4

Part of 2021 Argentina National Olympiad

Problems(3)

Argentina MO 2021 National Level 1 P4

Source:

Martu wants to build a set of cards with the following properties:
• Each card has a positive integer on it. • The number on each card is equal to one of

5

possible numbers. • If any two cards are taken and added together, it is always possible to find two other cards in the set such that the sum is the same.
Determine the fewest number of cards Martu's set can have and give an example for that number.

contestscombinatorics

Argentina MO 2021 National Level 2 P4

Source:

The sum of several positive integers, not necessarily different, all of them less than or equal to

10

S

. We want to distribute all these numbers into two groups such that the sum of the numbers in each group is less than or equal to

80.

Determine all values of

S

for which this is possible.

combinatoricscontests

Argentina MO 2021 National Level 3 P4

Source:

Find the real numbers

x, y, z

\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}, \frac{1}{y}+\frac{1}{z+x}=\frac{1}{3}, \frac{1}{z}+\frac{1}{x+y}=\frac{1}{4}.

algebraArgentinacontests