2013 Greece JBMO TST

Part of Greece JBMO TST

Subcontests

(4)

1

isosceles trapezoid in a greek jmbo tst from 2013

Given the circle

c(O,R)

O

R

), one diameter

AB

C

AB

. Consider circle

c_1(K,KO)

K

lies on the segment

OA

, and consider the tangents

CD,CO

C

c_1(K,KO)

KD

intersects circle

c(O,R)

E

Z

E

lies on the semicircle that lies also point

C

EC

CZ

AB

N

M

respectively. Prove that quadrilateral

EMZN

is an isosceles trapezoid, inscribed in a circle whose center lie on circle

c(O,R)

1

solve in N: p(x-2)=x(y-1) and x+y=21 , where p is prime

p

is a prime positive integer and

x,y

are positive integers, find , in terms of

p

(x,y)

that are solutions of the equation:

p(x-2)=x(y-1)

. (1) If it is also given that

x+y=21

, find all triplets

(x,y,p)

that are solutions to equation (1).

1

n points lie on a circle, no three chords are concurrent

n

different points lying on a circle, such that there are not three chords defined by that point that pass through the same interior point of the circle. a) Find the value of

n

, if the numbers of triangles that are defined using

3

of the n points is equal to

2n

b) Find the value of

n

, if the numbers of the intersection points of the chords that are interior to the circle is equal to

5n

1

(x+2/y)(y/x+2)>= 8 if, xy>0

If x,y<0 prove that

\left(x+\frac{2}{y} \right) \left(\frac{y}{x}+2 \right)\geq 8

. When do we have equality?