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Found problems: 2

2011 Bosnia and Herzegovina Junior BMO TST, 2
Tags: nonnegative , inequality , inequalities
Prove inequality, with $a$ and $b$ nonnegative real numbers: $\frac{a+b}{1+a+b}\leq \frac{a}{1+a} + \frac{b}{1+b} \leq \frac{2(a+b)}{2+a+b}$

2020 Bundeswettbewerb Mathematik, 4
Tags: combinatorics , inequalities , nonnegative , sum
In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number. Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.