MathDB
1992 Chile Classification / Qualifying NMO IV

Source:

October 7, 2021
algebranumber theorycombinatoricsgeometrychilean NMO

Problem Statement

p1. Suppose the number 2+10933+2−10933\sqrt[3]{2+\frac{10}{9}\sqrt3} + \sqrt[3]{2-\frac{10}{9}\sqrt3} is integer. Calculate it.
p2. A house A is located 300300 m from the bank of a 200200 m wide river. 600600 m above and 500500 m from the opposite bank is another house BB. A bridge has been built over the river, that allows you to go from one house to the other covering the minimum distance. What distance is this and in what place of the riverbank is at the bridge?
p3. Show that it is not possible to find 19921992 positive integers x1,x2,...,x1992x_1, x_2, ..., x_{1992} that satisfy the equation: ∑j=119922j−1xj1992=21992∏j=21992xj\sum_{j=1}^{1992}2^{j-1}x_j^{1992}=2^{1992}\prod_{j=2}^{1992}x_j
p4. The following division of positive integers with remainder has been carried out. Every cross represents one digit, and each question mark indicates that there are digits, but without specifying how many. Fill in each cross and question mark with the missing digits. https://cdn.artofproblemsolving.com/attachments/8/e/5658e77025210a46723b6d85abf5bdc3c7ad51.png
p5. Three externally tangent circles have, respectively, radii 11, 22, 33. Determine the radius of the circumference that passes through the three points of tangency.
p6. ∙\bullet It is possible to draw a continuous curve that cuts each of the segments of the figure exactly one time? It is understood that cuts should not occur at any of the twelve vertices. ∙\bullet What happens if the figure is on a sphere? https://cdn.artofproblemsolving.com/attachments/a/d/b998f38d4bbe4121a41f1223402c394213944f.png
p7. Given any △ABC\vartriangle ABC, and any point PP, let X1X_1, X2X_2, X3X_3 be the midpoints of the sides ABAB, BCBC, ACAC respectively, and Y1Y_1, Y2Y_2, Y3Y_3 midpoints of PCPC, PAPA, PBPB respectively. Prove that the segments X1Y1X_1Y_1, X2Y2X_2Y_2, X3Y3X_3Y_3 concur.
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