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≥0, then y=1−x;
if y≤1, then x=1+y;
if x≤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!⋅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. algebracombinatoricsnumber theorySoros Olympiad