2005 Peru MO (ONEM)

Part of Peru MO (ONEM)

Subcontests

(4)

1

same no of chips on each sqaure of 5x5 board, after many moves

On the grid board shown, a token is placed on each white space. https://cdn.artofproblemsolving.com/attachments/3/2/0060b2436edb0ce25160d2f94f379defef237c.png A move consists of choosing four squares on the board that form a "

T

" in any of the shapes shown below, and add a token to each of these four squares. https://cdn.artofproblemsolving.com/attachments/8/c/3890aed5289ec9ea2d147f8000a0422c233029.png Is it possible, after carrying out several moves, to get the

25

squares to have the same amount of chips?

1

sina sinb sin (a+b)/2=sin (60-a)sin (60-b)sin [60-((a+b)/2) ]

The measures, in degrees, of the angles ,

\alpha, \beta

\theta

are greater than

0

60

. Find the value of

\theta

knowing, also, that

\alpha + \beta = 2\theta

\sin \alpha \sin \beta \sin \theta = \sin(60 - \alpha ) \sin(60 - \beta) \sin(60 - \theta ).

1

cos^2 x - cos^4 x -sin2x + 2

p = (1- \cos x)(1+ \sin x)

q = (1+ \cos x)(1- \sin x)

, write the expression

\cos^2 x - \cos^4 x - \sin2x + 2

p

q

1

find an angle, related to 3 equal collinear segments + 3 equilateral triangles

A,B,C,D

, be four different points on a line

\ell

AB=BC=CD

. In one of the semiplanes determined by the line

\ell

P

Q

are chosen in such a way that the triangle

CPQ

is equilateral with its vertices named clockwise. Let

M

N

be two points of the plane be such that the triangles

MAP

NQD

are equilateral (the vertices are also named clockwise). Find the angle

\angle MBN