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June 15, 2019

An inequality with unit coefficients

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]Let $x_1,\ldots ,x_n$ be positive real numbers. Show that there exist $a_1,\ldots ,a_n\in\{-1,1\}$ such that:
\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2\]

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.22.4"][et_pb_column type="4_4" _builder_version="3.22.4"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.23.3" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Iberoamerican olympiad 2011[/et_pb_accordion_item][et_pb_accordion_item title="Topic" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Inequalities[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Easy[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]
Inequalities: An Approach Through Problems
by B.J. Venkatachala
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Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.23.3"]Try using induction.[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.23.3"]As the inequality is symmetric in x_1,x_2,\cdots x_n, introducing an order might help.[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"]Consider the inequality as f(x_1,x_2,\cdots x_n)\ge 0.[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.23.3"]

Let us assume that x_1\ge x_2\ge\cdots\ge x_n.   Claim a_{odd}=1, a_{even}=-1 works.   Proof: For n=1 Trivial.   For n=2 The inequality is equivalent to x_1^2-x_2^2\ge (x_1-x_2)^2. Expanding, this becomes x_1x_2\ge x_2^2 which is a consequence of x_1\ge x_2.   For n\ge 3   Consider a_1x_1^2+a_2x_2^2+\cdots +a_nx_n^2-(a_1x_1+a_2x_2+\cdots +a_nx_n)^2 as a function of x_1 (say f(x_1)). We need to show that the minimum of f is at least 0. Clearly, f is linear in x_1 and the coefficient of x_1 is 2(x_2-x_3+x_4-x_5\cdots +(-1)^nx_n). Due to the order chosen, this coefficient is non-negative. Hence the minimum is attained at the minimum of x_1, which is x_2. However, putting x_1=x_2, we are left with  a_3x_3^2+a_4x_4^2+\cdots +a_nx_n^2\ge (a_3x_3+a_4x_4+\cdots a_nx_n)^2 which is true from the induction hypothesis.   QED

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Similar Problems

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