Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation

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Problem B. 4057. (January 2008)

B. 4057. Solve the equation x⁶-6x+5=0.

(4 pont)

Deadline expired on February 15, 2008.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás: Ha x $\le$ 0, akkor x⁶-6x+5 $\ge$ 5>0. Ha x>0, akkor a számtani és mértani közepek között fennálló egyenlőtlenség értelmében

$\frac{x^6+1+1+1+1+1}{6}\ge \root{6}\of{x^6\cdot1\cdot1\cdot1\cdot1\cdot1},$

vagyis x⁶+5 $\ge$ 6x, és egyenlőség pontosan az x⁶=1 esetben áll fenn. Az egyenlet egyetlen valós megoldása tehát x=1.

Statistics:

185 students sent a solution.

4 points: 118 students.

3 points: 25 students.

2 points: 16 students.

1 point: 20 students.

0 point: 2 students.

Unfair, not evaluated: 4 solutionss.

Problems in Mathematics of KöMaL, January 2008

185 students sent a solution.
4 points:	118 students.
3 points:	25 students.
2 points:	16 students.
1 point:	20 students.
0 point:	2 students.
Unfair, not evaluated:	4 solutionss.