I haven't seen this problem so I don't know if this is my solution, or it has been found already. I'll use notadion $ld(x)=log_{2}(x)$, and $ln(x)=log_{e}(x)$: $(1+a_{1})(1+a_{2})\cdots (1+a_{n}) < 2$ $\Leftrightarrow$ $\sum_{i=1}^{n} ld(a_{i}+1) <1$ Let $f(x)=ld(x+1)$. Since f is concave, and $f(0)=0$ we have: $f(x)\le f'(0)x$ $\Rightarrow$ $ld(x)\le \frac{1}{ln(2)}x$ $\Rightarrow$ $\sum_{i=1}^{n} ld(a_{i}+1) \le \frac{1}{ln(2)}\sum_{i=1}^{n}a_{i} \le \frac{1}{2ln(2)}<1$

Left side equals $1+\sum_{i=1}^n a_i+\sum_{i < j =1}^n a_ia_j+....+a_1a_2....a_n$ obviously, that $\sum_{i_1<i_2<...<i_k=1}^{n} \prod_{j=1}^k a_{i_j} \leq (\sum_{i=1}^n a_i)^n $ for all $k=1,2,....,n$ => left side $ \leq 1+\sum_{i=1}^n a_i + (\sum_{i=1}^n a_i)^2+ ....+ (\sum_{i=1}^n a_i)^n = \sum_{i=0}^n \frac{1}{2^n} < 2$ => $QED$

amparvardi, I think TT2008 it's a race. See here: http://www.madzone.net/pictures/displayimage.php?album=17&pos=9 What do you mean when you write TT2008 Junior A-Level - P5?

arqady wrote: What do you mean when you write TT2008 Junior A-Level - P5? At Tournament of Towns (TT) there are two variants: easy (or training) and difficult they held whith interval 2 weeks. And two leagues : Junior and Senior.

Ovchinnikov Denis wrote: arqady wrote: What do you mean when you write TT2008 Junior A-Level - P5? At Tournament of Towns (TT) there are two variants: easy (or training) and difficult they held whith interval 2 weeks. And two leagues : Junior and Senior. Thank you, Ovchinnikov Denis! It 's a first time when I see that Tournament of Towns is TT.

Quote: It 's a first time when I see that Tournament of Towns is TT. Why not ?

Ovchinnikov Denis wrote: Quote: It 's a first time when I see that Tournament of Towns is TT. Why not ? Well, TT or ToT are quite common abbreviations for this contest.

Any other solutions?

Any other solutions?

The question is weak and my solution is same as #4