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~Questions~
31/5/98
Answer to be posted on 2/6/98
Number 1:
An acute-angled triangle ABC is given
in the plane.The circle with diameter AB intersects altitude CE and its
extension at M and N,and the circle with diameter AC intersects altitude
BD and its extension at P and Q.See fig.1
Prove that the points M,N,P,Q lie
on a common circle.
(From : USA Mathematical Olympiad
1990')
fig 1
Number 2:<!!>
Prove that for all natural number ,n.a>0
& not equal 1:
(1+a^2+a^4+.....+a^2n)/(a+a^3+...+a^2n-1)>(n+1)/n
ANSWER TO NUMBER 1:
Tri AED~Tri ABC, then:
DAxAC=BAxEA
Tri APD~Tri APC => AP=(DAxAC)^1/2
Tri AME~Tri AMB=> AM=(BAxEA)^1/2
=>AP=AM
AP=AQ=AM=AN
QED!
ANSWER TO NUMBER
2