Question 12
Prove that
$$
\frac{1}{1 \cdot 2013} + \frac{1}{2 \cdot 2012} + \frac{1}{3 \cdot 2011} + \dots + \frac{1}{2012 \cdot 2} + \frac{1}{2013 \cdot 1} \lt 1.
$$
Macedonian source version, translated in full:
Prove that
$$
\frac{1}{1 \cdot 2013} + \frac{1}{2 \cdot 2012} + \frac{1}{3 \cdot 2011} + \dots + \frac{1}{2012 \cdot 2} + \frac{1}{2013 \cdot 1} \lt 1.
$$
$$
\frac{1}{1 \cdot 2013} + \frac{1}{2 \cdot 2012} + \frac{1}{3 \cdot 2011} + \dots + \frac{1}{2012 \cdot 2} + \frac{1}{2013 \cdot 1} \lt 1.
$$
Macedonian source version, translated in full:
Prove that
$$
\frac{1}{1 \cdot 2013} + \frac{1}{2 \cdot 2012} + \frac{1}{3 \cdot 2011} + \dots + \frac{1}{2012 \cdot 2} + \frac{1}{2013 \cdot 1} \lt 1.
$$