2

Part of 1988 Vietnam National Olympiad

Problems(2)

Determine the coefficients

Source: Vietnam NMO 1988 Problem 2

Suppose P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0 be a real polynomial of degree

n > 2

with a_n \equal{} 1, a_{n\minus{}1} \equal{} \minus{}n, a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2} such that all the roots of

P

are real. Determine the coefficients

a_i

algebrapolynomialalgebra unsolved

tan A, tan B, tan C are roots of an equation

Source: Vietnam NMO 1988 Problem 5

ABC

is an acute triangle such that

\tan A

\tan B

\tan C

are the three roots of the equation x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0, where

q\neq 1

. Show that p \le \minus{} 3\sqrt 3 and

q > 1

trigonometryalgebra unsolvedalgebra