HMMT 二月 2010 · 冲刺赛 · 第 26 题

HMMT February 2010 — Guts Round — Problem 26

[ 15 ] Express the following in closed form, as a function of x : 2 2 2 2 2 2 2 2010 2 2009 2 2 sin ( x )+sin (2 x ) cos ( x )+sin (4 x ) cos (2 x ) cos ( x )+ · · · +sin (2 x ) cos (2 x ) · · · cos (2 x ) cos ( x ) .

解析

[ 15 ] Express the following in closed form, as a function of x : 2 2 2 2 2 2 2 2010 2 2009 2 2 sin ( x )+sin (2 x ) cos ( x )+sin (4 x ) cos (2 x ) cos ( x )+ · · · +sin (2 x ) cos (2 x ) · · · cos (2 x ) cos ( x ) . 2 2011 sin (2 x ) Answer: 1 − Note that 2 2011 4 sin ( x ) 2 2 2 2 2010 2 2009 2 2 sin ( x ) + sin (2 x ) cos ( x ) + · · · + sin (2 x ) cos (2 x ) · · · cos (2 x ) cos ( x ) 2 2 2 2 2010 2 2009 2 = (1 − cos ( x )) + (1 − cos (2 x )) cos ( x ) + · · · + (1 − cos (2 x )) cos (2 x ) · · · cos ( x ) , 2 2 2 2 2010 2 2 2 2010 which telescopes to 1 − cos ( x ) cos (2 x ) cos (4 x ) · · · cos (2 x ). To evaluate cos ( x ) cos (2 x ) · · · cos (2 x ), 2 multiply and divide by sin ( x ). We then get 2 2 2 2 2010 sin ( x ) cos ( x ) cos (2 x ) · · · cos (2 x ) 1 − . 2 sin ( x ) 2 sin (2 y ) 2 2 Using the double-angle formula for sin, we get that sin ( y ) cos ( y ) = . Applying this 2011 times 4 makes the above expression 2 2011 sin (2 x ) 1 − , 2 2011 4 sin ( x ) which is in closed form.