MathDB

3

Part of 2006 Moldova Team Selection Test

Problems(3)

Cyclic ineq with triangle sides

Source: Moldavian TST_1

3/6/2006
Let a,b,ca,b,c be sides of the triangle. Prove that a2(bc−1)+b2(ca−1)+c2(ab−1)≥0. a^2\left(\frac{b}{c}-1\right)+b^2\left(\frac{c}{a}-1\right)+c^2\left(\frac{a}{b}-1\right)\geq 0 .
inequalitiesalgebrapolynomialinequalities proposed
Sum a sin a/2>=p

Source: Moldova TST 2006 Test II problem 3

3/25/2006
Let a,b,ca,b,c be sides of a triangle and pp its semiperimeter. Show that a(p−b)(p−c)bc+b(p−c)(p−a)ac+c(p−a)(p−b)ab≥pa\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p
functioninequalitiesgeometrycircumcircletrigonometryabstract algebrainequalities proposed
Moldova tst iii, problem 3

Source: Moldova TST III

3/26/2006
Positive real numbers a,b,ca,b,c satisfy the relation abc=1abc=1. Prove the inequality: a+3(a+1)2+b+3(b+1)2+c+3(c+1)2≥3\frac{a+3}{(a+1)^{2}}+\frac{b+3}{(b+1)^{2}}+\frac{c+3}{(c+1)^{2}}\geq3.
inequalitiesinequalities proposed