1

Part of 2010 South East Mathematical Olympiad

Problems(2)

Source:

a,b,c\in\{0,1,2,\cdots,9\}

.The quadratic equation

ax^2+bx+c=0

has a rational root. Prove that the three-digit number

abc

is not a prime number.

quadraticsalgebranumber theory unsolvednumber theory

Source: 2010 China South East Mathematical Olympiad

ABC

is a triangle with a right angle at

C

M_1

M_2

are two arbitrary points inside

ABC

M

is the midpoint of

M_1M_2

. The extensions of

BM_1,BM

BM_2

AC

N_1,N

N_2

respectively. Prove that

\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}

inequalitiesgeometry unsolvedgeometry