Problem B. 5350. (December 2023)
B. 5350. \(\displaystyle a)\) Is it possible to find positive integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) such that the arithmetic mean of \(\displaystyle a\) and \(\displaystyle b\) is greater than the quadratic mean of \(\displaystyle c\) and \(\displaystyle d\), however, the geometric mean of \(\displaystyle a\) and \(\displaystyle b\) is smaller than the harmonic mean of \(\displaystyle c\) and \(\displaystyle d\)?
\(\displaystyle b)\) Is it possible to find positive integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) such that the geometric mean of \(\displaystyle a\) and \(\displaystyle b\) is greater than the quadratic mean of \(\displaystyle c\) and \(\displaystyle d\), however, the arithmetic mean of \(\displaystyle a\) and \(\displaystyle b\) is smaller than the harmonic mean of \(\displaystyle c\) and \(\displaystyle d\)?
Submitted by Bálint Hujter, Budapest
(3 pont)
Deadline expired on January 10, 2024.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. a) Vannak ilyen számok, például \(\displaystyle a=1\), \(\displaystyle b=9\), \(\displaystyle c=d=4\) esetén
\(\displaystyle \frac{a+b}{2} = \frac{1+9}{2} = 5 \quad > \quad 4 = \sqrt{\frac{4^2+4^2}2} = \sqrt{\frac{c^2+d^2}2}, \)
míg
\(\displaystyle \sqrt{ab} = \sqrt{1 \cdot 9} = 3 \quad < \quad 4 = \frac2{\frac1{4}+\frac1{4}} = \frac2{\frac1{c}+\frac1{d}}. \)
b) A következő egyenlőtlenség-lánc első és harmadik egyenlőtlenségét a feladat feltételei adják; a második egyenlőtlenség a mértani és számtani közép közti egyenlőtlenség \(\displaystyle a\) és \(\displaystyle b\) számokra; míg az utolsó egyenlőtlenség a harmonikus és négyzetes közép közti ismert egyenlőtlenség \(\displaystyle c\) és \(\displaystyle d\) számokra:
\(\displaystyle \sqrt{\frac{c^2+d^2}2} < \sqrt{ab} \leq \frac{a+b}{2} < \frac2{\frac1{c}+\frac1{d}} \leq \sqrt{\frac{c^2+d^2}2}. \)
Ellentmondást kaptunk, tehát nincsenek ilyen \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), \(\displaystyle d\) pozitív számok.
Statistics:
146 students sent a solution. 3 points: 105 students. 2 points: 22 students. 1 point: 2 students. 0 point: 3 students. Not shown because of missing birth date or parental permission: 10 solutions.
Problems in Mathematics of KöMaL, December 2023