MathDB

2020 PUMaC Algebra A6 / B8

Source:

1/1/2022

Given integer

n

, let

W_n

be the set of complex numbers of the form

re^{2qi\pi}

, where

q

is a rational number so that

q_n \in Z

and

r

is a real number. Suppose that p is a polynomial of degree

\ge 2

such that there exists a non-constant function

f : W_n \to C

so that

p(f(x))p(f(y)) = f(xy)

for all

x, y \in W_n

. If

p

is the unique monic polynomial of lowest degree for which such an

f

exists for

n = 65

, find

p(10)

.

algebra

2020 PUMaC Combinatorics A6 / B8

Source:

1/1/2022

In the country of Princetonia, there are an infinite number of cities, connected by roads. For every two distinct cities, there is a unique sequence of roads that leads from one city to the other. Moreover, there are exactly three roads from every city. On a sunny morning in early July, n tourists have arrived at the capital of Princetonia. They repeat the following process every day: in every city that contains three or more tourists, three tourists are picked and one moves to each of the three cities connected to the original one by roads. If there are

2

or fewer tourists in the city, they do nothing. After some time, all tourists will settle and there will be no more changing cities. For how many values of n from

1

to

2020

will the tourists end in a configuration in which no two of them are in the same city?

combinatorics

2020 PUMaC Geometry A6 / B8

Source:

12/31/2021

Triangle

ABC

has side lengths

13

,

14

, and

15

. Let

E

be the ellipse that encloses the smallest area which passes through

A, B

, and

C

. The area of

E

is of the form

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

, where

a

and

c

are coprime and

b

has no square factors. Find

a + b + c

.

geometry

2020 PUMaC NT A6 / B8

Source:

1/1/2022

Find the number of ordered pairs of integers

(x, y)

such that

2167

divides

3x^2 + 27y^2 + 2021

with

0 \le x, y \le 2166

.

number theory

A6/B8

Problems(4)