MathDB
1989 Chile Classification / Qualifying NMO I

Source:

October 6, 2021
algebrageometrycombinatoricsnumber theorychilean NMO

Problem Statement

p1. A wooden cube, the side of which measures 44 cm, is painted on its entire outer surface with color blue. Making horizontal and vertical cuts, we obtain 6464 cubes of 1 1 cm on each side. How many cubes have, respectively, 33,2 2, 1 1 and 00 blue faces?
p2. Find all the right triangles that have an integer leg and a integer hypotenuse knowing that the other leg has length 1989\sqrt{1989}.
Note: 1989=9×13×171989 = 9 \times 13 \times 17
p3. The length of the sides of an equilateral triangle is 55. From an interior point, draw segments perpendicular on each of the three sides. if the lengths of these segments are a,b,ca, b, c. Determine the value a+b+ca + b + c.
p4. Find the solutions of the radical equation: x+2x−1+x−2x−1=2\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt2
p5. Find integer numbers a,ba, b such that a2+b2=1989a^2 + b^2 = 1989.
p6. The letters a,b,c,da, b, c, d and f,g,hf, g, h denote positive integers. Determine them from the following puzzle, where each given key represents a four-digit integer that does not start with 00. As a help, a digit has been delivered in the puzzle. https://cdn.artofproblemsolving.com/attachments/2/c/29b7f8a63f7a79636d3c2a3e544ec654d11dbd.png
p7. Given a triangle ABCABC, choose P∈ABP \in AB, so that PP is closer to A A than to B B. Then three points are constructed: QQ, RR, SS, on the sides ACAC, CBCB, BABA, respectively, such that PQ∥BCPQ \parallel BC, QR∥ABQR \parallel AB, RS∥CARS \parallel CA. Calculate the maximum value that the area of the quadrilateral PQRSPQRS can take in terms of the area of the given triangle.
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