MathDB
2014 Chile Classification / Qualifying NMO Juniors XXVI

Source:

October 11, 2021
algebrageometrycombinatoricsnumber theorychilean NMO

Problem Statement

p1. Find 55 different odd numbers such that the product of any two of them are multiples each of the others.
p2. The following game is played on the 3×3 3\times 3 board of the figure. A move allowed is to choose one of the squares and change color (black to white, white to black) all those that are still attached to it, either diagonally or sharing one side (the chosen square does not change color). Determine if possible, through movements allowed, make all the squares in the given figure the same color. https://cdn.artofproblemsolving.com/attachments/f/d/151203e973a60bf73ff2e24c8e5ad18b44ae6a.png
p3. In an equilateral triangle ABCABC with side 22, side ABAB is extended to a point DD so that BB is the midpoint of ADAD. Let EE be the point on ACAC such that ∠ADE=15o\angle ADE = 15^o and take a point FF on ABAB so that ∣EF∣=∣EC∣|EF| = |EC|. Determine the area of the triangle AFEAFE.
p4. For each positive integer nn we consider S(n)S (n) as the sum of its digits. For example S(1234)=1+2+3+4=10S (1234) = 1 + 2 + 3 + 4 = 10. Calculate S(1)−S(2)+S(3)−S(4)+...−S(202)+S(203)S (1)- S (2) + S (3)- S (4) +...- S (202) + S (203)
p5. The four code words □∗⊗   ⊕×∙   ∗□∙   ⊕⋄⊕\Box * \otimes \,\,\, \oplus \times \bullet \,\,\, * \Box \bullet\,\,\, \oplus \diamond \oplus They are, in some order AMO   SUR   REO   MASAMO \,\,\, SUR \,\,\, REO \,\,\, MAS Decipher ⊗⋄□∗⊕×□∙⊕ \otimes \diamond \Box * \oplus \times \Box \bullet \oplus
p6. Consider a parallelogram ABCDABCD such that the angle ∠DAB\angle DAB is acute. Let GG be a point on the line ABAB different from B B and such that ∣BC∣=∣GC∣|BC| = |GC|, and let HH be a point on the line BCBC different from B B and such that ∣AB∣=∣AH∣|AB| = |AH|. Prove that the triangle GDHGDH is isosceles.
PS. Juniors P3, P4 were also proposed as [url=https://artofproblemsolving.com/community/c4h2693874p23392747]Seniors P3, harder P4.
Back to ProblemsView on AoPS