Problem C. 876. (December 2006)
C. 876. Solve the following simultaneous equations:
x+y=x2+2xy+y2,
x-y=x2-2xy+y2.
(5 pont)
Deadline expired on January 15, 2007.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. A két egyenletet átalakítva:
(1) | x+y=(x+y)2, |
(2) | x-y=(x-y)2. |
Mivel a=a2 akkor teljesül, ha a(a-1)=0, vagyis a=0 vagy a=1 esetén, ezért (1) akkor igaz, ha x+y=0 vagy x+y=1, (2) pedig akkor, ha x-y=0 vagy x-y=1. Ez négy esetet jelent, mind a négy esetben egy-egy megoldást kapunk: x1=y1=0, x2=0,5; y2=-0,5; x3=0,5, y3=0,5; x4=1, y4=0.
Statistics:
535 students sent a solution. 5 points: 346 students. 4 points: 7 students. 3 points: 28 students. 2 points: 90 students. 1 point: 50 students. 0 point: 7 students. Unfair, not evaluated: 7 solutionss.
Problems in Mathematics of KöMaL, December 2006