[et_pb_section fb_built="1" _builder_version="3.22.4" fb_built="1" _i="0" _address="0"][et_pb_row _builder_version="3.25" _i="0" _address="0.0"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.0.0"][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" _i="0" _address="0.0.0.0"]Understand the problem
[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2" _i="1" _address="0.0.0.1"]Find all positive integer solutions of the equation 
[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.22.4" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" hover_enabled="0" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="off" _builder_version="3.29.2" hover_enabled="0" _i="0" _address="0.1.0.0.0"]Croatia MO 2005
[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.29.2" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="off"]Number Theory
[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.29.2" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]6/10
[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.29.2" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="on"]Challenges and Thrills in Pre College Mathematics
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Start with hints
[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.29.2" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.29.2" hover_enabled="0" _i="1" _address="0.1.0.2.1"]Observe that the LHS is sort of a quadratic and the RHS is sort of a linear type.
Observe that the given equation is the same as (k!-1)(l! - 1) = m! + 1.
Now observe that it implies that m > k,l.
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.29.2" hover_enabled="0" _i="2" _address="0.1.0.2.2"]Given that m>l,k, Observe that the equation is symmetric in k and l.
WLOG, assume l $latex \geq$ k.
Now. there are two cases:
1. m > l = k.
2. m > l > k.
[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.29.2" hover_enabled="0" _i="3" _address="0.1.0.2.3"]1. m > l = k.
Observe that it turns out that k! = 2 + (k+1)...m.
Hence, k can be 0/1/2/3.
The only solution turns out to be (3,3,4) for (k,l,m).
[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.29.2" hover_enabled="0" _i="4" _address="0.1.0.2.4"]2. m > l > k.
l! = 1 + (k+1)...l + (k+1)...l...m.
Hence, it implies that l can only be = 1, but it doesn't give rise to any solution.
Hence, the only solution is (3,3,4).
[/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" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" _i="3" _address="0.1.0.3"]Watch video
[/et_pb_text][et_pb_code _builder_version="3.26.4" _i="4" _address="0.1.0.4"]https://apis.google.com/js/platform.js[/et_pb_code][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" _i="5" _address="0.1.0.5"]Connected Program at Cheenta
[/et_pb_text][et_pb_blurb title="Math Olympiad Program" url="https://cheenta.com/matholympiad/" url_new_window="on" image="https://cheenta.com/wp-content/uploads/2018/03/matholympiad.png" _builder_version="3.23.3" header_font="||||||||" header_text_color="#e02b20" header_font_size="48px" link_option_url="https://cheenta.com/matholympiad/" link_option_url_new_window="on" _i="6" _address="0.1.0.6"]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://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" _i="7" _address="0.1.0.7"][/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" _i="8" _address="0.1.0.8"]
Similar Problems
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