Problem B. 5195. (October 2021)
B. 5195. Prove that the inequality \(\displaystyle x^{p}\cdot y^{1-p}<x+y\) holds for every pair of positive real numbers \(\displaystyle (x,y)\), and all real numbers \(\displaystyle 0<p<1\).
(3 pont)
Deadline expired on November 10, 2021.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. A hatványfüggvény szigorú monotonitását használva \(\displaystyle x,y>0\) alapján kapjuk, hogy
\(\displaystyle x^{p}\cdot y^{1-p}<(x+y)^p\cdot (x+y)^{1-p}=x+y,\)
így a feladat állítása valóban teljesül.
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127 students sent a solution. 3 points: 96 students. 2 points: 9 students. 1 point: 6 students. 0 point: 11 students. Not shown because of missing birth date or parental permission: 1 solutions.
Problems in Mathematics of KöMaL, October 2021