2012 APMO

Part of APMO

Subcontests

(5)

1

APMO 2012 #2

Into each box of a

2012 \times 2012

square grid, a real number greater than or equal to

0

and less than or equal to

1

is inserted. Consider splitting the grid into

2

non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to

1

, no matter how the grid is split into

2

such rectangles. Determine the maximum possible value for the sum of all the

2012 \times 2012

numbers inserted into the boxes.

1

APMO 2012 #1

P

be a point in the interior of a triangle

ABC

D, E, F

be the point of intersection of the line

AP

BC

of the triangle, of the line

BP

CA

, and of the line

CP

AB

, respectively. Prove that the area of the triangle

ABC

6

if the area of each of the triangles

PFA, PDB

PEC

1

1

APMO 2012 #5

n

be an integer greater than or equal to

2

. Prove that if the real numbers

a_1 , a_2 , \cdots , a_n

a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n

\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2}

1

APMO 2012 #3

Determine all the pairs

(p , n )

of a prime number

p

and a positive integer

n

\frac{ n^p + 1 }{p^n + 1}

1

Show that AB/AC=BF/FC

ABC

be an acute triangle. Denote by

D

the foot of the perpendicular line drawn from the point

A

BC

M

the midpoint of

BC

H

the orthocenter of

ABC

E

be the point of intersection of the circumcircle

\Gamma

of the triangle

ABC

and the half line

MH

F

be the point of intersection (other than

E

ED

\Gamma

\tfrac{BF}{CF} = \tfrac{AB}{AC}

must hold.
(Here we denote

XY

the length of the line segment

XY