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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)=

=\bigl( (a+b)^2-(c+d)^2 \bigr) \bigl( (a-b)^2-(c-d)^2 \bigr)=

=(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:

114 students sent a solution.
4 points:97 students.
3 points:4 students.
2 points:2 students.
1 point:4 students.
0 point:7 students.

Problems in Mathematics of KöMaL, October 2008