2005 Taiwan TST Round 3

Part of TST Round 3

Subcontests

(3)

1

number of functions

\{1,2,\dots\>,n\}

P

f: P \to \{1,2,\dots\>,m\}

f(A\cap B)=\min (f(A), f(B)).

What is the relationship between the number of possible functions

f

\displaystyle \sum_{j=1}^m j^n

? There is a nice and easy solution to this. Too bad I did not think of it...

4

2

geometric inequality

P

be a point in the interior of

\triangle ABC

. The lengths of the sides of

\triangle ABC

a,b,c

, and the distance from

P

to the sides of

\triangle ABC

p,q,r

. Show that the circumradius

R

\triangle ABC

\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}.

When does equality hold?

not as hard as it seems

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?