676 Number Theory
and ifx= 2 k+1 withk>3, then
x^3 =( 2 k+ 1 )^3 =( 2 k− 1 )^3 +(k+ 4 )^3 +( 4 −k)^3 +(− 5 )^3 +(− 1 )^3.In this last case it is not hard to see that 2k− 1 ,k+ 4 ,| 4 −k|,5, and 1 are all less than
2 k+1. Ifx>3 is an arbitrary integer, then we writex=my, whereyis 4, 6, or an odd
number greater than 3, andmis an integer. If we expressy^3 =y 13 +y 23 +y^33 +y 43 +y^35 ,
thenx^3 =(my 1 )^3 +(my 2 )^3 +(my 3 )^3 +(my 4 )^3 +(my 5 )^3 , and the lemma is proved.
Returning to the problem, using the fact thatfis odd and what we proved before,
we see thatf(x)=xf ( 1 )for|x|≤3. Forx>4, suppose thatf(y)=yf ( 1 )for ally
with|y|<|x|. Using the lemma writex^3 =x 13 +x 23 +x^33 +x^34 +x^35 , where|xi|<|x|,
i= 1 , 2 , 3 , 4 ,5. After writing
x^3 +(−x 4 )^3 +(−x 5 )^3 =x 13 +x 23 +x^33 ,we applyfto both sides and use the fact thatfis odd and the condition from the statement
to obtain
(f (x))^3 −(f (x 4 ))^3 −(f (x 5 ))^3 =f(x 1 )^3 +f(x 2 )^3 +f(x 3 )^3.The inductive hypothesis yields
(f (x))^3 −(x 4 f( 1 ))^3 −(x 5 f( 1 ))^3 =(x 1 f( 1 ))^3 +(x 2 f( 1 ))^3 +(x 3 f( 1 ))^3 ;hence
(f (x))^3 =(x^31 +x 23 +x 33 +x 43 +x^35 )(f ( 1 ))^3 =x^3 (f ( 1 ))^3.Hencef(x)=xf ( 1 ), and the induction is complete. Therefore, the only answers to the
problem aref(x)=−xfor allx,f(x)=0 for allx, andf(x)=xfor allx. That these
satisfy the given equation is a straightforward exercise.
(American Mathematical Monthly, proposed by T. Andreescu)
706.The number on the left ends in a 0, 1 , 4 , 5 , 6 ,or 9, while the one on the right ends
ina0, 2 , 3 , 5 , 7 ,or 8. For equality to hold, bothxandzshould be multiples of 5, say
x= 5 x 0 andz= 5 z 0. But then 25x^20 + 10 y^2 = 3 · 25 z^2. It follows thatyis divisible
by 5 as well,y= 5 y 0. The positive integersx 0 ,y 0 ,z 0 satisfy the same equation, and
continuing we obtain an infinite descent. Since this is not possible, the original equation
does not have positive integer solutions.
707.It suffices to show that there are no positive solutions. Adding the two equations,
we obtain
6 (x^2 +y^2 )=z^2 +t^2.