MathDB

1

Part of 2005 Taiwan TST Round 3

Problems(2)

geometric inequality

Source: Taiwan 3rd TST, 1st independent study, problem 1

8/12/2005
Let PP be a point in the interior of △ABC\triangle ABC. The lengths of the sides of △ABC\triangle ABC is a,b,ca,b,c, and the distance from PP to the sides of △ABC\triangle ABC is p,q,rp,q,r. Show that the circumradius RR of △ABC\triangle ABC satisfies R≤a2+b2+c218pqr3.\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}. When does equality hold?
inequalitiesgeometrycircumcirclegeometry proposed
not as hard as it seems

Source: Taiwan 3rd TST 2005, 2nd independent study, problem 1

8/12/2005
A club provides 30 snacks to 18 members, and each member orders 3 different snacks. It is known that every snack is ordered by at least one member, and that any two members order at most one same snack. Is it possible to find 12 snacks, such that the snacks ordered by any member is not completely in these 12 snacks?
combinatorics proposedcombinatorics