April 2004 1

Geometric Inequalities

1. In a triangle ABC, let M be the midpoint of BC. Prove that

AM < \frac{1}{2}(AB + AC)

2. Prove that ∠ABC is acute if and only if AC^{2} < AB^{2} + BC^{2}

3. (1998 AMO) Triangle ABC has centroid G and area 1998. A line ℓ through G cuts ABC into two regions, with areas A_{1} and A_{2}. Find, with proof, the greatest possible value of |A_{2} − A_{1}|

4. Let R and r be the radii of the circumcircle and incircle of a triangle. Prove R ≥ 2r.

5. Find the greatest possible area of a quadrilateral with side lengths 1,4,7,8 (not necessarily in that order)

6. (1978 BMO) Find all points P inside the triangle ABC such that the product P L·P M ·P N is maximal. Here L, M and N are the feet of the perpendiculars from P onto BC, CA and AB.

7. Let P be a point in the interior of a triangle ABC and let P A, P B and P C intersect BC, CA and AB in D, E and F respectively. Prove that

[DEF] ≤ \frac{1}{4}[ABC]

where [XY Z] denotes the area of triangle XY Z.

8. (1961 IMO) Let a, b, c be the sides of a triangle and T its area. Prove: a^{2} + b^{2} + c^{2} ≥ 4\sqrt{3}T. When does equality hold?

In fact, it is possible to prove the stronger result

a^{2} + b^{2} + c^{2} ≥ (a − b)^{2} + (b − c)^{2} + (c − a)^{2} + 4\sqrt{3}T

9. Given a triangle ABC, let I be the centre of its inscribed circle. The internal bisectors of the angles A, B and C meet the opposite sides in A′, B′ and C′ respectively. Prove that

\frac{1}{4} < \frac{AI · BI · CI}{AA′ · BB′ · CC′} ≤ \frac{8}{27}