MathDB

p1. Is it possible to write

5

different fractions that add up to

1

, such that their numerators are equal to one and their denominators are natural numbers?
p2. The following is known about two numbers

x

and

y

: if

x\ge 0

, then

y = 1 -x

; if

y\le 1

, then

x = 1 + y

; if

x\le 1

, then

x = |1 + y|

. Find

x

and

y

.
p3. Five people living in different cities received a salary, some more, others less (

143

233

313

410

and

413

rubles). Each of them can send money to the other by mail. In this case, the post office takes

10\%

of the amount of money sent for the transfer (in order to receive

100

rubles, you need to send

10\%

more, that is,

110

rubles). They want to send money so that everyone has the same amount of money, and the post office receives as little money as possible. How much money will each person have using the most economical shipping method?
p4. a) List three different natural numbers

m

n

and

k

for which

m! = n! \cdot k!

. b) Is it possible to come up with

1999

such triplets?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement