2019 Greece JBMO TST

Part of Greece JBMO TST

Subcontests

(3)

1

concyclic points wanted, related to circumcircle, midpoint, perpendicular

Consider an acute triangle

ABC

AB>AC

inscribed in a circle of center

O

. From the midpoint

D

BC

(\ell)

perpendicular to side

AB

that intersects it at point

E

AO

intersects line

(\ell)

Z

, prove that points

A,Z,D,C

1

12 black unit squares in a 8x8 all white chessboard

8\times 8

chessboard where all

64

unit squares are at the start white. Prove that, if any

12

64

unit square get painted black, then we can find

4

4

rows that have all these

12

1

solve in positive integers: 3 \cdot 2^x +4 =n^2

Find all pairs of positive integers

(x,n)

that are solutions of the equation

3 \cdot 2^x +4 =n^2