9.7

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(2)

incenters not collinear, equal angles (I Soros Olympiad 1994-95 R1 9.7 10.7)

Source:

Given an acute triangle

ABC

\angle BAC <30^o

AC

AB

are taken respectively points

D

E

\angle BDC=\angle BDE = \angle ADE = 60^o

. Prove that the centers of the circles. inscribed in triangles

ADE

BDE

BCD

do not lie on the same line.

geometryincentercollinearequal angles

4 consecutive into 2 groups of 2 (I Soros Olympiad 1994-95 R2 9.7)

Source:

Four consecutive natural numbers are divided into two groups of

2

numbers. It is known that the product of numbers in one group is

1995

greater than the product of numbers in another group. Find these numbers.