MathDB

2

Part of 1981 Vietnam National Olympiad

Problems(2)

find all m

Source:

6/22/2006
Consider the polynomials f(p)=p12−p11+3p10+11p3−p2+23p+30;f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30; g(p)=p3+2p+m.g(p) = p^3 + 2p + m. Find all integral values of mm for which ff is divisible by gg.
algebrapolynomialcalculusintegrationnumber theory unsolvednumber theory
A^2/B >= 4pq/(p+q)^2.

Source: Vietnam MO 1981 P5

3/17/2011
Let p,qp, q be real numbers with 0<p<q0 < p < q and let t1,t2,⋯ ,tnt_1, t_2, \cdots, t_n be real numbers in the interval [p,q][p, q]. Denote by AA and BB the arithmetic means of t1,t2,⋯ ,tnt_1, t_2, \cdots, t_n and of t12,t22,⋯ ,tn2t_1^2, t_2^2,\cdots , t_n^2, respectively. Prove that A2B≥4pq(p+q)2.\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.
inequalitiesinequalities unsolved