Problem 5

Part of 2007 Mongolian Mathematical Olympiad

Problems(2)

single-player game in nxn table

Source: Mongolian MO 2007 Grade 11 P5

n\times n

table with non-negative real entries such that the sums of entries in each column and row are equal, a player plays the following game: The step of the game consists of choosing

n

cells, no two of which share a column or a row, and subtracting the same number from each of the entries of the

n

cells, provided that the resulting table has all non-negative entries. Prove that the player can change all entries to zeros.

find area of triangle formed by segments of circumcircle

Source: Mongolian MO 2007 Teachers P5

P

in the circumcircle

\omega

of an equilateral triangle

ABC

, prove that the segments

PA

PB

PC

form a triangle

T

R

be the radius of the circumcircle

\omega

d

be the distance between

P

and the circumcenter. Find the area of

T

geometrycircumcircle