[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="4.0"][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="4.1" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.1" hover_enabled="0"]American Mathematical Contest 2014, AMC 10A Problem 8
[/et_pb_accordion_item][et_pb_accordion_item title="Key Competency" _builder_version="4.1" hover_enabled="0" open="off"]This number theory problem is based on the concept that when a number is a perfect square
[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.1" hover_enabled="0" open="off"]5/10
[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.0.9" open="off"]Challenges and Thrills in Pre College MathematicsExcursion Of Mathematics [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="4.0.9" 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|0px|20px||" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]
, we have![\[\dfrac{n!(n+1)!}{2}\]](https://latex.artofproblemsolving.com/c/6/c/c6cadcea118b402a6a8d08b9ed33371afc4b43a0.png)
.Now what we have to do with this ?
[/et_pb_tab][et_pb_tab title="HINT 2" _builder_version="4.0.9"]Now we have to find which member has what uniform numbers from the given conversation .[/et_pb_tab][et_pb_tab title="HINT 3" _builder_version="4.1" hover_enabled="0"]After some simple manipulations , we have![\[\implies\dfrac{(n!)^2\cdot(n+1)}{2}\]](https://latex.artofproblemsolving.com/0/f/4/0f4a73e6e6c4cc24be518527f91d6a867e8f5b24.png)
![\[\implies (n!)^2\cdot\dfrac{n+1}{2}\]](https://latex.artofproblemsolving.com/2/1/3/2130d4422339b6c4b17ef79fc45d476fffa67ed2.png)
. Thus now the problem reduces to finding a value of such that 
is a perfect square.
[/et_pb_tab][et_pb_tab title="HINT 4" _builder_version="4.1" hover_enabled="0"]Since 
is a perfect square, we must also have 
be a perfect square. In order for to be a perfect square, 
must be twice a perfect square.[/et_pb_tab][et_pb_tab title="HINT 5" _builder_version="4.1"]Now check the options and see for what value of n , must be twice a perfect square. 
works, thus, 
and our desired answer is 
[/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" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]
Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/amc-8-american-mathematics-competition/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="4.0.9" 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"]
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
