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August 17, 2018

Real Surds - Problem 2 Pre RMO 2017

[et_pb_section bb_built="1"][et_pb_row][et_pb_column type="4_4"][et_pb_tabs _builder_version="3.12" active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" tab_text_color="#ffffff"][et_pb_tab _builder_version="3.12" title="Problem" use_background_color_gradient="off" background_color_gradient_start="#2b87da" background_color_gradient_end="#29c4a9" background_color_gradient_type="linear" background_color_gradient_direction="180deg" background_color_gradient_direction_radial="center" background_color_gradient_start_position="0%" background_color_gradient_end_position="100%" background_color_gradient_overlays_image="off" parallax="off" parallax_method="on" background_size="cover" background_position="center" background_repeat="no-repeat" background_blend="normal" allow_player_pause="off" background_video_pause_outside_viewport="on" tab_text_shadow_style="none" body_text_shadow_style="none"] Suppose \(a,b\) are positive real numbers such that \(a\sqrt{a}+b\sqrt{b}=183\). \(a\sqrt{b}+b\sqrt{a}=182\). Find \(\frac{9}{5}(a+b)\). [/et_pb_tab][et_pb_tab _builder_version="3.12" title="Hint 1" use_background_color_gradient="off" background_color_gradient_start="#2b87da" background_color_gradient_end="#29c4a9" background_color_gradient_type="linear" background_color_gradient_direction="180deg" background_color_gradient_direction_radial="center" background_color_gradient_start_position="0%" background_color_gradient_end_position="100%" background_color_gradient_overlays_image="off" parallax="off" parallax_method="on" background_size="cover" background_position="center" background_repeat="no-repeat" background_blend="normal" allow_player_pause="off" background_video_pause_outside_viewport="on" tab_text_shadow_style="none" body_text_shadow_style="none"] This problem will use the following elementary algebraic identity: $$ (x+y)^3 = x^3 + y^3 + 3x^2y + 3xy^2 $$ Can you identify what is x and what is y? [/et_pb_tab][et_pb_tab _builder_version="3.12" title="Hint 2" use_background_color_gradient="off" background_color_gradient_start="#2b87da" background_color_gradient_end="#29c4a9" background_color_gradient_type="linear" background_color_gradient_direction="180deg" background_color_gradient_direction_radial="center" background_color_gradient_start_position="0%" background_color_gradient_end_position="100%" background_color_gradient_overlays_image="off" parallax="off" parallax_method="on" background_size="cover" background_position="center" background_repeat="no-repeat" background_blend="normal" allow_player_pause="off" background_video_pause_outside_viewport="on" tab_text_shadow_style="none" body_text_shadow_style="none"] Set \( x = \sqrt a, y = \sqrt b \).  Then the given information translates to $$ x^3 + y^3 = 183 , x^2y + xy^2 = 182 $$ This implies \( (x + y)^3 \\ = ( \sqrt a + \sqrt b)^3 \\ = x^3 + y^3 + 3(x^2y + xy^2) \\ = 183 + 3 \times 182 \\=  729 \) Finally taking cube root on both sides, we have \( \sqrt a + \sqrt b = 9 \) [/et_pb_tab][et_pb_tab _builder_version="3.12" title="Hint 3" use_background_color_gradient="off" background_color_gradient_start="#2b87da" background_color_gradient_end="#29c4a9" background_color_gradient_type="linear" background_color_gradient_direction="180deg" background_color_gradient_direction_radial="center" background_color_gradient_start_position="0%" background_color_gradient_end_position="100%" background_color_gradient_overlays_image="off" parallax="off" parallax_method="on" background_size="cover" background_position="center" background_repeat="no-repeat" background_blend="normal" allow_player_pause="off" background_video_pause_outside_viewport="on" tab_text_shadow_style="none" body_text_shadow_style="none"] Note that \( \sqrt a b + a \sqrt b = 182 \\ \Rightarrow \sqrt a \sqrt b ( \sqrt a + \sqrt b ) = 182 \\ \Rightarrow \sqrt {ab} \times 9 = 182 \) So at this point we know \( ( \sqrt a + \sqrt b ) = 9, \sqrt{ab} = \frac{182}{9} \). It should be easy to find the value of \( \frac{9}(5) (a+b) \) from these relations.   [/et_pb_tab][et_pb_tab _builder_version="3.12" title="Final Answer" use_background_color_gradient="off" background_color_gradient_start="#2b87da" background_color_gradient_end="#29c4a9" background_color_gradient_type="linear" background_color_gradient_direction="180deg" background_color_gradient_direction_radial="center" background_color_gradient_start_position="0%" background_color_gradient_end_position="100%" background_color_gradient_overlays_image="off" parallax="off" parallax_method="on" background_size="cover" background_position="center" background_repeat="no-repeat" background_blend="normal" allow_player_pause="off" background_video_pause_outside_viewport="on" tab_text_shadow_style="none" body_text_shadow_style="none" tab_text_shadow_horizontal_length="0em" tab_text_shadow_vertical_length="0em" tab_text_shadow_blur_strength="0em" body_text_shadow_horizontal_length="0em" body_text_shadow_vertical_length="0em" body_text_shadow_blur_strength="0em"] \( a+b \\ = (\sqrt a + \sqrt b )^2  - 2 \sqrt{ab} \\ = 9^2 - 2 \times \frac {182}{9} \\ = \frac{365}{9} \) Hence \( \frac {9}{5}(a+b) =  \frac {9}{5} \times \frac {365}{9} = 73 \) [/et_pb_tab][/et_pb_tabs][/et_pb_column][/et_pb_row][/et_pb_section]

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