5

Part of 2007 Irish Math Olympiad

Problems(2)

weird inequality

Source: Ireland 2007

r

n

be nonnegative integers such that

r \le n

(a)

Prove that: \frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r} is an integer.

(b)

Prove that: \displaystyle\sum_{r\equal{}0}^{[n/2]}\frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}<2^{n\minus{}2} for all

n \ge 9

inequalitiesinequalities unsolved

quadratic polynomial

Source: Ireland 2007

a

b

are real numbers such that the quadratic polynomial f(x)\equal{}x^2\plus{}ax\plus{}b has no nonnegative real roots. Prove that there exist two polynomials

g,h

whose coefficients are nonnegative real numbers such that: f(x)\equal{}\frac{g(x)}{h(x)} for all real numbers

x

quadraticsalgebrapolynomialalgebra proposed