Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

# Problem A. 561. (April 2012)

A. 561. Show that

holds for all positive numbers a, b, cp.

(5 pont)

Deadline expired on May 10, 2012.

Solution 1. We show that

 (1)

Applying the AM-GM inequality to the fractions on the left-hand side, then to the numbers 3a+b and a+b+2c,

Rearranging (1), by the cyclic permutations of the variables we get

The statement of the problem is the sum of the last three lines above.

Solution 2.

Lemma. For A,p>0 we have

where

is Euler's Gamma-function. (See also the solution of Problem A. 493.)

Proof. Substituting u=At,

Applying the Lemma to A=3a+b, A=3b+c, A=3c+a, A=2a+b+c, A=2b+c+a, A=2c+a+b,

Remark. Both soutions show that equality holds only when a=b=c.

### Statistics:

 5 students sent a solution. 5 points: Ágoston Tamás, Janzer Olivér, Mester Márton, Omer Cerrahoglu, Strenner Péter.

Problems in Mathematics of KöMaL, April 2012