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.

Statistics:

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