MathDB

Round 1
p1. Ben the bear has an algorithm he runs on positive integers- each second, if the integer is even, he divides it by

2

, and if the integer is odd, he adds

1

. The algorithm terminates after he reaches

1

. What is the least positive integer n such that Ben's algorithm performed on n will terminate after seven seconds? (For example, if Ben performed his algorithm on

3

, the algorithm would terminate after

3

seconds:

3 \to 4 \to 2 \to 1

.)
p2. Suppose that a rectangle

R

has length

p

and width

q

, for prime integers

p

and

q

. Rectangle

S

has length

p + 1

and width

q + 1

. The absolute difference in area between

S

and

R

21

. Find the sum of all possible values of

p

.
p3. Owen the origamian takes a rectangular

12 \times 16

sheet of paper and folds it in half, along the diagonal, to form a shape. Find the area of this shape.
Round 2
p4. How many subsets of the set

\{G, O, Y, A, L, E\}

contain the same number of consonants as vowels? (Assume that

Y

is a consonant and not a vowel.)
p5. Suppose that trapezoid

ABCD

satisfies

AB = BC = 5

CD = 12

, and

\angle ABC = \angle BCD = 90^o

. Let

AC

and

BD

intersect at

E

. The area of triangle

BEC

can be expressed as

\frac{a}{b}

, for positive integers

a

and

b

with

gcd(a, b) = 1

. Find

a + b

.
p6. Find the largest integer

n

for which

\frac{101^n + 103^n}{101^{n-1} + 103^{n-1}}

is an integer.
Round 3
p7. For each positive integer n between

1

and

1000

(inclusive), Ben writes down a list of

n

's factors, and then computes the median of that list. He notices that for some

n

, that median is actually a factor of

n

. Find the largest

n

for which this is true.
p8. ([color=#f00]voided) Suppose triangle

ABC

has

AB = 9

BC = 10

, and

CA = 17

. Let

x

be the maximal possible area of a rectangle inscribed in

ABC

, such that two of its vertices lie on one side and the other two vertices lie on the other two sides, respectively. There exist three rectangles

R_1

R_2

, and

R_3

such that each has an area of

x

. Find the area of the smallest region containing the set of points that lie in at least two of the rectangles

R_1

R_2

, and

R_3

.
p9. Let

a, b,

and

c

be the three smallest distinct positive values of

\theta

satisfying

\cos \theta + \cos 3\theta + ... + \cos 2021\theta = \sin \theta+ \sin 3 \theta+ ... + \sin 2021\theta.

What is

\frac{4044}{\pi}(a + b + c)

?
[color=#f00]Problem 8 is voided.
PS. You should use hide for answers.Rounds 4-5 have been posted [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here . Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement