MathDB

Set 7
p19. Let circles

\omega_1

and

\omega_2

, with centers

O_1

and

O_2

, respectively, intersect at

X

and

Y

. A lies on

\omega_1

and

B

lies on

\omega_2

such that

AO_1

and

BO_2

are both parallel to

XY

, and

A

and

B

lie on the same side of

O_1O_2

. If

XY = 60

\angle XAY = 45^o

, and

\angle XBY = 30^o

, then the length of

AB

can be expressed in the form

\sqrt{a - b\sqrt2 + c\sqrt3}

, where

a, b, c

are positive integers. Determine

a + b + c

.
p20. If

x

is a positive real number such that

x^{x^2}= 2^{80}

, find the largest integer not greater than

x^3

.
p21. Justin has a bag containing

750

balls, each colored red or blue. Sneaky Sam takes out a random number of balls and replaces them all with green balls. Sam notices that of the balls left in the bag, there are

15

more red balls than blue balls. Justin then takes out

500

of the balls chosen randomly. If

E

is the expected number of green balls that Justin takes out, determine the greatest integer less than or equal to

E

.
Set 8
These three problems are interdependent; each problem statement in this set will use the answers to the other two problems in this set. As such, let the positive integers

A, B, C

be the answers to problems

22

23

, and

24

, respectively, for this set.
p22. Let

WXYZ

be a rectangle with

WX =\sqrt{5B}

and

XY =\sqrt{5C}

. Let the midpoint of

XY

M

and the midpoint of

YZ

N

. If

XN

and

W Y

intersect at

P

, determine the area of

MPNY

.
p23. Positive integers

x, y, z

satisfy

xy \equiv A \,\, (mod 5)

yz \equiv 2A + C\,\, (mod 7)

zx \equiv C + 3 \,\, (mod 9).

(Here, writing

a \equiv b \,\, (mod m)

is equivalent to writing

m | a - b

.) Given that

3 \nmid x

3 \nmid z

, and

9 | y

, find the minimum possible value of the product

xyz

.
p24. Suppose

x

and

y

are real numbers such that

x + y = A

xy =\frac{1}{36}B^2.

Determine

|x - y|

.
Set 9
p25. The integer

2017

is a prime which can be uniquely represented as the sum of the squares of two positive integers:

9^2 + 44^2 = 2017.

N = 2017 \cdot 128

can be uniquely represented as the sum of the squares of two positive integers

a^2 +b^2

, determine

a + b

.
p26. Chef Celia is planning to unveil her newest creation: a whole-wheat square pyramid filled with maple syrup. She will use a square flatbread with a one meter diagonal and cut out each of the five polygonal faces of the pyramid individually. If each of the triangular faces of the pyramid are to be equilateral triangles, the largest volume of syrup, in cubic meters, that Celia can enclose in her pyramid can be expressed as

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

where

a, b

and

c

are the smallest possible possible positive integers. What is

a + b + c

?
p27. In the Cartesian plane, let

\omega

be the circle centered at

(24, 7)

with radius

6

. Points

P, Q

, and

R

are chosen in the plane such that

P

lies on

\omega

Q

lies on the line

y = x

, and

R

lies on the

x

-axis. The minimum possible value of

PQ+QR+RP

can be expressed in the form

\sqrt{m}

for some integer

m

. Find m.
Set 10
Deja vu?
p28. Let

ABC

be a triangle with incircle

\omega

. Let

\omega

intersect sides

BC

CA

AB

D, E, F

, respectively. Suppose

AB = 7

BC = 12

, and

CA = 13

. If the area of

ABC

K

and the area of

DEF

\frac{m}{n}\cdot K

, where

m

and

n

are relatively prime positive integers, then compute

m + n

.
p29. Sebastian is playing the game Split! again, but this time in a three dimensional coordinate system. He begins the game with one token at

(0, 0, 0)

. For each move, he is allowed to select a token on any point

(x, y, z)

and take it off, replacing it with three tokens, one at

(x + 1, y, z)

, one at

(x, y + 1, z)

, and one at

(x, y, z + 1)

At the end of the game, for a token on

(a, b, c)

, it is assigned a score

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

. These scores are summed for his total score. If the highest total score Sebastian can get in

100

moves is

m/n

, then determine

m + n

.
p30. Determine the number of positive

6

digit integers that satisfy the following properties:

\bullet

All six of their digits are

1, 5, 7

, or

8

\bullet

The sum of all the digits is a multiple of

5

.
Set 11
p31. The triangular numbers are defined as

T_n =\frac{n(n+1)}{2}

. We also define

S_n =\frac{n(n+2)}{3}

. If the sum

\sum_{i=16}^{32} \left(\frac{1}{T_i}+\frac{1}{S_i}\right)= \left(\frac{1}{T_{16}}+\frac{1}{S_{16}}\right)+\left(\frac{1}{T_{17}}+\frac{1}{S_{17}}\right)+...+\left(\frac{1}{T_{32}}+\frac{1}{S_{32}}\right)

can be written in the form

a/b

, where

a

and

b

are positive integers with

gcd(a, b) = 1

, then find

a + b

.
p32. Farmer Will is considering where to build his house in the Cartesian coordinate plane. He wants to build his house on the line

y = x

, but he also has to minimize his travel time for his daily trip to his barnhouse at

(24, 15)

and back. From his house, he must first travel to the river at

y = 2

to fetch water for his animals. Then, he heads for his barnhouse, and promptly leaves for the long strip mall at the line

y =\sqrt3 x

for groceries, before heading home. If he decides to build his house at

(x_0, y_0)

such that the distance he must travel is minimized,

x_0

can be written in the form

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

, where

a, b, c, d

are positive integers,

b

is not divisible by the square of a prime, and

gcd(a, c, d) = 1

. Compute

a+b+c+d

.
p33. Determine the greatest positive integer

n

such that the following two conditions hold:

\bullet

n^2

is the difference of consecutive perfect cubes;

\bullet

2n + 287

is the square of an integer.
Set 12
The answers to these problems are nonnegative integers that may exceed

1000000

. You will be awarded points as described in the problems.
p34. The “Collatz sequence” of a positive integer n is the longest sequence of distinct integers

(x_i)_{i\ge 0}

with

x_0 = n

and

x_{n+1} =\begin{cases} \frac{x_n}{2} & if \,\, x_n \,\, is \,\, even \\ 3x_n + 1 & if \,\, x_n \,\, is \,\, odd \end{cases}.

It is conjectured that all Collatz sequences have a finite number of elements, terminating at

1

. This has been confirmed via computer program for all numbers up to

2^{64}

. There is a unique positive integer

n < 10^9

such that its Collatz sequence is longer than the Collatz sequence of any other positive integer less than

10^9

. What is this integer

n

?
An estimate of

e

gives

\max\{\lfloor 32 - \frac{11}{3}\log_{10}(|n - e| + 1)\rfloor, 0\}

points.
p35. We define a graph

G

as a set

V (G)

of vertices and a set

E(G)

of distinct edges connecting those vertices. A graph

H

is a subgraph of

G

if the vertex set

V (H)

is a subset of

V (G)

and the edge set

E(H)

is a subset of

E(G)

. Let

ex(k, H)

denote the maximum number of edges in a graph with

k

vertices without a subgraph of

H

. If

K_i

denotes a complete graph on

i

vertices, that is, a graph with

i

vertices and all

{i \choose 2}

edges between them present, determine

n =\sum_{i=2}^{2018} ex(2018, K_i).

An estimate of

e

gives

\max\{\lfloor 32 - 3\log_{10}(|n - e| + 1)\rfloor, 0\}

points.
p36. Write down an integer between

1

and

100

, inclusive. This number will be denoted as

n_i

, where your Team ID is

i

. Let

S

be the set of Team ID’s for all teams that submitted an answer to this problem. For every ordered triple of distinct Team ID’s

(a, b, c)

such that a, b, c ∈ S, if all roots of the polynomial

x^3 + n_ax^2 + n_bx + n_c

are real, then the teams with ID’s

a, b, c

will each receive one virtual banana.
If you receive

v_b

virtual bananas in total and

|S| \ge 3

teams submit an answer to this problem, you will be awarded

\left\lfloor \frac{32v_b}{3(|S| - 1)(|S| - 2)}\right\rfloor

points for this problem. If

|S| \le 2

, the team(s) that submitted an answer to this problem will receive

32

points for this problem.
PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777264p24369138]here.Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement