Mathematical and Physical Journal
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Problem A. 548. (December 2011)

A. 548. Prove that

\prod_{i=1}^n \left(1+\frac1{x_1+\ldots+x_i}\right) +
\prod_{i=1}^n \left(1-\frac1{x_i+\ldots+x_n}\right) \le n+1

holds for arbitrary real numbers x_1,\ldots,x_n\ge 1.

(5 pont)

Deadline expired on January 10, 2012.

Solution. Note that the factors on the left-hand side are all nonnegative.

Let S=x1+...+xn. From

1+\frac1{x_1+\ldots+x_i} \le 1+\frac1i = \frac{i+1}i


1-\frac1{x_i+\ldots+x_n} = 1-\frac1{S-x_1-\dots-x_{i-1}} \le
1-\frac1{S-(i-1)} = \frac{S-i}{S-i+1}

we get

\prod_{i=1}^n \left(1+\frac1{x_1+\ldots+x_i}\right) +
\prod_{i=1}^n \left(1-\frac1{x_i+\ldots+x_n}\right)

\le \prod_{i=1}^{n-1} \frac{i+1}i \cdot \left(1+\frac1S\right)
+ \prod_{i=1}^n \frac{S-i}{S-i+1} 
= n\cdot \frac{S+1}S + \frac{S-n}S = n+1.

Equality holds if and only if x1=...=xn-1=1.


3 students sent a solution.
5 points:Gyarmati Máté, Janzer Olivér, Omer Cerrahoglu.

Problems in Mathematics of KöMaL, December 2011