1

Part of 1985 Vietnam Team Selection Test

Problems(2)

Real number between two subsequences

Source: Vietnam TST 1985 Problem 1

(x_n)

of real numbers is defined by x_1\equal{}\frac{29}{10} and x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3} for all

n\ge 1

. Find a real number

a

(if exists) such that x_{2k\minus{}1}>a>x_{2k}.

algebra proposedalgebra

Inequality with sum of inradii in polygon

Source: Vietnam TST 1985 Problem 4

A convex polygon

A_1,A_2,\cdots ,A_n

is inscribed in a circle with center

O

R

O

lies inside the polygon. Let the inradii of the triangles A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n \minus{} 1}A_n be denoted by r_1,r_2,\cdots ,r_{n \minus{} 2}. Prove that r_1 \plus{} r_2 \plus{} ... \plus{} r_{n \minus{} 2}\leq R(n\cos \frac {\pi}{n} \minus{} n \plus{} 2).

inequalitiestrigonometrygeometry proposedgeometry