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