MathDB

1

ratio of triangle areas equal to ratio of chords, intersecting circles related

B,C

. Consider point

A

not lying on the line

BC

and draw the circles

C_1(K_1,R_1)

(with center

K_1

and radius

R_1

) and

C_2(K_2,R_2)

with chord

AB, AC

respectively such that their centers lie on the interior of the triangle

ABC

and also

R_1 \cdot AC= R_2 \cdot AB

. Let

T

be the intersection point of the two circles, different from

A

, and M be a random pointof line

AT

, prove that

TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}

3

1

no of subsets of {1,2,3,...,2003} with even number of elements

Consider the set

M=\{1,2,3,...,2003\}

. How many subsets of

M

with even number of elements exist?

5

1

diophantine x^3+y^3-2xy+x+y+2=0

Find integer solutions of

x^3+y^3-2xy+x+y+2=0

2

1

sum ...+ \sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}}

Calculate if

n\in N

with

n>2

the value of

B=\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}}

1

\sqrt{y^2-8x}+\sqrt{y^2+2x+5} when y=x+2, 1<y<3

If point

M(x,y)

lies on the line with equation

y=x+2

and

1<y<3

, calculate the value of

A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}

2003 Greece JBMO TST

Subcontests

ratio of triangle areas equal to ratio of chords, intersecting circles related

no of subsets of {1,2,3,...,2003} with even number of elements

diophantine x^3+y^3-2xy+x+y+2=0

sum ...+ \sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}}

\sqrt{y^2-8x}+\sqrt{y^2+2x+5} when y=x+2, 1<y<3

2003 Greece JBMO TST

Subcontests

ratio of triangle areas equal to ratio of chords, intersecting circles related

no of subsets of {1,2,3,...,2003} with even number of elements

diophantine x^3+y^3-2xy+x+y+2=0

sum ...+ \sqrt{1+\frac{1}{(n-1)^2}+\frac{1}{n^2}}

\sqrt{y^2-8x}+\sqrt{y^2+2x+5} when y=x+2, 1&lt;y&lt;3

\sqrt{y^2-8x}+\sqrt{y^2+2x+5} when y=x+2, 1<y<3