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2013 Chile Classification NMO Juniors

Part of Chile Classification NMO Juniors

Subcontests

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2013 Chile Classification / Qualifying NMO Juniors XXV

p1. Inside an equilateral triangle with side 33, 55 points are marked. Show that there are two of these points at a distance less than or equal to 3/23/2.
p2. On each of the squares of a 3×3 3\times 3 board write 00, or −1-1 or 1 1. Show that of all the and columns, there are two such that the sums of the corresponding cells are equal.
p3. Consider a grid board of 2012×20132012\times 2013 cells. How many squares are crossed by a diagonal line of the board?
Clarification: We say that a line crosses a square when it passes through it's interior.
p4. The numbers are written on a blackboard 1   2   3   4   5   6   7   8   9   101\,\,\,2\,\,\,3\,\,\,4\,\,\,5\,\,\,6\,\,\,7\,\,\,8\,\,\,9\,\,\,10 We want to place symbols ++ or −- in front of each number and consider the result of the corresponding sum. a) Is it possible to place the symbols so that the resulting sum is 300300? b) Is it possible to place the symbols so that the resulting sum is 00?
p5. Consider a triangle. Show that there are 44 points on the sides of the triangle forming a square.
p6. In the celebration of the 2525 years of the National Olympiad of Mathematics 100100 students participate. Every man in this town knows exactly 22 women from the east and each woman from the east knows exactly 22 men from the is. If the participants only dance with people they know, show that it is possible to have all participants dance at the same time.
PS. Harder versions of P1, P3 were posted as [url=https://artofproblemsolving.com/community/c4h2693432p23386337]Seniors P1, P3.