3

Part of 2023 Mongolian Mathematical Olympiad

Problems(2)

Source: Mongolian Mathematical Olympiad P3

Five girls and five boys took part in a competition. Suppose that we can number the boys and girls

1, 2, 3, 4, 5

such that for each

1 \leq i,j \leq 5

, there are exactly

|i-j|

contestants that the girl numbered

i

and the boy numbered

j

a_i

b_i

be the number of contestants that the girl numbered

i

knows and the number of contestants that the boy numbered

i

knows respectively. Find the minimum value of

\max(\sum\limits_{i=1}^5a_i, \sum\limits_{i=1}^5b_i)

. (Note that for a pair of contestants

A

B

A

B

doesn't mean that

B

A

and a contestant cannot know themself.)

combinatoricsExtremalgraph

An absolutely breathtaking number theory

Source: Mongolian Mathematical Olympiad P6

m

be a positive integer. We say that a sequence of positive integers written on a circle is good , if the sum of any

m

consecutive numbers on this circle is a power of

m

n \geq 2

be a positive integer. Prove that for any good sequence with

mn

numbers, we can remove

m

numbers such that the remaining

mn-m

numbers form a good sequence.
2. Prove that in any good sequence with

m^2

numbers, we can always find a number that was repeated at least

m

times in the sequence.

number theorySequences