Problem C. 1368. (September 2016)

C. 1368. Solve the equation

\(\displaystyle {[x]}^2+ {\{x\}}^2 + x^2 +2[x]\{x\}=4x-2x[x]-2x\{x\}-1, \)

where \(\displaystyle [x]\) denotes the greatest integer not greater than the number \(\displaystyle x\), and \(\displaystyle \{x\}\) denotes the difference between \(\displaystyle x\) and \(\displaystyle [x]\).

(5 pont)

Deadline expired on October 10, 2016.

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

Megoldás. Rendezzük át az egyenletet:

\(\displaystyle [x]^2+2[x]\{x\}+\{x\}^2+x^2=4x-2x\cdot([x]+\{x\})-1,\)

\(\displaystyle ([x]+\{x\})^2+x^2=-2x\cdot([x]+\{x\})+4x-1.\)

Mivel \(\displaystyle [x]+\{x\}=x\), így az egyenlet így írható: \(\displaystyle x^2+x^2=-2x^2+4x-1\). Rendezve: \(\displaystyle 4x^2-4x+1=0\), teljes négyzetté alakítva: \(\displaystyle (2x-1)^2=0\), amiből \(\displaystyle 2x-1=0\), és így \(\displaystyle x=\frac12\) következik.

Ellenőrzés: \(\displaystyle 0+\frac14+\frac14+2\cdot0\cdot\frac12=4\cdot\frac12-2\cdot\frac12\cdot0-2\cdot\frac12\cdot\frac12-1\), vagyis \(\displaystyle \frac12= \frac12\).

Statistics:

334 students sent a solution.
5 points:	282 students.
4 points:	21 students.
3 points:	13 students.
2 points:	6 students.
1 point:	1 student.
0 point:	4 students.
Unfair, not evaluated:	7 solutionss.

Problems in Mathematics of KöMaL, September 2016