How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# What are we learning ?

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Competency in Focus: Perfect square numbers  This problem from American Mathematics contest (AMC 10A, 2014) is based on the concept that when a number is a perfect square . [/et_pb_text][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_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

# Next understand the problem

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" hover_enabled="0" box_shadow_style="preset2"]Which of the following numbers is a perfect square? $\textbf{(A)}\ \dfrac{14!15!}2\qquad\textbf{(B)}\ \dfrac{15!16!}2\qquad\textbf{(C)}\ \dfrac{16!17!}2\qquad\textbf{(D)}\ \dfrac{17!18!}2\qquad\textbf{(E)}\ \dfrac{18!19!}2$ [/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 Mathematics Excursion Of Mathematics

[/et_pb_text][et_pb_tabs _builder_version="4.1"][et_pb_tab title="HINT 0" _builder_version="4.0.9"]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title="HINT 1" _builder_version="4.1" hover_enabled="0"]First of all look at the examples , see that   for all positive $n$, we have$$\dfrac{n!(n+1)!}{2}$$.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$$\dfrac{n!(n+1)!}{2}$$$$\implies\dfrac{(n!)^2\cdot(n+1)}{2}$$$$\implies (n!)^2\cdot\dfrac{n+1}{2}$$ . Thus now the problem reduces to  finding  a value of $n$ such that $(n!)^2\cdot\dfrac{n+1}{2}$ is a perfect square. [/et_pb_tab][et_pb_tab title="HINT 4" _builder_version="4.1" hover_enabled="0"]Since $(n!)^2$ is a perfect square, we must also have $\frac{n+1}{2}$ be a perfect square. In order for $\frac{n+1}{2}$ to be a perfect square,  $n+1$ 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 , $n+1$ must be twice a perfect square.   $n+1=18$ works, thus, $n=17$ and our desired answer is $\boxed{\textbf{(D)}\ \frac{17!18!}{2}}$ [/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"]

# Similar Problems

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