MathDB

Problem 2

Part of 1999 Mongolian Mathematical Olympiad

Problems(4)

4/5(1/a+1/b)+2/c>=27/(2(a+b+c)) if a>=8b/5>0 and a>=c>0 over R

Source: Mongolia 1999 Grade 8 P2

5/4/2021
Let a,b,ca,b,c be the real numbers with a≥85b>0a\ge\frac85b>0 and a≥c>0a\ge c>0. Prove the inequality 45(1a+1b)+2c≥272⋅1a+b+c.\frac45\left(\frac1a+\frac1b\right)+\frac2c\ge\frac{27}2\cdot\frac1{a+b+c}.
Inequalityinequalities
side expression is constant from a line

Source: Mongolia 1999 Grade 10 P2

5/5/2021
The rays l1,l2,…,ln−1l_1,l_2,\ldots,l_{n-1} divide a given angle ABCABC into nn equal parts. A line ll intersects ABAB at A1A_1, BCBC at An+1A_{n+1}, and lil_i at Ai+1A_{i+1} for i=1,…,n−1i=1,\ldots,n-1. Show that the quantity (1BA1+1BAn+1)(1BA1+1BA2+…+1BAn+1)−1\left(\frac1{BA_1}+\frac1{BA_{n+1}}\right)\left(\frac1{BA_1}+\frac1{BA_2}+\ldots+\frac1{BA_{n+1}}\right)^{-1}is independent of the line ll, and compute its value if ∠ABC=ϕ\angle ABC=\phi.
geometry
square as 10 noncongruent triangles

Source: Mongolia 1999 Teachers elementary level P2

5/5/2021
Can a square be divided into 1010 pairwise non-congruent triangles with the same area?
geometry
maximum 4-cycles in Cn digraph

Source: Mongolia 1999 Teachers secondary level P2

5/6/2021
Any two vertices A,BA,B of a regular nn-gon are connected by an oriented segment (i.e. either A→BA\to B or B→AB\to A). Find the maximum possible number of quadruples (A,B,C,D)(A,B,C,D) of vertices such that A→B→C→D→AA\to B\to C\to D\to A.
combinatorics