2004 Greece JBMO TST

Part of Greece JBMO TST

Subcontests

(4)

1

sum S=...+\frac{1}{1-x_{2004}^3}+...+\frac{1}{1-x_{2004}^{-3}}

x_1,x_2,...x_{2004},y_1,y_2,...y_{2004}

1

and are such that

x_ky_k=1

k=1,2,...,2004

. Calculate the sum

S=\frac{1}{1-x_1^3}+\frac{1}{1-x_2^3}+...+\frac{1}{1-x_{2004}^3}+\frac{1}{1-y_1^3}+\frac{1}{1-y_2^3}+...+\frac{1}{1-y_{2004}^3}

1

angle chasing in convex ABCD with <A=60^o, two symmetric given

ABCD

be a convex quadrilateral with

\angle A=60^o

E

Z

be the symmetric points of

A

BC

CD

respectively. If the points

B,D,E

Z

are collinear, then calculate the angle

\angle BCD

1

adding two 3-digits number to get a 4-digit number staring with 173

3

-digit number we replace with each other it's last two digits, and add the resulting number to the starting one, we find sum a

4

-digit number that starts with

173

. Which is the starting number?

1

if b^3+b <= a-a^3 for a,b>0 then b<a<1 and a^2+b^2<1

a,b

be positive real numbers such that

b^3+b\le a-a^3

. Prove that: i)

b<a<1

a^2+b^2<1