MathDB

1

2020 PUMaC Team 13

Will and Lucas are playing a game. Will claims that he has a polynomial

f

with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute

i

for the value of

f(i)

, starting from minute

1

. If Will is telling the truth, he will report

f(i)

. Otherwise, he will randomly and uniformly pick a positive integer from the range

[1,(i+1)!]

. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round

20

is

a/b

. If the prime factorization of

b

is

p_1^{e_1}... p_k^{e_k}

, determine the sum

\sum_{i=1}^{k} e_i

.

12

1

2020 PUMaC Team 12

Given a sequence

a_0, a_1, a_2, ... , a_n

, let its arithmetic approximant be the arithmetic sequence

b_0, b_1, ... , b_n

that minimizes the quantity

\sum_{i=0}^{n}(b_i -a_i)^2

, and denote this quantity the sequence’s anti-arithmeticity. Denote the number of integer sequences whose arithmetic approximant is the sequence

4

,

8

,

12

,

16

and whose anti-arithmeticity is at most

20

.

15

1

2020 PUMaC Team 15

Suppose that f is a function

f : R_{\ge 0} \to R

so that for all

x, y \in R_{\ge 0}

(nonnegative reals) we have that

f(x)+f(y) = f(x+y+xy)+f(x)f(y).

Given that

f\left(\frac{3}{5} \right) = \frac12

and

f(1) = 3

, determine

\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.

14

1

2020 PUMaC Team 14

Let

N

be the number of convex

27

-gons up to rotation there are such that each side has length

1

and each angle is a multiple of

2\pi/81

. Find the remainder when

N

is divided by

23

.

11

1

2020 PUMaC Team 11

Three (not necessarily distinct) points in the plane which have integer coordinates between

1

and

2020

, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is

a/b

in lowest terms. If the three points are collinear, the area of the degenerate triangle is

0

. Find

a + b

.

10

1

2020 PUMaC Team 10

Let

N

be the number of sequences of positive integers greater than

1

where the product of all of the terms of the sequence is

12^{64}

. If

N

can be expressed as

a(2^b)

), where

a

is an odd positive integer, determine

b

.

9

1

2020 PUMaC Team 9

Consider a regular

2020

-gon circumscribed into a circle of radius

1

. Given three vertices of this polygon such that they form an isosceles triangle, let

X

be the expected area of the isosceles triangle they create.

X

can be written as

\frac{1}{m \tan((2\pi)/n)}

where

m

and

n

are integers. Compute

m + n

.

8

1

2020 PUMaC Team 8

Let there be a tiger, William, at the origin. William leaps

1

unit in a random direction, then leaps

2

units in a random direction, and so forth until he leaps

15

units in a random direction to celebrate PUMaC’s 15th year. There exists a circle centered at the origin such that the probability that William is contained in the circle (assume William is a point) is exactly

1/2

after the

15

leaps. The area of that circle can be written as

A\pi

. What is

A

?

4

1

2020 PUMaC Team 4

Find the number of points

P \in Z^2

that satisfy the following two conditions: 1) If

Q

is a point on the circle of radius

\sqrt{2020}

centered at the origin such that the line

PQ

is tangent to the circle at

Q

, then

PQ

has integral length. 2) The x-coordinate of

P

is

38

.

6

1

2020 PUMaC Team 6

We say that a string of digits from

0

to

9

is valid if the following conditions hold: First, for

2 \le k \le 4

, no consecutive run of

k

digits sums to a multiple of

10

. Second, between any two

0

s, there are at least

3

other digits. Find the last four digits of the number of valid strings of length

2020

.

5

1

2020 PUMaC Team 5

Suppose two polygons may be glued together at an edge if and only if corresponding edges of the same length are made to coincide. A

3\times 4

rectangle is cut into

n

pieces by making straight line cuts. What is the minimum value of

n

so that it’s possible to cut the pieces in such a way that they may be glued together two at a time into a polygon with perimeter at least

2021

?

3

1

2020 PUMaC Team 3

Alice and Bob are playing a guessing game. Bob is thinking of a number n of the form

2^a3^b

, where a and b are positive integers between

1

and

2020

, inclusive. Each turn, Alice guess a number m, and Bob will tell her either

\gcd (m, n)

or

lcm (m, n)

(letting her know that he is saying that

gcd

or

lcm

), as well as whether any of the respective powers match up in their prime factorization. In particular, if

m = n

, Bob will let Alice know this, and the game is over. Determine the smallest number

k

so that Alice is always able to find

n

within

k

guesses, regardless of Bob’s number or choice of revealing either the

lcm

, or the

gcd

.

2

1

2020 PUMaC Team 2

Gary is baking cakes, one at a time. However, Gary’s not been having much success, and each failed cake will cause him to slowly lose his patience, until eventually he gives up. Initially, a failed cake has a probability of

0

of making him give up. Each cake has a

1/2

of turning out well, with each cake independent of every other cake. If two consecutive cakes turn out well, the probability resets to

0

immediately after the second cake. On the other hand, if the cake fails, assuming that he doesn’t give up at this cake, his probability of breaking on the next failed cake goes from p to

p + 0.5

. If the expected number of successful cakes Gary will bake until he gives up is

p/q

, for relatively prime

p, q

, find

p + q

.

1

2020 PUMaC Team 1

Consider a

2021

-by-

2021

board of unit squares. For some integer

k

, we say the board is tiled by

k

-by-

k

squares if it is completely covered by (possibly overlapping)

k

-by-

k

squares with their corners on the corners of the unit squares. What is the largest integer k such that the minimum number of

k

-by-

k

squares needed to tile the

2021

-by-

2021

board is exactly equal to

100

?

B6

1

2020 PUMaC Combinatorics B6

Billy the baker makes a bunch of loaves of bread every day, and sells them in bundles of size

1, 2

, or

3

. On one particular day, there are

375

orders,

125

for each bundle type. As such, Billy goes ahead and makes just enough loaves of bread to meet all the orders. Whenever Billy makes loaves, some get burned, and are not sellable. For nonnegative i less than or equal to the total number of loaves, the probability that exactly i loaves are sellable to customers is inversely proportional to

2^i

(otherwise, it’s

0

). Once he makes the loaves, he distributes out all of the sellable loaves of bread to some subset of these customers (each of whom will only accept their desired bundle of bread), without worrying about the order in which he gives them out. If the expected number of ways Billy can distribute the bread is of the form

\frac{a^b}{2^c-1}

, find

a + b + c

.

A4/B5

1

2020 PUMaC Combinatorics A4 / B5

Let

P

be the power set of

\{1, 2, 3, 4\}

(meaning the elements of P are the subsets of

\{1, 2, 3, 4\}

). How many subsets

S

of

P

are there such that no two distinct integers

a, b \in \{1, 2, 3, 4\}

appear together in exactly one element of

S

?

A3/B4

1

2020 PUMaC Combinatorics A3 / B4

Katie has a chocolate bar that is a

5

-by-

5

grid of square pieces, but she only wants to eat the center piece. To get to it, she performs the following operations: i. Take a gridline on the chocolate bar, and split the bar along the line. ii. Remove the piece that doesn’t contain the center. iii. With the remaining bar, repeat steps

1

and

2

. Determine the number of ways that Katie can perform this sequence of operations so that eventually she ends up with just the center piece.

A2/B3

1

2020 PUMaC Combinatorics A2 / B3

Cary has six distinct coins in a jar. Occasionally, he takes out three of the coins and adds a dot to each of them. Determine the number of orders in which Cary can choose the coins so that, eventually, for each number

i \in \{0, 1, . . . , 5\}

, some coin has exactly

i

dots on it.

A1/B2

1

2020 PUMaC Combinatorics A1 / B2

Joey is playing with a

2

-by-

2

-by-

2

Rubik’s cube made up of

8

1

-by-

1

-by-

1

cubes (with two of these smaller cubes along each of the sides of the bigger cubes). Each face of the Rubik’s cube is distinct color. However, one day, Joey accidentally breaks the cube! He decides to put the cube back together into its solved state, placing each of the pieces one by one. However, due to the nature of the cube, he is only able to put in a cube if it is adjacent to a cube he already placed. If different orderings of the ways he chooses the cubes are considered distinct, determine the number of ways he can reassemble the cube.

7

1

max area of triangle with vertices on concentric circles 2020 PUMaC Team 7

Let

X, Y

, and

Z

be concentric circles with radii

1

,

13

, and

22

, respectively. Draw points

A, B

, and

C

on

X

,

Y

, and

Z

, respectively, such that the area of triangle

ABC

is as large as possible. If the area of the triangle is

\Delta

, find

\Delta^2

.

A1

1

2020 PUMaC Individual Finals A1

Let

a_1, . . . , a_{2020}

be a sequence of real numbers such that

a_1 = 2^{-2019}

, and

a^2_{n-1}a_n = a_n-a_{n-1}

. Prove that

a_{2020} <\frac{1}{2^{2019} -1}

A3

1

2020 PUMaC Individual Finals A3

Let

n

be a positive integer, and let

F

be a family of subsets of

\{1, 2, ... , 2^n\}

such that for any non-empty

A\in F

there exists

B \in F

so that

|A| = |B| + 1

and

B \subset A

. Suppose that F contains all

(2^n - 1)

-element subsets of

\{1, 2, ... , 2^n\}

Determine the minimal possible value of

|F|

.

B3

1

tangent circles of BDX, CEX iff ABxAD<=4R^2 2020 PUMaC Individual Finals B3

Let

ABC

be a triangle and let the points

D, E

be on the rays

AB

,

AC

such that

BCED

is cyclic. Prove that the following two statements are equivalent:

\bullet

There is a point

X

on the circumcircle of

ABC

such that

BDX

,

CEX

are tangent to each other.

\bullet

AB \cdot AD \le 4R^2

, where

R

is the radius of the circumcircle of

ABC

.

A2

1

segment construction by straightedge + pencil 2020 PUMaC Individual Finals A2

Helen has a wooden rectangle of unknown dimensions, a straightedge, and a pencil (no compass). Is it possible for her to construct a line segment on the rectangle connecting the midpoints of two opposite sides, where she cannot draw any lines or points outside the rectangle?
Note: Helen is allowed to draw lines between two points she has already marked, and mark the intersection of any two lines she has already drawn, if the intersection lies on the rectangle. Further, Helen is allowed to mark arbitrary points either on the rectangle or on a segment she has previously drawn. Assume that only the four vertices of the rectangle have been marked prior to the beginning of this process.

4

4

5

4

4

4

3

2020 PUMaC Algebra A4 / B6

Let

P

be a

10

-degree monic polynomial with roots

r_1, r_2, . . . , r_{10} \ne

and let

Q

be a

45

-degree monic polynomial with roots

\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}

where

i < j

and

i, j \in \{1, ... , 10\}

. If

P(0) = Q(1) = 2

, then

\log_2 (|P(1)|)

can be written as

a/b

for relatively prime integers

a, b

. Find

a + b

.

2020 PUMaC Geometry A4 / B6

Let

C

be a circle centered at point

O

, and let

P

be a point in the interior of

C

. Let

Q

be a point on the circumference of

C

such that

PQ \perp OP

, and let

D

be the circle with diameter

PQ

. Consider a circle tangent to

C

whose circumference passes through point

P

. Let the curve

\Gamma

be the locus of the centers of all such circles. If the area enclosed by

\Gamma

is

1/100

the area of

C

, then what is the ratio of the area of

C

to the area of

D

?

2020 PUMaC NT A4 / B6

Given two positive integers

a \ne b

, let

f(a, b)

be the smallest integer that divides exactly one of

a, b

, but not both. Determine the number of pairs of positive integers

(x, y)

, where

x \ne y

,

1\le x, y, \le 100

and

\gcd(f(x, y), \gcd(x, y)) = 2

.

A3/B5

3

2020 PUMaC Algebra A3 / B5

Let

\{x\} = x- \lfloor x \rfloor

. Consider a function f from the set

\{1, 2, . . . , 2020\}

to the half-open interval

[0, 1)

. Suppose that for all

x, y,

there exists a

z

so that

\{f(x) + f(y)\} = f(z)

. We say that a pair of integers

m, n

is valid if

1 \le m, n \le 2020

and there exists a function

f

satisfying the above so

f(1) = \frac{m}{n}

. Determine the sum over all valid pairs

m, n

of

{m}{n}

.

2020 PUMaC Geometry A3 / B5

Let

ABCD

be a cyclic quadrilateral with circumcenter

O

and radius

10

. Let sides

AB

,

BC

,

CD

, and

DA

have midpoints

M, N, P

, and

Q

, respectively. If

MP = NQ

and

OM + OP = 16

, then what is the area of triangle

\vartriangle OAB

?

2020 PUMaC NT A3 / B5

Find the sum (in base

10

) of the three greatest numbers less than

1000_{10}

that are palindromes in both base

10

and base

5

.

A2/B4

3

2020 PUMaC Algebra A2 / B4

Let

C

denote the curve

y^2 =\frac{x(x+1)(2x+1)}{6}

. The points

(1/2, a)

,

(b, c)

, and

(24, d)

lie on

C

and are collinear, and

ad < 0

. Given that

b, c

are rational numbers, find

100b^2 + c^2

.

2020 PUMaC Geometry A2 / B4

Hexagon

ABCDEF

has an inscribed circle

\Omega

that is tangent to each of its sides. If

AB = 12

,

\angle FAB = 120^o

, and

\angle ABC = 150^o

, and if the radius of

\Omega

can be written as

m +\sqrt{n}

for positive integers

m, n

, find

m + n

.

2020 PUMaC NT A2 / B4

How many ordered triples of nonzero integers

(a, b, c)

satisfy

2abc = a + b + c + 4

?

A1/B3

3

2020 PUMaC Algebra A1 / B3

Let

f(x) =\frac{x+a}{x+b}

satisfy

f(f(f(x))) = x

for real numbers

a, b

. If the maximum value of a is

p/q

, where

p, q

are relatively prime integers, what is

|p| + |q|

?

2020 PUMaC Geometry A1 / B3

Let

\gamma_1

and

\gamma_2

be circles centered at

O

and

P

respectively, and externally tangent to each other at point

Q

. Draw point

D

on

\gamma_1

and point

E

on

\gamma_2

such that line

DE

is tangent to both circles. If the length

OQ = 1

and the area of the quadrilateral

ODEP

is

520

, then what is the value of length

PQ

?

2020 PUMaC NT A1 / B3

Compute the last two digits of

9^{2020} + 9^{2020^2}+ ... + 9^{2020^{2020}}

2020 Princeton University Math Competition

Subcontests

2020 PUMaC Team 13

2020 PUMaC Team 12

2020 PUMaC Team 15

2020 PUMaC Team 14

2020 PUMaC Team 11

2020 PUMaC Team 10

2020 PUMaC Team 9

2020 PUMaC Team 8

2020 PUMaC Team 4

2020 PUMaC Team 6

2020 PUMaC Team 5

2020 PUMaC Team 3

2020 PUMaC Team 2

2020 PUMaC Team 1

2020 PUMaC Combinatorics B6

2020 PUMaC Combinatorics A4 / B5

2020 PUMaC Combinatorics A3 / B4

2020 PUMaC Combinatorics A2 / B3

2020 PUMaC Combinatorics A1 / B2

max area of triangle with vertices on concentric circles 2020 PUMaC Team 7

2020 PUMaC Individual Finals A1

2020 PUMaC Individual Finals A3

tangent circles of BDX, CEX iff ABxAD<=4R^2 2020 PUMaC Individual Finals B3

segment construction by straightedge + pencil 2020 PUMaC Individual Finals A2

2020 PUMaC Algebra A4 / B6

2020 PUMaC Geometry A4 / B6

2020 PUMaC NT A4 / B6

2020 PUMaC Algebra A3 / B5

2020 PUMaC Geometry A3 / B5

2020 PUMaC NT A3 / B5

2020 PUMaC Algebra A2 / B4

2020 PUMaC Geometry A2 / B4

2020 PUMaC NT A2 / B4

2020 PUMaC Algebra A1 / B3

2020 PUMaC Geometry A1 / B3

2020 PUMaC NT A1 / B3

2020 Princeton University Math Competition

Subcontests

2020 PUMaC Team 13

2020 PUMaC Team 12

2020 PUMaC Team 15

2020 PUMaC Team 14

2020 PUMaC Team 11

2020 PUMaC Team 10

2020 PUMaC Team 9

2020 PUMaC Team 8

2020 PUMaC Team 4

2020 PUMaC Team 6

2020 PUMaC Team 5

2020 PUMaC Team 3

2020 PUMaC Team 2

2020 PUMaC Team 1

2020 PUMaC Combinatorics B6

2020 PUMaC Combinatorics A4 / B5

2020 PUMaC Combinatorics A3 / B4

2020 PUMaC Combinatorics A2 / B3

2020 PUMaC Combinatorics A1 / B2

max area of triangle with vertices on concentric circles 2020 PUMaC Team 7

2020 PUMaC Individual Finals A1

2020 PUMaC Individual Finals A3

tangent circles of BDX, CEX iff ABxAD&lt;=4R^2 2020 PUMaC Individual Finals B3

segment construction by straightedge + pencil 2020 PUMaC Individual Finals A2

2020 PUMaC Algebra A4 / B6

2020 PUMaC Geometry A4 / B6

2020 PUMaC NT A4 / B6

2020 PUMaC Algebra A3 / B5

2020 PUMaC Geometry A3 / B5

2020 PUMaC NT A3 / B5

2020 PUMaC Algebra A2 / B4

2020 PUMaC Geometry A2 / B4

2020 PUMaC NT A2 / B4

2020 PUMaC Algebra A1 / B3

2020 PUMaC Geometry A1 / B3

2020 PUMaC NT A1 / B3

tangent circles of BDX, CEX iff ABxAD<=4R^2 2020 PUMaC Individual Finals B3