Baltic Way 1993 · Problem 9

Algebra

Solve the system of equations:

\left\{\begin{array}{l} x^{5}=y+y^{5} \\ y^{5}=z+z^{5} \\ z^{5}=t+t^{5} \\ t^{5}=x+x^{5} . \end{array}\right.

Change pool

When you’re ready

Review material becomes available with the next Daily.

Review

Topics

Polynomials · Algebraic manipulation

Solutions

Solution

Solution: Adding all four equations we get $x+y+z+t=0$ . On the other hand, the numbers $x, y, z, t$ are simultaneously positive, negative or equal to zero. Thus, $x=y=z=t=0$ is the only solution.

Contest context

Results from Baltic Way 1993

8 teams

Mean score: 4.6 / 5
Scores of 4 or 5: 7 / 8
Estonia: 5 / 5

Score distribution

All team scores

Team	Score
Poland	5 / 5
Latvia	5 / 5
Estonia	5 / 5
Sweden	2 / 5
Lithuania	5 / 5
Finland	5 / 5
Iceland	5 / 5
Denmark	5 / 5

Elsewhere

AoPS · Baltic Way 1993
Official results