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

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
$$f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.$$

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.23.3" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.23.3"]Prove (by induction on $m$) that if $n\ge m$ then $f(n)\ge m$. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.23.3"]Note that hint 1 implies that $f(n+1)>f(n)$. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"]Note that hint 2 implies that $n+1>f(n)$, i.e. $f(n)\le n$. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.23.3"]Hint 3, combined with hint 1 gives $f(n)=n$ for all $n\in\mathbb{N}$. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.22.4" 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" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

# Similar Problems

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