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RMO 2019 Problem 6 Solution

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="4.0" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Suppose 91 distinct positive integers greater than 1 are given such that there are at least 456 pairs among
them which are relatively prime. Show that one can find four integers a, b, c, d among them such that
gcd(a, b) = gcd(b, c) = gcd(c, d) = gcd(d, a) = 1.

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.0" 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="4.0"]Regional Math Olympiad, 2019 Problem 6

[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="4.0" open="off"]Combinatorics

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8/10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.29.2" open="off"]Challenges and Thrills in Pre College Mathematics[/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" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.0"]The clue: Consider the numbers as points in a graph, and some edges between the points based on their gcd. Let us consider a graph G with 91 vertices (91 distinct numbers) and connect each pair of these vertices. which corresponds to co-prime pairs implies at least 456 edges i.e. e \( \geq\) 456. Here e = number of edges. Now getting four number a,b, c, d such that gcd (a, b) = gcd (b, c) = gcd(c,d) = gcd (d,a) = 1 implies existence at a cycle of length 4.  [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0"]We will follow the contradiction method. Let us assume there is no cycle of length '4'. Our aim is to find a bound on e, which contradicts the 456 bound on e. Let vertex set of G be V =  {\(v_1, v_2,...,v_{91} \)}, let degree of \( v_i = d_i\). Now, observe that the number of vertex pairs {\( v_k, v_l\)}, adjacent to \(v_i\) is \({d_i}\choose{2}\). Now observe that a vertex pair {\( v_k, v_l\)} cannot be common to two vertices say \(v_i\) and \(v_j\) both, because if so, \( v_i, v_k, v_j, v_l, v_i\) will form a cycle.[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.0"]

So, the sum of all such vertex pairs corresponding to a vertex over all the vertex set \( \leq\) the total number of vertex pairs. \( \sum_{i=1}^{91} ({d_i}\choose{2}) \leq {{91}\choose{2}} \rightarrow \) \( \sum_{i=1}^{91} ({d_i}^2 - d_i)\leq 91.90  \rightarrow \)  \( \sum_{i=1}^{91} {d_i}^2 \leq 2e + {{91}\choose{2}}\) -> (1)  as \( \sum_{i=1}^{91} d_i = 2e \).

We will try to find a lower bound on  \( \sum_{i=1}^{91} {d_i}^2 \) w.r.t e.

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By RMS - AM inequality, \( \sum_{i=1}^{91} {d_i}^2 \geq  \frac{( \sum_{i=1}^{91} {d_i})^2 }{91} = \frac{4e^2}{91} \) -> (2) Using (1) and (2), we get \( \frac{2e^2}{91} - e \leq 91.45 \) From here, by solving we get \( e \leq 91x5 =455\). This gives the contradiction.

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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%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/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" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Similar Problems

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