MathDB

1999 Chile Classification NMO Seniors

Part of Chile Classification NMO

Subcontests

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1999 Chile Classification / Qualifying NMO Seniors XI

p1. Prove that in every group of nn people (n>2n> 2), there are always at least 22 people who they have the same number of friends within the group.
p2. A foreman observes an operator placing in a hole 33 centimeters in diameter a 22 centimeter plug and a 1 1 centimeter in diameter, and it is suggested that you insert another two plugs to ensure a perfect fit. If these two new plugs are of diameter dd, determine the location of the dowels and find the value dd. https://cdn.artofproblemsolving.com/attachments/5/6/45c234b78ad274c94b4472996c05e84135dd79.png
p3. At a round table there is a king with 55 of his subjects. Initially, the king He has 66 coins and his subjects have none. To distribute the 66 tokens, the king proposes the following way: each person can pass signs to their neighbors, but every time they do so they must pass the same amount of tokens to each of them. This repeats until no one can do what you proposed the king. Is it possible in this way to ensure that each of the members of the table has only one coin?
p4. A number is said to be cototudo if the sum of its divisors, excluding its own number is greater than. For example, 1212 is cototudo since 1+2+3+4+6=16>121 + 2 + 3 + 4 + 6 = 16> 12. Show that there are infinite numbers of cototudo.
p5. Three friends stand at different vertices of a rectangle, and move according to the following rule: ''In each turn only one of them moves in the direction parallel to the determined line for the other two friends. '' Prove that the three friends cannot reach simultaneously three midpoints of the sides of the triangle.
p6. Given a positive integer nn, find all positive integers kk with the following property: There is a group of nn people in which each one is friends with other kk people in the group.