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Cono Sur Olympiad
2019 Cono Sur Olympiad
1
1
Part of
2019 Cono Sur Olympiad
Problems
(1)
2019 Cono Sur Mathematical Olympiad, P1
Source:
8/28/2019
Martin has two boxes
A
A
A
and
B
B
B
. In the box
A
A
A
there are
100
100
100
red balls numbered from
1
1
1
to
100
100
100
, each one with one of these numbers. In the box
B
B
B
there are
100
100
100
blue balls numbered from
101
101
101
to
200
200
200
, each one with one of these numbers. Martin chooses two positive integers
a
a
a
and
b
b
b
, both less than or equal to
100
100
100
, and then he takes out
a
a
a
balls from box
A
A
A
and
b
b
b
balls from box
B
B
B
, without replacement. Martin's goal is to have two red balls and one blue ball among all balls taken such that the sum of the numbers of two red balls equals the number of the blue ball.\\ What is the least possible value of
a
+
b
a+b
a
+
b
so that Martin achieves his goal for sure? For such a minimum value of
a
+
b
a+b
a
+
b
, give an example of
a
a
a
and
b
b
b
satisfying the goal and explain why every
a
a
a
and
b
b
b
with smaller sum cannot accomplish the aim.
combinatorics