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Select Problem from various Math Olympiad ខែ​មេសា 10, 2009

Posted by psvjupiter in គណិតវិទ្យា, អូឡាំព្យាដ.

The following problems are the selecting problems I select from math link. I only select suitable level, not the too high level. So hope we can solve together.

1. a b c are 3 positive, a+b+c=1.
Prove that a^{2}/b+b^{2}/c+c^{2}/a \geq 3(a^{2}+b^{2}+c^{2})
(Croatia 2008 Team Selection)

2. In triangle ABC, D is a point on AB and E is a point on AC such that BE and CD are bisectors of ∠B and ∠C respectively. Let Q,M and N be the feet of perpendiculars from the midpoint P of DE onto BC, AB and AC, respectively. Prove that PQ = PM + PN. (Singapore Team Selection 2008)

3. Let (O) be a circle, and let ABP be a line segment such that A,B lie on (O) and P is a point outside (O). Let C be a point on (O) such that PC is tangent to (O) and let D be the point on (O) such that CD is a diameter of (O) and intersects AB inside (O). Suppose that the lines DB and OP intersect at E. Prove that AC is perpendicular to CE. (Singapore Team Selection 2008)

4. Prove that \displaystyle \frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca,
where a, b, c is the side of triangle. (Greece National Math Olympiad 2007)

If you have our National Math Contest this year please post it. 😀
I will post the solution when I am free and I can find it!




1. psvjupiter - ខែ​មេសា 10, 2009

Hint for problem 1: Multiply left side with a+b+c, since a+b+c=1
so a^{2}/b+b^{2}/c+c^{2}/a=(a^{2}/b+b^{2}/c+c^{2}/a)(a+b+c)
i) prove that a^{3}/b+ ab +b^{3}/c+ bc+c^{3}/a+ ac >=a^{2}+b^{2}+c^{2}
ii) prove that a^{2}/bc+b^{2}/ca+c^{2}/ab >=ab+bc+ca


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