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2005 Chile Classification / Qualifying NMO Juniors XVII

Source:

October 9, 2021
algebrageometrynumber theorycombinatoricschilean NMO

Problem Statement

p1. Let aa, bb and cc real numbers. If a>ba> b and c>dc> d, prove that ac+bd>ad+bcac + bd> ad + bc.
p2. Prove that the difference of the cubes of two positive integers cannot be 20052005.
p3. In the sequence of digits 2005724815826...2005724815826... each term from the fifth is the last digit of the sum of the previous four. Prove that 20062006 never appears in this sequence.
p4. All the cells on a board of 100×100100 \times 100 are painted in various colors such that no pair of squares with a common side or vertex can have the same color. Determine the minimum number of colors with which it is possible to make such a coloring.
p5. Let ABCABC be a triangle and DD be a point inside it. Prove that AD+DB<AC+CBAD + DB <AC + CB.
p6. Find the sum of the numbers from 1 1 to 20002000 that have no 99s and no 00s in their digits.
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