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September 6, 2019

The best exponent for an inequality

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]Let $a,b,c,d$ be positive real numbers such that $abcd=1$.Find with proof that $x=3 $ is the minimal value for which the following inequality holds:
\[a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}\]

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27.4" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27.4" _i="0" _address="0.1.0.0.0" hover_enabled="0"]

Albania IMO TST 2013 [/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27.4" _i="1" _address="0.1.0.0.1" open="off" hover_enabled="0"]Inequalities [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27.4" _i="2" _address="0.1.0.0.2" open="off" hover_enabled="0"]Medium [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27.4" _i="3" _address="0.1.0.0.3" open="off" hover_enabled="0"]Inequalities by BJ Venkatachala [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"]

Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27.4" _i="1" _address="0.1.0.2.1" hover_enabled="0"]Choose specific values of a,b,c,d to show that no value of x less than 3 works. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27.4" _i="2" _address="0.1.0.2.2" hover_enabled="0"]Choosing a=b=c=t, the inequality becomes 3t^x+\frac{1}{t^{3x}}\ge \frac{3}{t}+t^3. Note that, as t\to\infty, the LHS grows as O(t^x) and the RHS grows as O(t^3). For the LHS to be always greater than the RHS, it should be of a higher order. Hence, $x\ge 3$. Now show that the inequality is true for x=3. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27.4" _i="3" _address="0.1.0.2.3" hover_enabled="0"]Note that the RHS can be rewritten as abc+bcd+cda+bda. This reminds us of the AM-GM inequality. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27.4" _i="4" _address="0.1.0.2.4" hover_enabled="0"]Indeed, abc+bcd+cda+bda\le \frac{a^3+b^3+c^3}{3}+\frac{d^3+b^3+c^3}{3}+\frac{a^3+d^3+c^3}{3}+\frac{a^3+b^3+d^3}{3}=a^3+b^3+c^3+d^3. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]

Watch video

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Connected Program at Cheenta

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Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://www.cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark" _i="7" _address="0.1.0.7"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" _i="8" _address="0.1.0.8"]

Similar Problems

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