MathDB

1

externally tangent circles inside a rectangle (Greece JBMO TST 2007 p3)

ABCD

be a rectangle with

AB=a >CD =b

. Given circles

(K_1,r_1) , (K_2,r_2)

with

r_1<r_2

tangent externally at point

K

and also tangent to the sides of the rectangle, circle

(K_1,r_1)

tangent to both

AD

and

AB

, circle

(K_2,r_2)

tangent to both

AB

and

BC

. Let also the internal common tangent of those circles pass through point

D

. (i) Express sidelengths

a

and

b

in terms of

r_1

and

r_2

. (ii) Calculate the ratios

\frac{r_1}{r_2}

and

\frac{a}{b}

. (iii) Find the length of

DK

in terms of

r_1

and

r_2

.

1

computational in a triangle with given angles

Let

ABC

be a triangle with

\angle A=105^o

and

\angle C=\frac{1}{4} \angle B

. a) Find the angles

\angle B

and

\angle C

b) Let

O

be the center of the circumscribed circle of the triangle

ABC

and let

BD

be a diameter of that circle. Prove that the distance of point

C

from the line

BD

is equal to

\frac{BD}{4}

.

4

1

sum with last term \sqrt{1+(8\cdot 1003^2-1)/(2005^2 2007^2)}

Calculate the sum

S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}

2

1

n(n+3) a perfect square => 3 not a multiple of 3

Let

n

be a positive integer such that

n(n+3)

is a perfect square of an integer, prove that

n

is not a multiple of

3

.

2007 Greece JBMO TST

Subcontests

externally tangent circles inside a rectangle (Greece JBMO TST 2007 p3)

computational in a triangle with given angles

sum with last term \sqrt{1+(8\cdot 1003^2-1)/(2005^2 2007^2)}

n(n+3) a perfect square => 3 not a multiple of 3

2007 Greece JBMO TST

Subcontests

externally tangent circles inside a rectangle (Greece JBMO TST 2007 p3)

computational in a triangle with given angles

sum with last term \sqrt{1+(8\cdot 1003^2-1)/(2005^2 2007^2)}

n(n+3) a perfect square =&gt; 3 not a multiple of 3

n(n+3) a perfect square => 3 not a multiple of 3