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290
(b) Deduce that if both q and t are odd then f is irreducible.
(Generalization of Question 1, IMO 1993)
A219. Proposed by Mohammed Aassila, Universite Louis Pasteur,
Strasbourg, France.
Solve the following system:
3 x + 1
x
= 4 y + 1
y
xy + yz + zx = 1 :
= 5 z + 1
A220. Proposed by Waldemar Pompe, student, University of
Warsaw, Poland.
P is an interior point of triangle ABC. D, E, F are the feet of the
perpendicularsfrom P to the lines BC, CA, AB, respectively. Let Q be the
interior point of triangle ABC such that
\ACP = \BCQ and \BAQ = \CAP :
Prove that \DEF =90 if and only if Q is the orthocentre of triangle BDF.
Challenge Board Problems
Editor: David Savitt, Department of Mathematics, Harvard University,
1 Oxford Street, Cambridge, MA, USA 02138
C79. Proposed by Mohammed Aassila, Universite Louis Pasteur,
Strasbourg, France.
Let f : R + ! R + be a non-increasing function, and assume that there
are two constants p>0and T >0such that
Z 1
for all t 2 R + .Prove that
for all t T .
t
f (s) p+1 ds Tf(0) p f(t)
f (t) f(0)
T + pt
T + pT
C80. Suppose a1, a2, ::: ,amaretranspositions in Sn (the symmetric
group on n elements) such that a1a2 am =1. Show that if the ai generate
Sn, then m 2n , 2.
, 1 p
z
;