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Iran MO (2nd Round)
2017 Iran MO (2nd Round)
5
5
Part of
2017 Iran MO (2nd Round)
Problems
(1)
Combinatorics from Iran MO 2017
Source: 2017 Iran MO 2nd round P5
4/21/2017
There are five smart kids sitting around a round table. Their teacher says: "I gave a few apples to some of you, and none of you have the same amount of apple. Also each of you will know the amount of apple that the person to your left and the person to your right has." The teacher tells the total amount of apples, then asks the kids to guess the difference of the amount of apple that the two kids in front of them have.
a
)
a)
a
)
If the total amount of apples is less than
16
16
16
, prove that at least one of the kids will guess the difference correctly.
b
)
b)
b
)
Prove that the teacher can give the total of
16
16
16
apples such that no one can guess the difference correctly.
combinatorics
Iran