MathDB

p5. Find the number of three-digit integers that contain at least one

0

5

. The leading digit of the three-digit integer cannot be zero.
p6. What is the sum of the solutions to

\frac{x+8}{5x+7} =\frac{x+8}{7x+5}

p7. Let

BC

be a diameter of a circle with center

O

and radius

4

. Point

A

is on the circle such that

\angle AOB = 45^o

. Point

D

is on the circle such that line segment

OD

intersects line segment

AC

E

and

OD

bisects

\angle AOC

. Compute the area of

ADE

, which is enclosed by line segments

AE

and

ED

and minor arc

AD

.
p8. William is a bacteria farmer. He would like to give his fiance

2021

bacteria as a wedding gift. Since he is an intelligent and frugal bacteria farmer, he would like to add the least amount of bacteria on his favorite infinite plane petri dish to produce those

2021

bacteria. The infinite plane petri dish starts off empty and William can add as many bacteria as he wants each day. Each night, all the bacteria reproduce through binary fission, splitting into two. If he has infinite amount of time before his wedding day, how many bacteria should he add to the dish in total to use the least number of bacteria to accomplish his nuptial goals?
PS. You should use hide for answers Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement