MathDB

1

inequality on acute triangles

ABC

be an acute triangle with

\omega,S

, and

R

being its incircle, circumcircle, and circumradius, respectively. Circle

\omega_{A}

is tangent internally to

S

at

A

and tangent externally to

\omega

. Circle

S_{A}

is tangent internally to

S

at

A

and tangent internally to

\omega

. Let

P_{A}

and

Q_{A}

denote the centers of

\omega_{A}

and

S_{A}

, respectively. Define points

P_{B}, Q_{B}, P_{C}, Q_{C}

analogously. Prove that

8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; ,

with equality if and only if triangle

ABC

is equilateral.

4

1

primitive polyominoes

An animal with

n

cells is a connected figure consisting of

n

equal-sized cells[1].
A dinosaur is an animal with at least

2007

cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
(1) Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with

n

cells, one with

n+1

cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

3

1

on parts of a partition of S into two classes( |S|=n^2+n-1)

Let

S

be a set containing

n^{2}+n-1

elements, for some positive integer

n

. Suppose that the

n

-element subsets of

S

are partitioned into two classes. Prove that there are at least

n

pairwise disjoint sets in the same class.

2

1

cover all grid points by an infinite family of discs

A square grid on the Euclidean plane consists of all points

(m,n)

, where

m

and

n

are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least

5

?

1

sequence (.) eventually becomes constant.

Let

n

be a positive integer. Define a sequence by setting

a_{1}= n

and, for each

k > 1

, letting

a_{k}

be the unique integer in the range

0\leq a_{k}\leq k-1

for which

a_{1}+a_{2}+...+a_{k}

is divisible by

k

. For instance, when

n = 9

the obtained sequence is

9,1,2,0,3,3,3,...

. Prove that for any

n

the sequence

a_{1},a_{2},...

eventually becomes constant.

2007 USAMO

Subcontests

inequality on acute triangles

primitive polyominoes

on parts of a partition of S into two classes( |S|=n^2+n-1)

cover all grid points by an infinite family of discs

sequence (.) eventually becomes constant.