2012 Greece JBMO TST

Part of Greece JBMO TST

Subcontests

(4)

1

2 equal circles, 1 symmetric point, 1 altitude, perpendicularity wanted

ABC

be an acute triangle with

AB<AC<BC

, inscribed in circle

c(O,R)

O

R

O_1

be the symmetric point of

O

AC

c_1(O_1,R)

BC

Z

. If the extension of the altitude

AD

intersects the cicrumscribed circle

c(O,R)

E

EC

is perpendicular on

AZ

1

integer solutions of x+y+z=2013 so that xyz becomes max

x,y,z

are positive integers and satisfy the equation

x+y+z=2013

. (E) a) Find the number of the triplets

(x,y,z)

that are solutions of the equation (E). b) Find the number of the solutions of the equation (E) for which

x=y

. c) Find the solution

(x,y,z)

of the equation (E) for which the product

xyz

becomes maximum.

1

p^2+2q^2+334=[p^2,q^2] , where p,q coprime

Find all pairs of coprime positive integers

(p,q)

p^2+2q^2+334=[p^2,q^2]

[p^2,q^2]

is the leact common multiple of

p^2,q^2

1

(a-b-c)+(b^2+c^2-bc)=4c^2 (abc-\frac{a^2}{4}-b^2c^2)

Find all triplets of real

(a,b,c)

that solve the equation

a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)