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September 1, 2018

I.S.I. Entrance Solution Sequence of isosceles triangles -2018 Problem 6

[et_pb_section bb_built="1" admin_label="Blog Hero" _builder_version="3.0.82" use_background_color_gradient="on" background_color_gradient_start="rgba(114,114,255,0.24)" background_color_gradient_end="#ffffff" background_blend="multiply" custom_padding="0|0px|0|0px|false|false" animation_style="slide" animation_direction="top" animation_intensity_slide="2%" locked="off" next_background_color="#ffffff"][et_pb_row custom_width_px="1280px" custom_padding="27px|0px|27px|0px" custom_margin="|||" _builder_version="3.0.82" background_size="initial" background_position="top_left" background_repeat="repeat"][et_pb_column type="4_4"][et_pb_text _builder_version="3.12.2" text_text_color="#474ab6" text_line_height="1.9em" background_size="initial" background_position="top_left" background_repeat="repeat" text_orientation="center" max_width="540px" module_alignment="center" locked="off"] Let, \( a \geq b \geq 0 \) be real numbers such that for all natural number n, there exist triangles of side lengths \( a^n,b^n,c^n \)  Prove that the triangles are isosceles. [/et_pb_text][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section bb_built="1" admin_label="Blog" _builder_version="3.0.82" custom_margin="|||" custom_padding="0px|0px|21px|0px|false|false" prev_background_color="#000000" next_background_color="#f7f8fc"][et_pb_row use_custom_width="on" custom_width_px="960px" custom_padding="0|0px|24px|0px|false|false" _builder_version="3.0.82" background_size="initial" background_position="top_left" background_repeat="repeat"][et_pb_column type="4_4"][et_pb_tabs _builder_version="3.12.2"][et_pb_tab title="Hint 1 - Triangular Inequality" _builder_version="3.12.2" 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" 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"] If a, b, c are sides of a triangle, triangular inequality assures that difference of two sides is lesser than the third side. Since \( a \ge b \ge c > 0 \), hence using triangular inequality we have a - b < c. Infact for all n, \( a^n - b^n < c^n \) [/et_pb_tab][et_pb_tab title="Hint 2 - Factor and estimate" _builder_version="3.12.2" 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" 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"] We have \( a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + ... + b^{n-1} ) < c^n \) Replacing every a by b in the left hand side, we make the expression to the left even smaller. i.e. \((a-b)(b^{n-1} + b^{n-2}b + ... + b^{n-1} ) \le (a-b)(a^{n-1} + a^{n-2}b + ... + b^{n-1} ) < c^n \) Hence \( (a-b) \times n \times b^{n-1} < c^n \) [/et_pb_tab][et_pb_tab title="Hint 3 - Final Steps" _builder_version="3.12.2" 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" 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"] Now notice \( (a-b) < \frac {c^n}{n\times b^{n-1}} = \frac{c}{n} \times \frac {c^{n-1}}{b^{n-1}} = \frac {c}{n} (\frac{c}{b})^{n-1}\) Clearly \( \frac{c}{b} \le 1 \) by given hypothesis. Hence \( a-b \le \frac{c}{n} \) for all n. But letting n go to infinity, we see that a and b can be made arbitrarily close to each other. This implies a=b. Hence each triangle in the sequence is isosceles [/et_pb_tab][/et_pb_tabs][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section bb_built="1" admin_label="Footer" _builder_version="3.0.82" background_color="#f7f8fc" custom_padding="0px|0px|2px|0px|false|false" animation_style="zoom" animation_direction="bottom" animation_intensity_zoom="6%" animation_starting_opacity="100%" saved_tabs="all" prev_background_color="#ffffff"][et_pb_row use_custom_gutter="on" gutter_width="2" custom_padding="24px|0px|145px|0px|false|false" _builder_version="3.0.82" background_size="initial" background_position="top_left" background_repeat="repeat"][et_pb_column type="1_2"][et_pb_text _builder_version="3.12.2" text_text_color="#7272ff" header_font="|on|||" header_text_color="#7272ff" header_font_size="36px" header_line_height="1.5em" background_size="initial" background_position="top_left" background_repeat="repeat" custom_margin="||20px|" animation_style="slide" animation_direction="bottom" animation_intensity_slide="10%"]

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