MathDB

Set 6
B16. Let

\ell_r

denote the line

x + ry + r^2 = 420

. Jeffrey draws the lines

\ell_a

and

\ell_b

and calculates their single intersection point.
B17. Let set

L

consist of lines of the form

3x + 2ay = 60a + 48

across all real constants a. For every line

\ell

L

, the point on

\ell

closest to the origin is in set

T

. The area enclosed by the locus of all the points in

T

can be expressed in the form nπ for some positive integer

n

. Compute

n

.
B18. What is remainder when the

2020

-digit number

202020 ... 20

is divided by

275

?
Set 7
B19. Consider right triangle

\vartriangle ABC

where

\angle ABC = 90^o

\angle ACB = 30^o

, and

AC = 10

. Suppose a beam of light is shot out from point

A

. It bounces off side

BC

and then bounces off side

AC

, and then hits point

B

and stops moving. If the beam of light travelled a distance of

d

, then compute

d^2

.
B20. Let

S

be the set of all three digit numbers whose digits sum to

12

. What is the sum of all the elements in

S

?
B21. Consider all ordered pairs

(m, n)

where

m

is a positive integer and

n

is an integer that satisfy

m! = 3n^2 + 6n + 15,

where

m! = m \times (m - 1) \times ... \times 1

. Determine the product of all possible values of

n

.
Set 8
B22. Compute the number of ordered pairs of integers

(m, n)

satisfying

1000 > m > n > 0

and

6 \cdot lcm(m - n, m + n) = 5 \cdot lcm(m, n)

.
B23. Andrew is flipping a coin ten times. After every flip, he records the result (heads or tails). He notices that after every flip, the number of heads he had flipped was always at least the number of tails he had flipped. In how many ways could Andrew have flipped the coin?
B24. Consider a triangle

ABC

with

AB = 7

BC = 8

, and

CA = 9

. Let

D

lie on

\overline{AB}

and

E

lie on

\overline{AC}

such that

BCED

is a cyclic quadrilateral and

D, O, E

are collinear, where

O

is the circumcenter of

ABC

. The area of

\vartriangle ADE

can be expressed as

\frac{m\sqrt{n}}{p}

, where

m

and

p

are relatively prime positive integers, and

n

is a positive integer not divisible by the square of any prime. What is

m + n + p

?
Set 9
This set consists of three estimation problems, with scoring schemes described.
B25. Submit one of the following ten numbers:

3 \,\,\,\, 6\,\,\,\, 9\,\,\,\, 12\,\,\,\, 15\,\,\,\, 18\,\,\,\, 21\,\,\,\, 24\,\,\,\, 27\,\,\,\, 30.

The number of points you will receive for this question is equal to the number you selected divided by the total number of teams that selected that number, then rounded up to the nearest integer. For example, if you and four other teams select the number

27

, you would receive

\left\lceil \frac{27}{5}\right\rceil = 6

points.
B26. Submit any integer from

1

1,000,000

, inclusive. The standard deviation

\sigma

of all responses

x_i

to this question is computed by first taking the arithmetic mean

\mu

of all responses, then taking the square root of average of

(x_i -\mu)^2

over all

i

. More, precisely, if there are

N

responses, then

\sigma =\sqrt{\frac{1}{N} \sum^N_{i=1} (x_i -\mu)^2}.

For this problem, your goal is to estimate the standard deviation of all responses.
An estimate of

e

gives

\max \{ \left\lfloor 130 ( min \{ \frac{\sigma }{e},\frac{e}{\sigma }\}^{3}\right\rfloor -100,0 \}

points.
B27. For a positive integer

n

, let

f(n)

denote the number of distinct nonzero exponents in the prime factorization of

n

. For example,

f(36) = f(2^2 \times 3^2) = 1

and

f(72) = f(2^3 \times 3^2) = 2

. Estimate

N = f(2) + f(3) +.. + f(10000)

.
An estimate of

e

gives

\max \{30 - \lfloor 7 log_{10}(|N - e|)\rfloor , 0\}

points.

PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777391p24371239]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Sets 6-9

Problems(1)