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

Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]Find, with justification, all positive real numbers   $a,b,c$   satisfying the system of equations:    $$a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b.$$[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.27.4" hover_enabled="0" _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="3.27.4" 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="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]SMO (senior)-2014 stage 2 problem 2

[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="on"]Number Theory[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27.4" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Easy [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Excursion in Mathematics

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27.4" 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="3.27" hover_enabled="0" _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="3.27.4" hover_enabled="0" _i="1" _address="0.1.0.2.1"]Given all three relations are cyclic and symmetric . So without loss of generality it can be assumed that $a \geq b \geq c >0$ .     Then proceed .               [ Note $(0, 0, 0)$ can't be a solution since $a , b , c$ are positive reals .] [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27.4" hover_enabled="0" _i="2" _address="0.1.0.2.2"]So $a \sqrt b = a + c \Rightarrow a(\sqrt b - 1) = c \ [and \ we \ have \ a \geq c] \Rightarrow ( \sqrt b - 1 ) \leq 1 \Rightarrow b \leq 4$[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27.4" hover_enabled="0" _i="3" _address="0.1.0.2.3"]Similarly $b \sqrt c = b + a \Rightarrow b(\sqrt c - 1) = a \ [and \ we \ have \ a \geq b ] \Rightarrow \sqrt c - 1 \geq 1 \Rightarrow c \geq 4$[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27.4" hover_enabled="0" _i="4" _address="0.1.0.2.4"]

Till now we have $b \leq 4 \ and \ c \geq 4$ , but we assumed that $b \geq c$ . So it is clear that $b = c =4$  $\Rightarrow a = 4$ also. So the only triplet $(a , b , c)$ is $(4,4 ,4)$ .  [/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" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]