2004 Serbia Team Selection Test

Part of Serbia Team Selection Test

Subcontests

(3)

1

Serbia and Montenegro TST 2004

a

b

c

be real numbers such that

abc=1

. Prove that the most two of numbers

2a-\frac{1}{b},\ 2b-\frac{1}{c},\ 2c-\frac{1}{a}

are greater than

1

1

interesting geometry problem

Let ABCD be a square and K be a circle with diameter AB. For an arbitrary point P on side CD, segments AP and BP meet K again at points M and N, respectively, and lines DM and CN meet at point Q. Prove that Q lies on the circle K and that AQ : QB = DP : PC.

1

Beautiful polynomial

P(x)

be a polynomial of degree

n

whose roots are

i-1, i-2,\cdot\cdot\cdot, i-n

i^2=-1

R(x)

S(x)

be the polynomials with real coefficients such that

P(x)=R(x)+iS(x)

. Show that the polynomial

R

n

real roots. (R. Stanojevic)