2017 Saudi Arabia JBMO TST

Part of Saudi Arabia JBMO TST

Subcontests

(8)

1

concyclic wanted, orthocenter and circumcircle related

ABC

be a triangle inscribed in the circle

(O)

, with orthocenter

H

. Let d be an arbitrary line which passes through

H

(O)

P

Q

. Draw diameter

AA'

(O)

A'P

A'Q

BC

K

L

, respectively. Prove that

O, K, L

A'

1

1/(a^2b+16)+1/(b^2c+16)+1/(c^2a+16)>=1/8 if a,b,c>0, a+b+c=6$

a,b,c>0

a+b+c=6

\frac{1}{a^2b+16}+\frac{1}{b^2c+16}+\frac{1}{c^2a+16} \ge \frac{1}{8}.

1

p^2 + 5pq + 4q^2 is a perfect square.

Find all pairs of prime numbers

(p, q)

p^2 + 5pq + 4q^2

is a perfect square.

4

4

4

1

Bishop paths

A chessboard has 64 cells painted black and white in the usual way. A bishop path is a sequence of distinct cells such that two consecutive cells have exactly one common point. At least how many bishop paths can the set of all white cells be divided into?

4