[et_pb_section fb_built="1" _builder_version="3.22.4" fb_built="1" _i="0" _address="0"][et_pb_row _builder_version="3.25" _i="0" _address="0.0"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.0.0"][et_pb_text _builder_version="3.22.4" 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_padding="20px|20px|20px|20px" _i="0" _address="0.0.0.0"]Understand the problem
[/et_pb_text][et_pb_text _builder_version="3.27" 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"]Positive real numbers 
and 
have the property that![\[\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100\]](https://latex.artofproblemsolving.com/3/9/0/390863333afc338a0d3a3157a3e2be1d552a68c2.png)
and all four terms on the left are positive integers, where log denotes the base 10 logarithm. What is 
?
[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.27" _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" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]2019 AMC 12A Problems/Problem 15
[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="off"]logarithm, diophantine equation[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Medium[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Mathematical Circles (Russian Experience)[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.27" 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
[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.2.1"]Givenboth \( \sqrt {\log a} , \sqrt {\log b} \) are positive integers . \( \Rightarrow \) both \( \log a ,\log b \) are perfect squares .similarly , both \( \log {\sqrt a} , \log {\sqrt b} \) are positive integers.\( \Rightarrow \) both \( a ,b \) are perfect squares .[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.2.2"]so $$ \log a = m^2 , \log b = n^2 $$ where \( m,n \in {Z^+} \)\( \Rightarrow a= 10^{m^2} , b= 10^{n^2} \) and as both \( a, b \) are perfect squares \( \\ \Rightarrow 10^{m^2} ,10^{n^2}\) are both perfect squares i.e \( 10^{m^2} = p^2,10^{n^2} =q^2 \) , where \( p,q \in {Z^+} \) . \( \\ \Rightarrow \frac {m^2}{2} , \frac {n^2}{2} \) are integers \( \\ \Rightarrow 2|m^2 , 2|n^2 \) \( \\ \Rightarrow 2|m, 2|n \) (as \(2\) is a prime number ) \( \\ \) so now \( a= 10^{4x^2} , b= 10^{4y^2} \) can be put in the original equation , where \( x,y \in {Z^+} \) . [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.2.3"] Now to get the solution from the derived equation i.e. \( 2x + 2y + 2x^2 + 2y^2 =100 \) multiply both the sides by \( 2 \) and then add \( 2 \) in both sides to arrive at \( (2x+1)^2 + (2y+1)^2 = 202 \) .[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27" hover_enabled="0" _i="4" _address="0.1.0.2.4"]Now use trial and error method to express \( 202 \) as a sum of two odd perfect squares .Finally the only way i.e. \( 9^2 + 11^2 = 202 \) . So without loss of generality it can be written that \( (2x+1) = 9 , (2y+1)= 11 \)So \( a= 10^{64} , b= 10^{100} \) and \( ab = 10^{164} \)[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.22.4" 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" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" _i="7" _address="0.1.0.7"]Connected Program at Cheenta
[/et_pb_text][et_pb_blurb title="Math Olympiad Program" url="https://cheenta.com/matholympiad/" url_new_window="on" image="https://cheenta.com/wp-content/uploads/2018/03/matholympiad.png" _builder_version="3.23.3" header_font="||||||||" header_text_color="#e02b20" header_font_size="48px" link_option_url="https://cheenta.com/matholympiad/" link_option_url_new_window="on" _i="8" _address="0.1.0.8"]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://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="9" _address="0.1.0.9"][/et_pb_button][et_pb_text _builder_version="3.22.4" 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="10" _address="0.1.0.10"]
Similar Problems
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