of lower bounds of

S

\ell\in L

if and only if for all

s\in S

\ell\leq s

. Then the infinimum of

S

\max(L)

f(x)

, what is the infinimum of the leading coefficient (the coefficient of the

x^3

1

2018 PUMaC Live Round Estimation 1

2

2018

grid is completely covered by non-overlapping L-tiles (see diagram below) and

1

1

squares. If the L-tiles can be rotated and flipped, there are a total of

M

such tilings. [asy] size(1cm); draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle); draw((0,1)--(1,1)--(1,0)); [/asy] What is

\ln M?

Give your answer as an integer or decimal. If your answer is

A

and the correct answer is

C

, then your score will be

\max\{\lfloor7.5-\tfrac{|A-C|^{1.5}}{20}\rfloor,0\}.

1

2018 PUMaC Live Round 2.3

20

indistinguishable pairs of socks in a laundry bag. She pulls them out one at a time. After pulling out

30

socks, the expected number of unmatched socks among the socks that she has pulled out can be expressed in simplest form as

\tfrac{m}{n}

m+n

1

2018 PUMaC Live Round 2.2

ABC

be a triangle with side lengths

13,14,15

. The points on the interior of

ABC

with distance at least

1

from each side are shaded. The area of the shaded region can be written in simplest form as

\tfrac{m}{n}

m+n

1

2018 PUMaC Live Round 2.1

Compute the period (i.e. length of the repeating part) of the decimal expansion of

\tfrac{1}{729}

1

2018 PUMaC Live Round 1.3

Let a sequence be defined as follows:

a_0=1

n>0

a_n

\tfrac{1}{3}a_{n-1}

\tfrac{1}{9}a_{n-1}

with probability

\tfrac{1}{2}

. If the expected value of

\textstyle\sum_{n=0}^{\infty}a_n

can be expressed in simplest form as

\tfrac{p}{q}

p+q

1

2018 PUMaC Live Round 1.2

Define a function given the following

2

\qquad

p

f(p)=p+1

\qquad

2) for positive integers

a

b

f(ab)=f(a)\cdot f(b)

. For how many positive integers

n\leq 100

f(n)

3

1

2018 PUMaC Live Round Live Round 1.1

Find the number of pairs of real numbers

(x,y)

x^4+y^4=4xy-2

1

2018 PUMaC Live Round Calculus 1

Freddy the king of flavortext has an infinite chest of coins. For each number

p

in the interval

[0, 1]

, Freddy has a coin that has probability

p

of coming up heads. Jenny the Joyous pulls out a random coin from the chest and flips it 10 times, and it comes up heads every time. She then flips the coin again. If the probability that the coin comes up heads on this 11th flip is

\frac{p}{q}

for two integers

p, q

p + q

.
Note: flavortext is made up