Problem K. 65.

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K. 65. x is a real number, such that $x+\frac{1}{x}=5$ . Determine the exact values of $x^2+\frac{1}{x^2}$ and $x^3+\frac{1}{x^3}$ .

(6 points)

Deadline expired.

Sorry, the solution is published in Hungarian only.

Megoldás: Használjuk fel, hogy $\left(x+{1\over x}\right)^2=x^2+2\cdot x\cdot{1\over x}+{1\over x^2}=x^2+{1\over x^2}+2$ . Ebből adódik, hogy $x^2+{1\over x^2}=\left(x+{1\over x}\right)^2-2=5^2-2=23$ .

Használjuk fel, hogy $\left(x+{1\over x}\right)^3=x^3+3\cdot x^2\cdot{1\over x}+3\cdot x\cdot{1\over x^2}+{1\over x^3}=$

$=x^3+3x+{3\over x}+{1\over x^3}=x^3+{1\over x^3}+3\cdot\left(x+{1\over x}\right)$ .

Ebből adódik, hogy $x^3+{1\over x^3}=\left(x+{1\over x}\right)^3-3\cdot\left(x+{1\over x}\right)=5^3-3\cdot5=110$ .

Statistics on problem K. 65.

198 students sent a solution.

6 points: 151 students.

5 points: 6 students.

4 points: 14 students.

3 points: 7 students.

2 points: 4 students.

0 point: 10 students.

Unfair, not evaluated: 6 solutions.

Problems in Mathematics of KöMaL, December 2005

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