Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 266. (November 2010)

K. 266. Let a, b, c, d denote positive integers. Given that , show that always lies between and .

(6 pont)

Deadline expired on December 10, 2010.

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

Megoldás. Ha $\displaystyle \frac ab < \frac cd$, akkor $\displaystyle ad<bc$, továbbá $\displaystyle \frac{a+c}{b+d}-\frac ab=\frac{ab+cb-ab-ad}{b(b+d)}>0$ és $\displaystyle \frac{cb-ad}{b^2+bd}<\frac{cb-ad}{bd}=\frac ab - \frac cd$. Az egyenlőtlenség-sorozatból következik a bizonyítandó állítás: $\displaystyle \frac ab < \frac{a+c}{b+d} < \frac cd$.

Statistics:

 170 students sent a solution. 6 points: 113 students. 5 points: 25 students. 4 points: 5 students. 3 points: 2 students. 2 points: 2 students. 1 point: 5 students. 0 point: 12 students. Unfair, not evaluated: 6 solutionss.

Problems in Mathematics of KöMaL, November 2010