Problem B. 4113. (October 2008)
B. 4113. Prove that if a, b, c, d are integers and a+b+c+d=0 then 2(a4+b4+c4+d4)+8abcd is a square number.
(4 pont)
Deadline expired on November 17, 2008.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás: Vegyük észre, hogy
(a+b+c+d)(a+b-c-d)(a-b+c-d)(a-b-c+d)=
=(a2+b2-c2-d2+2ab-2cd)(a2+b2-c2-d2-2ab+2cd)=
=(a2+b2-c2-d2)2-(2ab-2cd)2=
=a4+b4+c4+d4-2a2b2-2a2c2-2a2d2-2b2c2-2b2d2-2c2d2+8abcd.
Ezt a 0-val egyenlő számot az S=2(a4+b4+c4+d4)+8abcd számból kivonva kapjuk, hogy
S=a4+b4+c4+d4+2a2b2+2a2c2+2a2d2+2b2c2+2b2d2+2c2d2=
=(a2+b2+c2+d2)2,
ami valóban egy egész szám négyzete.
Statistics:
113 students sent a solution. 4 points: 96 students. 3 points: 4 students. 2 points: 2 students. 1 point: 4 students. 0 point: 7 students.
Problems in Mathematics of KöMaL, October 2008