on a blackboard. One operation is to delete and replace it by 
; or by 
; or by 
if 
. The final goal of Azambuja is to write the number 
after performing a finite number of operations. Show that if the initial number written is 
, then Azambuja cannot reach his goal.Do you really need a hint? Just try it yourself! [/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.23.3"]It is always a good idea to try using the invariance principle in such problems. [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.23.3"]Make a change of variables to see patterns. [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"]Note that the operation is restricted to rational numbers. Hence, writing $latex q=\frac{r}{s}=(r,s)$ could help. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.23.3"]Let us denote by $latex q_n$ the number on the board at the $latex n$th step. We shall use the new variable $latex a_n=2q_n-1$ (just to simplify the denominator). Clearly, $latex a_{n+1}$ is either $latex a_n\pm 2$ or $latex -\frac{1}{a_n}$. Writing $latex a_n=(r_n,s_n)$, this means that $latex (r_{n+1},s_{n+1})$ is either $latex (r_n \pm 2s_n,s_n)$ or $latex (-s_n,r_n)$. Thus, we need to find out if $latex (-1008,1009)$ is reachable starting from $latex (-1,1)$. However, (odd, odd) pairs can produce only other (odd,odd) pairs under this operation, and $latex (-1008,1009)$ is an (even, even) pair. Hence $latex \frac{1}{2018}$ cannot be reached starting from 0. [/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"]
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%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark"][/et_pb_button][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="50px||50px" custom_padding="20px|20px|20px|20px"]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
Cheenta students shine at the Purple Comet Math Meet 2025 organized by Titu Andreescu and Jonathan Kanewith top national and global ranks.
Celebrate the success of Cheenta students in the Stanford Math Tournament. The Unified Vectors team achieved Top 20 in the Team Round.
