MathDB

2004 Chile Classification NMO Seniors

Part of Chile Classification NMO

Subcontests

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2004 Chile Classification / Qualifying NMO Seniors XVI

p1. Investigate whether there exists a positive integer NN such that if the first digit of its decimal expression, results in a number pp such that NN is 5858 times pp.
p2. Influenced by the triumphs of Gonz alez and Massu, in a course of a mixed school they decided to organize a tennis tournament. nn girls and 2n2n boys participated, each pair of participants from the 3n3n students, fought in exactly one match. After the tournament, the ratio between the number of victories of the girls and the number of victories of the boys, resulting in this ratio being the same at 7:57: 5. Determine n and show that in every game in which a girl was confronted with a boy, the girl was the winner.
p3. In a certain country there is direct rail communication between any pair of cities, but trains only travel in one direction. Prove that one such city exists, from which can be reached in any other city by passing at most through an intermediate one.
p4. Find at least one quartet of positive integers m,n,p,qm, n, p, q that are distinct such that satisfy the relations: m+n=p+q m + n = p + q m+n3=p+q3>2004\sqrt{m} + \sqrt[3]{n} =\sqrt{p} + \sqrt[3]{q}> 2004
p5. Let ABCDABCD be a convex quadrilateral such that each of its sides is less than 2020. Prove that for any point P inside the quadrilateral, at least one of the distances PAPA, PBPB, PCPC, PDPD is less than 1515. Can it be confirmed that at least one of these distances is less than 1414?
p6. On the blackboard a teacher wrote several zeros, ones and two. Then with these numbers He did the following operation: he erased two different numbers from each other and instead of them on the blackboard he He wrote a different number than the ones deleted (for example, if a 11 and a 22 are deleted, then a 00 is written). After doing this operation several times, a single number was written on the blackboard. Investigate if this number depends on the order in which the erasures were carried out.
p7. A citizen in a certain country is considered as a person of class ABC1ABC_1 if his monthly income exceeds a certain level XX established by the Ministry of Finance; otherwise the citizen belongs to class C2C_2. By an ancient custom of that country, the men of class ABC1ABC_1. They only marry women of class C2C_2. Furthermore, it is known that the monthly income of all citizens of the country are different from each other. Prove that in this case it is possible to define XX such that there will be an equal number of men of class ABC1ABC_1 and women of class C2C_2.