!
!
""#!
$%&'($%&)*+!$%,%-)N(!
/0oo2!34t6o7t!3o08s:!);e=7>?4t4es!2o0!!
&@o!N7ABe0s!36ose!+7A!)s!C;e!
*D%EF)!%D+)N%!
E;4Ge0s4t>t!/o?4tHI;4I>!8e!*>t>?7;J>!
!
KC,(K!LM!N(D+(N!
De@4s!N!*?>0O!*o??ePe!
!
""# ! #"
""# =$ !
#
#
1
p+p
1
q+ q
!
!
$ $
"$
"$
#$
" " % and # " + & + # # + & '
.
" #
"%
#%
#
"
"
Proof.
(a)
(b)
p
q
p + 1p
q
1
q + 1q
p
1
(a) 1 ! 4 pq "
2
1 1
+ ! 4,
p q
!
!
1$
(b) 2 # p + & + 2 # q +
p%
"
"
2
!
1$
1
'
p+ +q+
&
#
q%
p
"
2
1$
' (1 + 4)2 = 25 .
&
q%
!
;<=<;<>?<
1.
Problema 20/3(c), CC Olympiada MatemGtica <spaHola (1I(3-(K), ,l.mpiada Matem4tica 6spa8ola
en el :iglo ;;, ;eal Sociedad MatemGtica <spaHola, Madrid, 2000.
Math% Mag. !" (2011) 22(. !"#$%&'(%)*+,-t/',-0'1('2'331' !Mathematical 2ssociation of 2merica