Write a full solution.

1.) In a party of 100 people, each people are known by even number of people. Prove that there exists a set of 3 people that are known by the same number of people.

2.) Answer the following:

2.1) An ordinary deck of cards consists of 52 cards - 4 suits, 13 cards each suits. Take out all cards with \(J,Q,K,A\). Find the number of different ways to choose 4 cards from the leftover such that the sum of all 4 numbers on chosen cards is 30, and none of the cards chosen have the same suit.

2.2) Find the number of different base-4 \(n\)-digit numbers such that

- there exists the digit 0 on any place (0 can be at the first digit)
- there are even number of digit 0
- there are no more than 1 number of digit 1

3.) Let \(k,n\) be a natural number. Let \(A = \{1,2,\dots,kn\}\). Find the least number to choose the elements from \(A\) such that we guarantee that there exists \(k\) consecutive numbers within the chosen number.

4.) Write down each digits from 1 to 9 in the \(14 \times 14\) table, such that the adjacent cells (horizontal, vertical, diagonal) must be coprime to each other. Prove that there exists 1 digit that appears on the table at least 33 times.

5.) Let \(S\) be a set of 8-digit number of all possible permutations of 1,2,3,4,6,7,8,9. Let the number from set \(S\) **nicely arranged** if there exists a pair adjacent digits that has an absolute difference of 5. Compare the number of **nicely arranged** numbers and not **nicely arranged**.

(If possible, find the number of both **nicely arranged** and not **nicely arranged**.)

Example: 13**72**6489 is **nicely arranged** because the adjacent number 72 have a difference of 5, but 13468792 is not **nicely arranged**.

This note is a part of Thailand Math POSN 2nd round 2015

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## Comments

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TopNewestI think 1) is wrong. Place the 100 people in a line and label them 1,2,3... Group them like this, [1,2],[3,4],[5,6]... Now if every group knows the everyone from the groups after them, for example, [51,52] know [53,54],[55,56]... Then all people are known by an even number of people and no three people are known by the same number of people.

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The original question has a lot of ambiguity though. I think it has to be mutual relationship that \(A\) knows \(B\) then \(B\) also has to know \(A\). I'm changing what I stated so that the question will be right.

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