MathDB

Problem 3

Part of 1999 Mongolian Mathematical Olympiad

Problems(5)

area ratio, configuration with three squares

Source: Mongolia 1999 Grade 8 P3

5/4/2021
Three squares ABB1B2,BCC1C2,CAA1A2ABB_1B_2,BCC_1C_2,CAA_1A_2 are constructed in the exterior of a triangle ABCABC. In the exterior of these squares, another three squares A1B2B3B4,B1C2C3C4,C1A2A3A4A_1B_2B_3B_4,B_1C_2C_3C_4,C_1A_2A_3A_4 are constructed. Prove that the area of a triangle with sides C3A4,A3B4,B3C4C_3A_4,A_3B_4,B_3C_4 is 1616 times the area of △ABC\triangle ABC.
geometry
hard inequality: sin < CAM + sin < CBM <= 2 / sqrt(3)

Source: Bulgaria 97

8/5/2004
I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one! If MM is the centroid of a triangle ABCABC, prove that the following inequality holds: sin⁡∠CAM+sin⁡∠CBM≤23.\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}. The equality occurs in a very strange case, I don't remember it.
inequalitiestrigonometrygeometryrectangleanalytic geometrygeometry proposed
NT sequence existence

Source: Mongolia 1999 Grade 10 P3

5/5/2021
Does there exist a sequence (an)n∈N(a_n)_{n\in\mathbb N} of distinct positive integers such that:
(i) an<1999na_n<1999n for all nn; (ii) none of the ana_n contains three decimal digits 11?
number theorySequences
minimum length connecting vertices of rectangle

Source: Mongolia 1999 Teachers elementary level P3

5/5/2021
At each vertex of a 4×54\times5 rectangle there is a house. Find the path of the minimum length connecting all these houses.
geometryoptimizationrectangle
maximum sum of sequences

Source: Mongolia 1999 Teachers secondary level P3

5/6/2021
Let (an)n=1∞(a_n)^\infty_{n=1} be a non-decreasing sequence of natural numbers with a20=100a_{20}=100. A sequence (bn)(b_n) is defined by bm=min⁡{n∣an≥m}b_m=\min\{n|an\ge m\}. Find the maximum value of a1+a2+…+a20+b1+b2+…+b100a_1+a_2+\ldots+a_{20}+b_1+b_2+\ldots+b_{100} over all such sequences (an)(a_n).
Sequencealgebrainequalities