MathDB

3

Part of 2010 Kosovo National Mathematical Olympiad

Problems(3)

Kosovo MO 2010 Grade 9, Problem 3

Source: Kosovo MO 2010 Grade 9, Problem 3

6/7/2021
Prove that in any polygon, there exist two sides whose radio is less than 22.(Essentialy if a1≥a2≥...≥ana_1\geq a_2\geq...\geq a_n are the sides of a polygon prove that there exist i,j∈{1,2,..,n}i,j\in\{1,2,..,n\} so that i<ji<j and aiaj<2\frac {a_i}{a_j}<2).
algebra
Kosovo MO 2010 Grade 11, Problem 3

Source: Kosovo MO 2010 Grade 11, Problem 3

6/7/2021
Arrange the numbers cos⁡2,cos⁡4,cos⁡6\cos 2, \cos 4, \cos 6 and cos⁡8\cos 8 from the biggest to the smallest where 2,4,6,82,4,6,8 are given as radians.
algebra
Kosovo MO 2010 Grade 12, Problem 3

Source: Kosovo MO 2010 Grade 12, Problem 3

6/7/2021
Let n∈Nn\in \mathbb{N}. Prove that the polynom p(x)=x2n−2x2n−1+3x2n−2−...−2nx+2n+1p(x)=x^{2n}-2x^{2n-1}+3x^{2n-2}-...-2nx+2n+1 doesn't have real roots.
algebra