7,8,9 |
|  | $[$ Text problems (various) $]$ |  |

A father was asked how old his two sons were. The father replied that if you add the sum of their ages to the product of their ages, you get 34.

How old are the sons?
Let one son be $m$ years old, and the other $-n$ years old. According to the condition, $m n + m + n = 34$. From this, $(m+1)(n+1) = 35$. $m+1$ and $n+1$ are divisors of 35, different from 1. Therefore, one of these numbers is 5, and the other is 7.

## [ Proof of identities. Transformations of expressions ] Problem 104092 Topics: [ $\frac{\text { Numerical inequalities. Comparisons of numbers. }}{\text { Irrational inequalities }}$ ]

Compare the numbers without using a calculator: $\sqrt{2006}+\sqrt{2005+\sqrt{2006}}$ and $\sqrt{2005}+\sqrt{2006+\sqrt{2005}}$.

## Solution

Consider the difference between these numbers:

$\sqrt{2006}+\sqrt{2005+\sqrt{2006}}-(\sqrt{2005}+\sqrt{2006+\sqrt{2005}})=$

$(\sqrt{2006}-\sqrt{2005})-(\sqrt{2005+\sqrt{2006}}-\sqrt{2006+\sqrt{2005}})=$

$=\frac{1}{\sqrt{2006}+\sqrt{2005}}+\frac{1+\sqrt{2006}-\sqrt{2005}}{\sqrt{2006+\sqrt{2005}}+\sqrt{2005+\sqrt{2006}}}>0$

since the first fraction is greater than the second. Indeed, the numerator of the first fraction is greater than the numerator of the second, and the denominator is smaller.

## Answer

The first number is greater.