MathDB
2000 Chile Classification / Qualifying NMO Seniors XII

Source:

October 12, 2021
algebrageometrynumber theorycombinatoricschilean NMO

Problem Statement

p1. Prove that it is possible to color, with 55 colors, the cells of a checkered board of 2000×20002000 \times 2000, so that the colors of any square and its four neighbors are all different (neighbor means a box that has one side in common with another).
p2. There are 2525 pieces of cheese, all of different weight. Decide if it is always possible to cut one of the pieces in two parts and then place the 2626 pieces in 22 packages such that the two packages they weigh the most and each of the parts that are cut , are in different packages.
p3. A square ABCDABCD, with center OO and side 1 1, rotates an angle α\alpha about OO. Determine the common area of the original square and the resulting square.
p4. Prove that ∑n=120001n3+3n2+2n<14\sum_{n = 1}^{2000}\frac{1}{n^3 + 3n^2 + 2n}< \frac14
p5. Let {an}\{a_n\} be a sequence with the following properties: ∙\bullet a0=1,a1=4a_0 = 1, a_1 = 4 ∙\bullet an+2=2an+1−ana_{n+2} = 2a_{n + 1}-a_n ∙\bullet a2n+2=2an−an+1a_{2n + 2} = 2a_n-a_{n + 1} Prove that the number 20002000 is not in the sequence.
p6. Let PP be a part of the plane with an area greater than 1 1. ∙\bullet Prove that there are two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in PP such that the values of (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are both integers. ∙\bullet Is the above statement true if the area of PP is 1 1?
Note: PP is not necessarily connected, that is, it can be made up of several more small ''parts''. In such a case, we say that its area is the sum of the areas of its ''parts''.
p7. In Chile Chico, the Andean, Central and Baha peoples participate in the XX-tlon Olympic, consisting of XX tests to be disputed. In each test, the score for first place is higher than the score for second place, and the latter is higher than for third place. Also, the score, and the scores are positive integers. The summed XX-tlon scores were: ∙\bullet Andean People: 2222 pts. ∙\bullet Pueblo de la Baha: 1515 pts. ∙\bullet Central Town: 1414 pts. If it is known that the Central people won the shooting: a) How many tests were disputed? b) Who came out second in the long jump?
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