Mathematical and Physical Journal
for High Schools
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Problem K. 65. (December 2005)

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 pont)

Deadline expired on January 10, 2006.

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

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.


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