MathDB

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names
Set 4
R4.16 / P1.4 Adam and Becky are building a house. Becky works twice as fast as Adam does, and they both work at constant speeds for the same amount of time each day. They plan to finish building in

6

days. However, after

2

days, their friend Charlie also helps with building the house. Because of this, they finish building in just

5

days. What fraction of the house did Adam build?
R4.17 A bag with

10

items contains both pencils and pens. Kanye randomly chooses two items from the bag, with replacement. Suppose the probability that he chooses

1

pen and

1

pencil is

\frac{21}{50}

. What are all possible values for the number of pens in the bag?
R4.18 / P2.8 In cyclic quadrilateral

ABCD

\angle ABD = 40^o

, and

\angle DAC = 40^o

. Compute the measure of

\angle ADC

in degrees. (In cyclic quadrilaterals, opposite angles sum up to

180^o

.)
R4.19 / P2.6 There is a strange random number generator which always returns a positive integer between

1

and

7500

, inclusive. Half of the time, it returns a uniformly random positive integer multiple of

25

, and the other half of the time, it returns a uniformly random positive integer that isn’t a multiple of

25

. What is the probability that a number returned from the generator is a multiple of

30

?
R4.20 / P2.7 Julia is shopping for clothes. She finds

T

different tops and

S

different skirts that she likes, where

T \ge S > 0

. Julia can either get one top and one skirt, just one top, or just one skirt. If there are

50

ways in which she can make her choice, what is

T - S

?
Set 5
R5.21 A

5 \times 5 \times 5

cube’s surface is completely painted blue. The cube is then completely split into

1 \times 1 \times 1

cubes. What is the average number of blue faces on each

1 \times 1 \times 1

cube?
R5.22 / P2.10 Find the number of values of

n

such that a regular

n

-gon has interior angles with integer degree measures.
R5.23

4

positive integers form an geometric sequence. The sum of the

4

numbers is

255

, and the average of the second and the fourth number is

102

. What is the smallest number in the sequence?
R5.24 Let

S

be the set of all positive integers which have three digits when written in base

2016

and two digits when written in base

2017

. Find the size of

S

.
R5.25 / P3.12 In square

ABCD

with side length

13

, point

E

lies on segment

CD

. Segment

AE

divides

ABCD

into triangle

ADE

and quadrilateral

ABCE

. If the ratio of the area of

ADE

to the area of

ABCE

4 : 11

, what is the ratio of the perimeter of

ADE

to the perimeter of

ABCE

?
Set 6
R6.26 / P6.25 Submit a decimal n to the nearest thousandth between

0

and

200

. Your score will be

\min (12, S)

, where

S

is the non-negative difference between

n

and the largest number less than or equal to

n

chosen by another team (if you choose the smallest number,

S = n

). For example, 1.414 is an acceptable answer, while

\sqrt2

and

1.4142

are not.
R6.27 / P6.27 Guang is going hard on his YNA project. From

1:00

AM Saturday to

1:00

AM Sunday, the probability that he is not finished with his project

x

hours after

1:00

AM on Saturday is

\frac{1}{x+1}

. If Guang does not finish by 1:00 AM on Sunday, he will stop procrastinating and finish the project immediately. Find the expected number of minutes

A

it will take for him to finish his project. An estimate of

E

will earn

12 \cdot 2^{-|E-A|/60}

points.
R6.28 / P6.28 All the diagonals of a regular

100

-gon (a regular polygon with

100

sides) are drawn. Let

A

be the number of distinct intersection points between all the diagonals. Find

A

. An estimate of

E

will earn

12 \cdot \left(16 \log_{10}\left(\max \left(\frac{E}{A},\frac{A}{E}\right)\right)+ 1\right)^{-\frac12}

0

points if this expression is undefined.
R6.29 / P6.29 Find the smallest positive integer

A

such that the following is true: if every integer

1, 2, ..., A

is colored either red or blue, then no matter how they are colored, there are always 6 integers among them forming an increasing arithmetic progression that are all colored the same color. An estimate of

E

will earn

12 min \left(\frac{E}{A},\frac{A}{E}\right)

points or

0

points if this expression is undefined.
R6.30 / P6.30 For all integers

n \ge 2

, let

f(n)

denote the smallest prime factor of

n

. Find

A =\sum^{10^6}_{n=2}f(n)

. In other words, take the smallest prime factor of every integer from

2

10^6

and sum them all up to get

A

. You may find the following values helpful: there are

78498

primes below

10^6

9592

primes below

10^5

1229

primes below

10^4

, and

168

primes below

10^3

. An estimate of

E

will earn

\max \left(0, 12-4 \log_{10}(max \left(\frac{E}{A},\frac{A}{E}\right)\right)

0

points if this expression is undefined.
PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here, and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement