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C. 876. Solve the following simultaneous equations:

x+y=x2+2xy+y2,

x-y=x2-2xy+y2.

(5 points)

Deadline expired on 15 January 2007.


Google Translation (Sorry, the solution is published in Hungarian only.)

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 on problem C. 876.
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 solutions.


  • Problems in Mathematics of KöMaL, December 2006

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