<s>[INST] Olympiad-like Inequality Problem Let
$$ A := \frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times\cdots\times \frac{2013}{2014};$$
let
$$ B := \frac{2}{3} \times \frac{4}{5} \times \frac{6}{7} \times\cdots \times \frac{2012}{2013};$$
and let
$$ C := \frac{1}{ \sqrt{2014} }. $$
Find the relations between $A, B,$ and $C$.
 [/INST] Let fractions in $A:= a_1, a_2, a_3, \ldots, a_{1007} $
Let fractions in $B:=b_1, b_2, b_3, \ldots , b_{1006} $
But $ a_1 < b_1, a_2 < b_2, a_3 < b_3 \ldots a_{1006} < b_{1006}, a_{1007} < 1$
Multiplying all these, $ A < B $
Observe that $ A \times B = \frac{1}{2014} = C^2 $
So $A, C, B $ are positive numbers forming increasing in Geometric Progression.
$$\therefore  A < C < B $$
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